{"text":"\\section{Introduction}\nLet $\\A$ be a Banach algebra (over the complex field $\\mathbb{C}$), and $\\U$ be a Banach $\\A$-bimodule. A linear map $\\delta:\\A\\rightarrow\\U$ is called a \\textit{derivation} if $\\delta (ab)=a\\delta(b)+\\delta (a)b$ holds for all $a,b\\in\\A$. For any $x\\in\\U$, the map $id_x:\\A\\rightarrow\\U$ given by $id_x(a)=ax-xa$ is a continuous derivation called \\textit{inner derivation}. The set of all continuous derivations from $\\A$ into $\\U$ is denoted by $Z^{1}(\\A,\\U)$ while $N^{1}(\\A,\\U)$ denotes the set of all inner derivations from $\\A$ into $\\U$. $Z^{1}(\\A,\\U)$ is a linear subspace $\\mathbb{B}(\\A,\\U)$ (the space of all continuous linear maps from $\\A$ into $\\U$) and $N^{1}(\\A,\\U)$ is a linear subspace of $Z^{1}(\\A,\\U)$. We denote by $H^{1}(\\A,\\U)$, the quotient space $\\frac{Z^{1}(\\A,\\U)}{N^{1}(\\A,\\U)}$ which is called \\textit{the first cohomology group of $\\A$ with coefficients in $\\U$}. We call the the first cohomology group of $\\A$ with coefficients in $\\A$ briefly by first cohomology group of $\\A$. Derivations and cohomology group are important subjects in the study of Banach algebras. Among the most important problems related to them are these questions; Under what conditions a derivation $\\delta:\\A\\rightarrow\\U$ is continuous? Under what conditions one has $H^{1}(\\A,\\U)=(0)$? (i.e., every continuous derivation from $\\A$ into $\\U$ is inner). Also the calculation of $H^{1}(\\A,\\U)$, up to isomorphism of linear spaces, is a considerable problem.\n\\par \nThe problem of continuity of derivations is related to the subject of automatic continuity which is an important subject in mathematical analysis. Many studies have been performed in this regard and it has a long history. We may refer to \\cite{da} for more information which is a detailed source in this subject. Here we only review the most important obtained results concerning the automatic continuity of derivations. Johnson and Sinclair in \\cite{john1} have shown that every derivation on a semisimple Banach algebra is continuous. Ringrose in \\cite{rin} showed that every derivation from a $C^{*}$-algebra $\\A$ into Banach $\\A$-bimodule $\\U$ is continuous. In \\cite{chr}, Christian proved that every derivation from a nest algebra on Hilbert space $\\mathcal{H}$ into $\\mathbb{B}(\\mathcal{H})$ is continuous. Additionally, some results on automatic continuity of derivations on prime Banach algebras have been established by Villena in \\cite{vi1} and \\cite{vi2} .\n\\par \nThe study of the first cohomology group of Banach algebras is also a considerable topic which may be used to study the structure of Banach algebras. Johnson in \\cite{john2}, using the first cohomology group, has defined amenable Banach Algebras and then various types of amenability defined by using the first cohomology group. We may refer the reader to \\cite{run} for more information. Also among the interesting problems in the theory of derivations is either characterizing algebras on which every continuous derivation is inner, that is, the first  cohomology group is trivial or characterizing the first cohomology group of Banach algebras up to vector spaces isomorphism. Sakai in \\cite{sa} showed that every continuous derivation on a $W^{*}$-algebra is inner. Kadison \\cite{ka} proved that every derivation of a $C^{*}$-algebra on a Hilbert space $\\mathcal{H}$ is spatial (i.e., it is of the form $a\\mapsto ta-at$ for $t\\in \\mathbb{B}(\\mathcal{H})$) and in particular, every derivation on a von Neumann algebra is inner. Some results have also been obtained in the case of  non-self-adjoint operator algebras. Christian \\cite{chr} showed that every continuous derivation on a nest algebra on $\\mathcal{H}$ to itself and to $\\mathbb{B}(\\mathcal{H})$ is inner and then this result generalized to some other forms among which we may refer to \\cite{li} and the references therein. Gilfeather and Smith have calculated the first cohomology group of some operator algebras called joins(\\cite{gil1}, \\cite{gil2}). In \\cite{do}, the cohomology group of operator algebras called seminest algebras has been calculated. All of these operator algebras and nest algebras have a structure like triangluar Banach algebras, so motivated by these studies, Forrest and Marcoux in \\cite{for} verified the first cohomology group of triangular Banach algebras. The first cohomology group of triangular Banach algebras is not necessarily zero and in \\cite{for}, in fact it has been calculated under some special conditions and using the results, various examples of Banach algebras with non-trivial cohomology have been given and then the first cohomology group of those examples has been computed. Also in \\cite{for}, some results concerning the automatic continuity of derivations on triangluar Banach algebras are presented. A generalization of triangluar Banach algebras is module extensions of Banach algebras which Zhang \\cite{zh} has studied the weak amenability of them and then Medghalchi and Pourmahmood in \\cite{med} computed the first cohomology group of module extensions of Banach algebras and using those results, they gave various examples of Banach algebras  with non-trivial cohomology group and in fact they calculated the first cohomology group of the given examples.\n\\par \nAnother class of Banach algebras which considered during the last thirty years, is the class of algebras obtained by a special product called $\\theta$-Lau product. This product is firstly introduced by Lau \\cite{lau} for a special class of Banach algebras which are pre-dual of von Neumann algebras where the dual unit element (the unit element of the dual) is a multiplicative linear functional. Afterwards, various studies have been performed to it. For instance, Monfared in \\cite{mn} has verified the structure of this special product and in \\cite{gha} the amenability of Banach algebras equipped with the same product has been studied. For more information about this product the reader may refer to \\cite{gha}, \\cite{mn} and references therein.  \n\\par \nLet $\\mathcal{B}$ be a Banach algebra such that $\\mathcal{B} =\\A\\oplus\\U$ (as direct sum of Banach spaces) where $\\A$ is a closed subalgebra of $\\mathcal{B}$ and $\\U$ is a closed ideal in $\\mathcal{B}$. In this case, we say that $\\mathcal{B}$ is \\textit{semidirect product} of $\\A$ and $\\U$ and write $\\mathcal{B}=\\A\\ltimes \\U$. Semidirect product Banach algebras appear in the study of many classes of Banach algebras. For instance, in the strong Wedderburn decomposition of a Banach algebra $\\mathcal{B}$, it is assumed that $\\mathcal{B}=\\A\\ltimes Rad\\mathcal{B}$ where $Rad\\mathcal{B}$ is the Jacobson radical of $\\mathcal{B}$ and $\\A $ is a closed subalgebra of $\\mathcal{B}$ with $\\A\\cong \\frac{\\mathcal{B}}{Rad\\mathcal{B}}$ or for example, in \\cite{da2} using the structure of semidirect product, the authors have studied the amenability of measure algebras since every measure algebra has a decomposition to semidirect product of Banach algebras. In \\cite{tho2}, Thomas has verified the necessary conditions of decomposition of a commutative Banach algebra to semidirect product of a closed subalgebra and a principal ideal. We also may refer to \\cite{ba}, \\cite{ber} and \\cite{wh} where semidirect product Banach algebras are studied from different points of view. Equivalently, one may discuss semidirect product Banach algebras as follows; Let $\\A$ and $\\U$ be Banach algebras such that $\\U$ is a Banach $\\A$-bimodule with compatible actions and appropriate norm. Consider the multiplication on $\\A\\times\\U$ given by \n\\[(a,x)(b,y)=(ab,ay+xb+xy)\\quad\\quad  ((a,b)\\in \\A\\times\\U).\\]\nIt can be shown that with this multiplication and $\\l^{1}$-norm, $\\A\\times\\U$ is a Banach algebra where $\\A$ is a closed subalgebra of this Banach algebra and $\\U$ is a closed ideal of it. So in fact this Banach algebra is equivalent to $\\A\\ltimes\\U$. By considering different module actions or algebra multiplications, it can be seen that $\\A\\ltimes\\U$ is a generalization of direct products of Banach algebras, tivial extension Banach algebras, triangular Banach algebras or $\\theta$-Lau product Banach algebras. In this paper we consider semidirect product Banach algebras as mentioned above and study the derivations on this special product of Banach algebras. In fact we establish various results concerning the automatic continuity of derivations on $\\A\\ltimes\\U$ and the first cohomology group of it and we present these results in special cases of $\\A\\ltimes\\U$ and obtain various examples of Banach algebras with automatically continuous derivations and trivial first cohomology group or compute their first cohomology group.\n\\par\nThis paper is organized as follows. In section 2, we investigate the definition of semidirect product Banach algebras and we show that this product is a generalization of various types of Banach algebras. In section 3, the structure of derivations on semidirect product Banach algebras will be discussed and using that we obtain several results about the automatic continuity of derivations on these Banach algebras and also verify the decomposition of derivations into the sum of a continuous derivation and another derivation. In section 4, we consider the fist cohomology group of semidirect product Banach algebras and compute it under some different conditions and establish various results in this context. In section 5, we apply the obtained results in sections 3,4 to some special cases of semidirect products of Banach algebras. In fact we investigate the automatic continuity of the derivations and the first cohomology group of direct products of Banach algebras, module extension Banach algebras and $\\theta$-Lau products of Banach algebras and  establish some various results about the derivations on these Banach algebras. \n\\par \nAt the end of this section we introduce some used notations and expressions in the paper.\n\\par \nIf $\\mathcal{X}$ and $\\mathcal{Y}$ are Banach spaces, for a linear transform $T:\\mathcal{X}\\rightarrow \\mathcal{Y}$, define the separating space $\\mathfrak{S}(T)$ as \n\\[\\mathfrak{S}(T):=\\{y\\in \\mathcal{Y}\\, \\mid \\, \\text{there is}\\, \\, \\{x_n\\}\\subseteq \\mathcal{X}\\,\\, \\text{with}\\, \\, x_n\\rightarrow 0 , \\, T(x_n)\\rightarrow y\\} .\\]\nBy closed graph theorem, $T$ is continuous if and only if $\\mathfrak{S}(T)=(0)$.\n\\par \nLet $\\A$ be a Banach algebra and $\\U$ be a Banach $\\A$-bimodule. By $Z(\\A)$, we mean the center of $\\A$. Consider the set $ann_{\\A}\\U$ as\n\\[ann_{\\A}\\U:=\\{a\\in\\A\\, \\mid \\, a\\U=\\U a=(0)\\}.\\]\nIf $\\mathcal{N}$ is an $\\A$-submodule of $\\U$, we put \n\\[(\\mathcal{N}:\\U)_{\\A}:=\\{a\\in\\A \\, \\mid \\, a\\U\\subseteq \\mathcal{N}, \\, \\U a\\subseteq \\mathcal{N}\\}. \\] \nIt is clear that if $\\mathcal{N}=(0)$, then $((0):\\U)_{\\A}=ann_{\\A}\\U$. Let $\\U$ and $\\mathcal{V}$ be Banach $\\A$-bimodules. A linear map $\\phi :\\U\\rightarrow \\mathcal{V}$ is said to be a \\textit{left $\\A$-module homomorphism} if $\\phi (ax)=a\\phi (x)$ whenever $a\\in \\A$ and $x\\in\\U$ and it is a \\textit{right $\\A$-module homomorphism} if $\\phi (xa)=\\phi (x) a\\quad (a\\in \\A,x\\in\\U)$. The linear map $\\phi$ is called \\textit{$\\A$-module homomorphism}, if $ \\phi$ is both of left and right $\\A$-module homomorphism. The set of all continuous $\\A$-module homomorphisms from $\\U$ into $\\mathcal{V}$ is denoted by $Hom_{\\A}(\\U,\\mathcal{V})$. Note that if the spaces are the same, we just write $Z^{1}(\\A), N^{1}(\\A), H^{1}(\\A), Hom_{\\A}(\\U)$\n\\section{Semidirect products of Banach algebras}\nIn this section we introduce the notion of semidirect produts of Banach algebras and give some properties of this concept.\\\\\nLet $\\A$ and $\\U$ be Banach algebras such that $\\U$ is a Banach $\\A$-bimodule with\n\\[\\parallel ax \\parallel \\leq \\parallel a \\parallel \\parallel x \\parallel, \\quad \\parallel xa \\parallel \\leq \\parallel x \\parallel \\parallel a \\parallel \\quad\\quad (a\\in \\A, x\\in \\U), \\] \nand compatible actions, that is \n\\[ (a.x)y=a.(xy),\\,\\,\\, (xy).a=x(y.a), \\,\\,\\, (x.a)y=x(a.y)  \\quad\\quad (a \\in \\A, \\, x,y\\in \\U). \\]\nIf we equip the set $\\A \\times \\U$ with the usual $\\mathbb{C}$-module structure, then the multiplication \n\\[ (a,x)(b,y)=(ab, a.y+x.b+xy) \\]\nturns $\\A \\times \\U$ into an associative algebra. \n\\par \nIn continue we also denote the module actions by $ax$ and $xa$.\n\\par \nThe \\textit{semidirect product} of Banach algebras $\\A$ and $\\U$, denoted by $\\A \\ltimes \\U$, is defined as the space $\\A \\times \\U$ with the above algebra multiplication and with the norm \n\\[ \\parallel (a,x) \\parallel = \\parallel a \\parallel + \\parallel x \\parallel . \\]\nThe semidirect product $\\A \\ltimes \\U$ is a Banach algebra.\n\\begin{rem}\\label {1}\nIn $\\A \\ltimes \\U$ we identify $\\A \\times \\lbrace 0 \\rbrace$ with $\\A$, and $\\lbrace 0 \\rbrace \\times \\U $ with $\\U$. Then $\\A$ is a closed subalgebra while $\\U$ is a closed ideal of $\\A \\ltimes \\U$, and\n\\[ \\A \\ltimes \\U \/ \\U \\cong \\A \\quad (isometric \\, \\, isomorphism). \\]\nIndeed $\\A \\ltimes \\U$ is equal to direct sum of $\\A$ and $\\U$ as Banach spaces.\n\\par\nConversely, let $\\mathcal{B}$ be a Banach algebra which has the form $\\mathcal{B}=\\A \\oplus \\U$ as Banach spaces direct sum, where $\\A$ is a closed subalgebra of $\\mathcal{B}$ and $\\U$ is a closed ideal in $\\mathcal{B}$. In this case, the product of two elements $a+x$ and $b+y$ of  $\\mathcal{B}=\\A \\oplus \\U$ is given by \n\\[(a+x)(b+y)=ab+(ay+xb+xy), \\] where $ab\\in \\A$ and $ay+xb+xy\\in \\U$. Also the norm on $\\mathcal{B}$ is equivalent to the one given by \n\\[ \\parallel a+x \\parallel =\\parallel a \\parallel + \\parallel  x \\parallel \\quad\\quad (a\\in \\A, x \\in \\U)).\\]\nOn the other hand $\\A$ and $\\U$ are Banach algebras such that  $\\U$ is a Banach $\\A$-bimodule with compatible actions and\n\\[\\parallel ax \\parallel \\leq \\parallel a \\parallel \\parallel x \\parallel, \\quad \\parallel xa \\parallel \\leq \\parallel x \\parallel \\parallel a \\parallel \\quad\\quad (a\\in \\A, x\\in \\U). \\] \nBy above arguments we have \n\\[ \\mathcal{B}\\cong \\A \\ltimes \\U , \\]\nas isomorphism of Banach algebras.\n\\end{rem}\nThe following examples provides various\ntypes of semidirect product af Banach algebras.\n \\begin{exm}\nLet $\\A$ be a Banach algebra. Then the unitization of $\\A$ denoted by $\\A^{\\#}$ is in fact the semidirect product  $\\mathbb{C}\\ltimes \\A$.\n\\end{exm}\n\\begin{exm}\\label{dp}\nSuppose that the action  $\\A$ on  $\\U$ is trivial, that is, $ \\A\\U=\\U\\A=(0)$, then we obtain the usual $l^{1}$-direct product of Banach algebras $\\A$ and $\\U$. In this case, $\\A \\ltimes \\U= \\A \\times \\U$.\n\\end{exm}\n\\begin{exm}\\label{me}\nLet the algebra multiplication on $\\U$ be the trivial action, that is, $\\U^2=(0)$, then $\\A\\ltimes \\U$ is same as the module extension (or trivial extension) of $\\A$ by $\\U$ which we denote by $T(\\A,\\U)$.\n\\end{exm}\n\\begin{exm}\\label{tri}\nLet $\\A$ and $\\mathcal{B}$ be Banach algebras and $\\mathcal{M}$ be a Banach $(\\A,\\mathcal{B})$-bimodule. The triangular Banach algebra introduced in \\cite{for} is \n$Tri(\\A, \\mathcal{M}, \\mathcal{B}):=\\begin{pmatrix}\n \\A & \\mathcal{M}  \\\\\n   0 &\\mathcal{B} \n\\end{pmatrix}$ with the usual matrix operations and $l^1$-norm. If we trun $\\mathcal{M}$ into a  Banach $\\A \\times\\mathcal{B}$-bimodule with the actions \n$$(a,b)m=am\\quad ,\\quad m(a,b)=mb\\quad\\quad ((a,b)\\in \\A\\times\\mathcal{B}, m\\in \\mathcal{M} )$$\n( $\\A \\times \\mathcal{B}$  is considered with $l^1$-norm), then $Tri(\\A, \\mathcal{M}, \\mathcal{B})\\cong T(\\A\\times\\mathcal{B},\\mathcal{M})$ as isomorphism of Banach algebras. Therefore triangular Banach algebra is an example of semidirect products of Banach algebras.\n\\end{exm}\n\\begin{rem}\nLet $\\A$ and $\\U$ be Banach algebras such that $\\U$ is a Banach $\\A$-bimodule with the compatible actions and norm, and  let $\\overline{\\A\\U}=\\U$ or $\\overline{\\U\\A}=\\U$. Let the Banach algebra $\\A$ be the direct sum of its closed ideals $I_1$ and $I_2$, that is, $\\A=I_1 \\oplus I_2$. If $I_2\\U=\\U I_1=(0)$, then $\\U^2=(0)$. Because if $a\\in I_1$, $b\\in I_2$ and $x,y\\in \\U$ are arbitrary, then by the associativity we have $$x[(a+b)y]=[x(a+b)]y.$$\nSo $$x(ay)=(xb)y.$$\nIf $\\overline{\\A\\U}=\\U$ and we put $b=0$ in the above equation, then $xy=0$ and analogously, if $\\overline{\\U\\A}=\\U$, letting $a=0$ gives  $xy=0$.\n\\par \nBy the above arguments, hypothesis and according to Example \\ref{tri}, in this case we have \n\\begin{equation*}\n\\A\\ltimes \\U=T(\\A,\\U)\\cong\\begin{pmatrix}\n I_1 & \\U \\\\\n   0 &I_2\n\\end{pmatrix}\n\\end{equation*}\n as isomorphism of Banach algebras.\n\\end{rem}\n\\begin{exm}\\label{la}\nLet $\\A$ and $\\U$ be Banach algebras, $\\theta\\in \\Delta(\\A)$ where $\\Delta(\\A)$ is the set of all non-zero characters of $\\A$. With the following module actions, $\\U$ becomes a Banach  $\\A$-bimodule:\n$$ax=xa=\\theta(a)x\\quad\\quad (a\\in \\A ,x\\in \\U).$$\nThe norm and the actions on $\\U$ are compatible and one can consider $\\A\\ltimes \\U$ with the multiplication\n$$(a,x)(b,y)=(ab,\\theta(a)y+\\theta(b)x+xy).$$\nIn this case $\\A\\ltimes \\U$ is the $\\theta$-Lau product which is introduced in \\cite{lau}. \n\\end{exm}\n\\begin{exm}\nLet $G$ be a locally compact group. Then $M(G)=l^{1}(G)\\ltimes M_{c}(G)$ where $M_{c}(G)$ is a subspace of $M(G)$ containing all continuous measures such that for $\\mu\\in M(G)$, $\\mu\\in M_c(G)$ if and only if $\\mu(\\{s\\})=0$ $(s\\in G)$. We denote the subspace of discrete measures by $M_d({G})$ which is isomorphic to $l^{1}(G)$ and \n\\begin{equation*}M_{d}(G)=\\{\\mu =\\sum{\\alpha_{s}\\delta_{s}}:\\Vert\\mu\\Vert =\\sum_{s\\in G}{\\vert \\alpha_{s}\\vert }<\\infty \\}.\n\\end{equation*}\nIndeed $l^{1}(G)$ is a closed subspace of $M(G)$ and $M_c(G)$ is a closed ideal of $M(G)$. If $G$ is discrete, then $M(G)=l^{1}(G)$ and $M_{c}(G)=\\{0\\}$. But if $G$ is not discrete, then $M_{c}(G)\\neq\\{0\\}$ \n\\end{exm}\nIt is possible for an algebra $\\mathcal{B}$ with a closed ideal $\\mathcal{I}$, not to exist some closed subalgebra $\\A$ of $\\mathcal{B}$ such that $\\mathcal{B}=\\A\\ltimes \\mathcal{I}$. In the following we give an example of such a Banach algebra.\n\\begin{exm}\nLet $\\mathcal{B}:=C([0,1])$ be the Banach algebra of continuous complex-valued functions on $[0,1]$ and let $\\mathcal{I}:=\\{f\\in \\mathcal{B} :f(0)=f(1)=0\\}$. $\\mathcal{I}$ is a closed ideal of $\\mathcal{B}$. If $\\A $ is a closed subalgebra of $\\mathcal{B}$ satisfying  $\\mathcal{B}=\\A \\oplus \\mathcal{I}$ as Banach spaces direct sum, then for $f\\in \\mathcal{I}$ and $g\\in \\A$ with $f(x)+g(x)=x$ for $x\\in [0,1]$, we have $g(0)=0$, $g(1)=1$ and $g-g^2\\in \\A\\cap \\mathcal{I}$. But yet $g-g^2\\neq 0$.\n\\end{exm}\nNote that $\\A\\ltimes\\U$ is commutative if and only if both $\\A$ and $\\U$ are commutative Banach algebras and $\\U$ is a commutative $\\A$-bimodule.\n\\par \nIn the rest of this section we introduce some special maps which are used in next sections.\n\\par \nFor $a\\in\\A$ define the map $r_{a}:\\U\\rightarrow\\U$ by $r_{a}(x)=xa-ax$. Some properties of this map are given in the following remark.\n\\begin{rem}\\label{inn1}\nFor $a\\in\\A$, consider the map $r_{a}:\\U\\rightarrow\\U$.\n\\begin{enumerate}\n\\item[(i)]\n$r_a$ is a derivation on $\\U$.\n\\item[(ii)]\nFor every $b\\in\\A$ and $x\\in\\U$,\n\\[r_{a}(bx)=br_{a}(x)+id_{a}(b)x \\quad \\text{and} \\quad r_{a}(xb)=r_{a}(x)b+x id_{a}(b).\\]\n\\item[(iii)]\nFor $a\\in \\A$, if $id_{a}=0$, then $r_a$ is an $\\A$-bimodule homomorphism. Also if $ann_{\\A}\\U =(0)$ and $r_a$ is an $\\A$-bimodule homomorphism, then $id_{a}=0$.\n\\end{enumerate}\n\\end{rem}\nAlso inner derivations on $\\U$ have significant properties considered in determining the first cohomology group of $\\A\\ltimes\\U$ which are given in the next remark.\n\\par \nNote that both of inner derivations from $\\A$ to $\\U$ and inner derivations from $\\U$ to $\\U$  are denoted by $id_{x}$. So in order to avoid confusion, we denote by $id_{\\A , x}$, the inner derivations  from $\\A$ to $\\U$ while $id_{\\U , x}$ denotes the inner derivations from $\\U$ to $\\U$.\n\\begin{rem}\\label{inn2}\nFor $x_0\\in\\U$, consider the inner derivation $id_{\\U , x_0}:\\U\\rightarrow\\U$.\n\\begin{enumerate}\n\\item[(i)]\nFor every $a\\in\\A$ and $x\\in\\U$, \n\\[id_{\\U ,x_0}(ax)=a \\, id _{\\U ,x_0}(x)+id _{\\A , x_0}(a)x \\quad \\text{and} \\quad id_{\\U ,x_0}(xa)=id _{\\U ,x_0}(x)a+x \\, id _{\\A, x_0}(a).\\]\n\\item[(ii)]\nFor $x_{0}\\in \\U$, if $id_{\\A, x_0}=0$, then $id_{\\U , x_0}$ is an $\\A$-bimodule homomorphism. If $ann_{\\U}\\U =(0)$ and $id_{\\U , x_0}$ is an $\\A$-bimodule homomorphism, then $id_{\\A, x_0}=0$.\n\\end{enumerate}\n\\end{rem}\nThis following sets play an important role in determining the first cohomology group of $\\A\\ltimes\\U$.\n\\[R_{\\A}(\\U):=\\{r_{a}: \\U\\rightarrow\\U \\,  \\mid  \\, a\\in \\A\\};\\]\n\\[C_{\\A}(\\U):=\\{r_{a} :\\U\\rightarrow\\U \\,  \\mid  \\,  id_{a}=0 \\, \\, (a\\in \\A)\\};\\]\n\\[I(\\U):=\\{id_{\\U, x}:\\U\\rightarrow\\U \\,  \\mid  \\,  id_{\\A, x}=0 \\, \\, (x\\in \\U)\\};\\]\n\\indent In view of the above remarks, the set $R_{\\A}(\\U)$ is a linear subspace of $Z^{1}(\\U)$. Also $C_{\\A}(\\U)$ is a linear subspace of $Hom_{\\A}(\\U) \\cap R_{\\A}(\\U)$ and $I(\\U)$ is a linear subspace of $Hom_{\\A}(\\U) \\cap N^{1}(\\U)$. Indeed, we have the following inclusion linear subspaces;\n\\[ C_{\\A}(\\U)+I(\\U)\\subseteq Hom_{\\A}(\\U) \\cap (R_{\\A}(\\U)+N^{1}(\\U))\\subseteq Hom_{\\A}(\\U) \\cap Z^{1}(\\U).\\]\nIf $\\A$ is commutative, then $R_{\\A}(\\U)= C_{\\A}(\\U)$ and if $\\U$ is a commutative $\\A$-bimodule, then $N^{1}(\\U)=I(\\U)$. If $\\U ^{2}=(0)$, then $Z^{1} (\\U)=\\mathbb{B}(\\U)$ and $N^{1}(\\U)=I(\\U )=(0)$.\n\\section{Derivations on $\\A \\ltimes \\U$}\nIn this section we determine the structure of derivations on $\\A\\ltimes \\U$. According to which, we get some results concerning the automatic continuity of derivations on $\\A\\ltimes \\U$. Also we use the results of this section to determine the first cohomology group of $\\A\\ltimes \\U$ in the next sections.\n\\par \nThroughout this section we always assume that $\\A$ and $\\U$ are Banach algebras where $\\U$ is a Banach $\\A$-bimodule with the compatible actions and norm (As in the section 2). In other cases the conditions will be specified.\n\\par \nIn the following theorem the structure of derivations on $\\A\\ltimes \\U$ are determined.\n\\begin{thm}\\label{asll}\nLet $D:\\A\\ltimes \\U\\rightarrow \\A\\ltimes \\U$ be a map. Then the following conditions are equivalent.\n\\begin{enumerate}\n\\item[(i)] $D$ is a derivation.\n\\item[(ii)] \n\\[D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad (a\\in \\A,x\\in \\U)\\]\nsuch that \n\\begin{enumerate}\n\\item[(a)] \n$\\delta_1:\\A\\rightarrow\\A$ is a derivation.\n\\item[(b)]\n$\\delta_2:\\A\\rightarrow \\U$ is a derivation.\n\\item[(c)]\n$\\tau_1 :\\U\\rightarrow \\A$ is an $\\A$-bimodule homomorphism such that \n$$\\tau_1(xy)=0,$$\nfor all $x,y\\in \\U$.\n\\item[(d)]\n$\\tau_2 :\\U\\rightarrow \\U$ is a linear map such that for every $a\\in \\A$ and $x,y\\in \\U$ satisfies the following conditions\n\\begin{eqnarray*}\n\\tau_2 (ax)&=&a\\tau_2 (x)+\\delta_1 (a)x+\\delta_2 (a)x; \\\\\n\\tau_2 (xa)&=&\\tau_2 (x)a+x\\delta_1 (a)+x\\delta_2 (a); \\\\\n\\tau_2(xy)&=&x\\tau_1(y)+\\tau_1(x)y+x\\tau_2(y)+\\tau_2(x)y.\n\\end{eqnarray*}\n\\end{enumerate}\n\\end{enumerate}\nMoreover, $D$ is an inner derivation if and only if $\\delta_1 ,\\delta_2$ are inner derivations, $\\tau_1 =0$ and if $\\delta_1 =ad_{a_0}$ and $ \\delta_2 =ad_{x_0}$, then $\\tau_2 =ad_{x_0}+r_{a_0}$.\n\\end{thm}\n\\begin{proof}\n$(i)\\implies (ii)$. Since $D$ is a linear map and $\\A\\ltimes \\U$ is the direct sum of the linear spaces $\\A$ and $\\U$, there are some linear maps $\\delta_1:\\A\\rightarrow \\A$, $\\delta_2:\\A\\rightarrow \\U$, $\\tau_1 :\\U\\rightarrow \\A$ and  $\\tau_2 :\\U\\rightarrow \\U$ such that for any $(a,x)\\in \\A\\ltimes \\U$ we have \n$$D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x)).$$\nBy applying $D$ on the equality $(a,0)(b,0)=(ab,0)$ we conclude that $\\delta_1$ and $\\delta_2$ are derivations. Analogously, applying $D$ on the equalities \n\\[ (a,0)(0,x)=(0,ax),\\,\\,\\, (0,x)(a,0)=(0,xa) \\,\\,\\, and \\,\\,\\, (0,x)(0,y)=(0,xy),\\]\n establishes the desired properties for the maps $\\tau_1$ and $\\tau_2$ given in parts $(c)$ and $(d)$ respectively.\n$(ii)\\implies (i)$ Clear.\n\\par \nThe equivalent conditions of inner-ness of $D$ can be obtained by a straightforward calculation.\n\\end{proof}\nIn the sequel for a derivation $D$ on $\\A\\ltimes \\U$, we always assume that \n$$D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad ((a,x)\\in \\A\\ltimes \\U)$$ in which the mentioned maps satisfy the conditions of the preceding theorem.\n\\par \nBy Theorem \\ref{asll}, if $D$ is an inner derivation on $\\A\\ltimes \\U$, then $\\tau_1 =0$. So this question is of interest for a given derivation $D$, under what conditions one has $\\tau_1 =0$?\nBy part $(ii)-(c)$ of Theorem \\ref{asll} if $\\U^2=\\U$ ($\\overline{\\U^2}=\\U$, if $D$ is continuous), then $\\tau_1 =0$. If $\\U$ has a bounded approximate identity, then by Cohen's factorization theorem we have $\\U^2=\\U$ and therefore in this case $\\tau_1 =0$. All unital Bnach algebras, $C^{*}$-algebras and group algebras have bounded approximate identity. Also if $\\U$ is a simple Banach algebra, then $\\U^{2}=\\U$.\n\\par \n The following corollary follows from Theorem \\ref{asll}.\n \\begin{cor}\\label{tak}\n Suppose that $\\delta_1:\\A\\rightarrow \\A$,  $\\delta_2:\\A\\rightarrow \\U$, $\\tau_1 :\\U\\rightarrow \\A$ and $\\tau_2 :\\U\\rightarrow \\U$ are linear  maps.\n \\begin{enumerate}\n \\item[(i)]\n $D:\\A\\ltimes\\U\\rightarrow \\A\\ltimes\\U $ defined by $D((a,x))=(\\delta_1(a),0)$ is a derivation if and only if $\\delta_1$ is a derivation and $\\delta_1 (\\A)\\subseteq ann_{\\A}\\U$. In this case if $\\delta_1 =id_{a_0}$ where $aa_0-a_0 a\\in ann_{\\A}\\U \\, \\, (a\\in \\A) $ and for any $x\\in \\U$, $a_0x =xa_0$, then $D$ is inner.\n \\item[(ii)]\n  $D:\\A\\ltimes\\U\\rightarrow \\A\\ltimes\\U $ with $D((a,x))=(0,\\delta_2(a))$ is a derivation if and only if $\\delta_2$ is a derivation and $\\delta_2 (A)\\subseteq ann_{\\U}\\U$. Moreover, if $\\delta_2 =id_{\\A, x_0}$ is inner, $ax_0 - x_0a\\in ann_{\\U}\\U$ (for all $a\\in \\A$) and $x_0\\in Z(\\U)$, then $D$ is inner.\n \\item[(iii)]\n$D:\\A\\ltimes\\U\\rightarrow \\A\\ltimes\\U $ with $D((a,x))=(\\tau_1(x),0)$ is a derivation if and only if $\\tau_1(xy)=0, x\\tau_1(y)+\\tau_1(x)y=0 (x,y\\in\\U)$. In this case $D$ is inner if and only if $\\tau_1 =0$.\n\\item[(iv)] $D:\\A\\ltimes\\U\\rightarrow \\A\\ltimes\\U $ with $D((a,x))=(0,\\tau_2(x))$ is a derivation if and only if $\\tau_2$ is a derivation and also an $\\A$-bimodule homomorphism. In this case $D$ is inner if and only if $\\tau_2(x)=r_{a_0}(x)+id_{\\U, x_0}(x)$ where $a_0\\in Z(\\A)$ and $ax_0 =x_0 a$ for all $a\\in \\A$.\n\\end{enumerate}\n \\end{cor}\nIn the following we give an example showing that the condition $\\tau_1 =0$ does not hold in general.\n\\begin{exm}\nLet $\\mathcal{B}$ be a Banach algebra and $\\A:=T(\\mathcal{B},\\mathcal{B})$ and let $\\U :=\\mathcal{B}$. The Banach algebra $\\U$ becomes a Banach $\\A$-bimodule by the following compatible module actions:\n\\[(a,b)x=ax,\\quad  x(a,b)= xa\\quad\\quad (a,b\\in \\A , x\\in \\U).\\]\nNow consider the Banach algebra $T(\\A,\\U)$ and define the map $\\tau_1:\\U\\rightarrow \\A$ by \n$$\\tau_1(x)=(0,x)\\quad\\quad (x\\in \\U)$$\nThen $\\tau_1\\neq 0$ is an $\\A$-bimodule homomorphism with $\\tau_1(\\U ^2)=(0)$ and for any $x,y\\in \\U$ we have \n$$x\\tau_1(y)+\\tau_1(x)y=x(0,y)+(0,x)y=0$$\nNow by Corollary \\ref{tak} it can be seen that the map $D$ on $T(\\A,\\U)$ defined by $D((a,b),x)=(\\tau_1(x),0)$ is a derivation in which $\\tau_1\\neq 0$.\n\\end{exm}\nIn this example $\\tau_1\\neq 0$ but $x\\tau_1(y)+\\tau_1(x)y=0$ $ (x,y\\in\\U)$. The next example shows that $\\tau_1$ does not satisfy the condition $x\\tau_1(y)+\\tau_1(x)y=0 \\,\\, (x,y\\in\\U)$ in general.\n\\begin{exm}\nLet $\\A$ be a Banach algebra, $\\mathcal{C}$ be a Banach $\\A$-bimodule and $\\gamma :\\mathcal{C}\\rightarrow \\A$ be a nonzero $\\A$-bimodule homomorphism such that \n\\[ c\\gamma (c')+\\gamma (c)c' =0\\]\nfor all $c,c'\\in\\mathcal{C}$.\n\\\\\nLet $\\U :=\\A\\times \\mathcal{C}$. With the usual actions, $\\U$ is a Banach $\\A$-bimodule. Consider the multiplication on $\\U$ given by\n\\[(x,y)(x',y')=(xx',0)\\quad\\quad ((x,y),(x',y')\\in\\U). \\]\nWith this multiplication $\\U$ is a Banach algebra such that the multiplication on $\\U$ is compatible with its module actions. Now we may consider the Banach algebra $\\A\\ltimes\\U$. Define the maps $\\tau_1:\\U\\rightarrow\\A$ and $\\tau_2:\\U\\rightarrow\\U$ by \n\\[\\tau_1 ((x,y))=\\gamma(y)\\quad , \\quad \\tau_2((x,y))=(-\\gamma (y),0).\\]\n$\\tau_1$ and $\\tau_2$ are $\\A$-bimodule homomorphisms such that $\\tau_1(\\U ^2)=0$, $\\tau_1\\,,\\tau_2\\neq 0$ and \n\\[0=\\tau_2((x,y)(x',y'))=(x,y)\\tau_1 ((x',y'))+(x,y)\\tau_2((x',y'))+\\tau_1 ((x,y)) (x',y')+\\tau_2((x,y))(x',y').\\]\nBy Theorem \\ref{asll} it can be seen that $D=(\\tau_1,\\tau_2)$ is a derivation on $\\A\\ltimes\\U$. If we assume further that $\\A$ is a unital algebra, then for any $y,y'\\in\\mathcal{C}$ with $\\gamma (y)\\neq 0$,\n\\begin{eqnarray*}\n(0,y)\\tau_1((1,y'))+\\tau_1((0,y))(1,y')&=&(0,y)\\gamma (y')+\\gamma (y)(1,y')\\\\&=&(\\gamma(y),y\\gamma (y')+\\gamma(y)y')\\\\&=&(\\gamma (y),0)\\neq 0.\n\\end{eqnarray*}\n \\end{exm}\nIn the following we investigate the automatic continuity of derivations on $\\A\\ltimes\\U$. It is clear that a derivation $D$ on $\\A\\ltimes \\U$ is continuous if and only if the maps $\\delta_1 , \\delta_2 , \\tau_1$ and $\\tau_2$ are continuous. \n\\par \n In the next theorem we state the relation between separating spaces of $\\delta_i$ and $\\tau_i$ ($i=1,2)$ and then by using this theorem, we obtain some results concerning the automatic continuity of the derivations on $\\A\\ltimes\\U$.\n\\begin{thm}\\label{joda}\nLet $D$ be a derivation on $\\A\\ltimes\\U$ such that \n$$D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad ((a,x)\\in \\A\\ltimes \\U),$$\nthen\n\\begin{enumerate}\n\\item[(i)]\n$\\mathfrak{S} (\\tau_1)$ is an ideal in $\\A$. If $\\tau_2$ is continuous, then $\\mathfrak{S} (\\tau_1)\\subseteq ann_{\\A}\\U$ and if $\\tau_2(\\U)\\subseteq ann_{\\U}\\U$, then $\\mathfrak{S} (\\tau_1)\\subseteq(\\mathfrak{S} (\\tau_2):\\U)_\\A$.\n\\item[(ii)]\n$\\mathfrak{S} (\\tau_2)$ is an $\\A$-sub-bimodule of $\\U$ and if $\\tau_1$ is continuous or $\\tau_1 (\\U)\\subseteq ann_{\\A}\\U$, then $\\mathfrak{S} (\\tau_2)$ is an ideal in $\\U$.\n\\item[(iii)]\n$\\mathfrak{S} (\\delta_1)$ is an ideal in $\\A$ and if $\\delta_2$ is continuous or $\\delta_2 (\\A)\\subseteq ann_{\\U}\\U$, then $\\mathfrak{S} (\\delta_1)\\subseteq(\\mathfrak{S} (\\tau_2):\\U)_\\A$.\n\\item[(iv)]\n$\\mathfrak{S} (\\delta_2)$ is an $\\A$-subbimodule of $\\U$ and if $\\delta_1$ is continuous or $\\delta_1 (\\A)\\subseteq ann_{\\A}\\U$, then $\\mathfrak{S} (\\delta_2)\\subseteq(\\mathfrak{S} (\\tau_2):\\U)_\\U$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\n$(i)$ Let $a\\in \\mathfrak{S} (\\tau_1)$. Thus there is a sequence $\\{x_n\\}$ in $\\U$ such that \n$x_n\\rightarrow 0$ and $ \\tau_1(x_n)\\rightarrow a$.\nFor any $b\\in \\A$, $bx_n\\rightarrow 0$, so $\\tau_1(bx_n)=b\\tau_1 (x_n)\\rightarrow ba $ and hence $ba\\in \\mathfrak{S} (\\tau_1)$. Similarly, $ab\\in \\mathfrak{S} (\\tau_1)$. Therefore $\\mathfrak{S} (\\tau_1)$ is an ideal in $\\A$.\\\\\n Now for every $x\\in \\U$ we have  \n$$\\tau_2(xx_n)=x\\tau_1(x_n)+x\\tau_2(x_n)+\\tau_1(x)x_n+\\tau_2(x)x_n$$\nand\n$$\\tau_2(x_nx)=x_n\\tau_1(x)+x_n\\tau_2(x)+\\tau_1(x_n)x+\\tau_2(x_n)x.$$\nIf $\\tau_2$ is continuous, taking limit of the above equations gives $ax=xa=0$ ($x\\in \\U$). So $a\\in ann_{\\A}\\U$. Hence $\\mathfrak{S} (\\tau_1)\\subseteq ann_{\\A}\\U$. \n\\par \nIf $\\tau_2 (\\U)\\subseteq ann_{\\U}\\U$, by taking limit of the above equations we get \n$$\\tau_2(xx_n)\\rightarrow xa\\quad  \\text{and} \\quad \\tau_2(x_n x)\\rightarrow ax.$$ So $ax,xa\\in \\mathfrak{S} (\\tau_2)$ and hence $\\mathfrak{S} (\\tau_1)\\subseteq(\\mathfrak{S} (\\tau_2):\\U)_\\A$.\n\\\\\n$(ii)$ If $x_n\\rightarrow 0$ and $\\tau_2(x_n)\\rightarrow x$, then for any $a\\in \\A$ we have \n$$\\tau_2(ax_n)=a\\tau_2(x_n)+\\delta_1(a)x_n+\\delta_2(a)x_n$$\nand\n$$\\tau_2(x_{n}a)=\\tau_2(x_n)a+x_n\\delta_1(a)+x_n\\delta_2(a).$$\nBy taking limits of these equations we get \n$$\\tau_2(ax_n)\\rightarrow ax\\quad\\text{and}\\quad \\tau_2(x_{n}a)\\rightarrow xa.$$\nSo $ax,xa\\in \\mathfrak{S} (\\tau_2)$ and hence $\\mathfrak{S} (\\tau_2)$ is an $\\A$-subbimodule of $\\U$.\n\\par \nFor any $y\\in \\U$ we have\n$$\\tau_2(yx_n)=\\tau_1(y)x_n+\\tau_2(y)x_n+y\\tau_1(x_n)+y\\tau_2(x_n)$$\n$$\\tau_2(x_n y)=\\tau_1(x_n)y+\\tau_2(x_n)y+x_n\\tau_1(y)+x_n\\tau_2(y).$$\nIf $\\tau_1$ is continuous or $\\tau_1(\\U)\\subseteq ann_{\\A} \\U$, by taking limit we obtain \n$$\\tau_2(yx_n)\\rightarrow yx\\quad \\text{and}\\quad \\tau_2(x_{n}y)\\rightarrow xy.$$\nTherefore $xy,yx\\in \\mathfrak{S} (\\tau_2)$ and $\\mathfrak{S} (\\tau_2)$ is an ideal.\n\\\\\n$(iii)$ Let $a_n\\rightarrow 0 $ and $\\delta_1 (a_n)\\rightarrow a$. Since $\\delta_1$ is a derivation, it follows that $\\mathfrak{S} (\\delta_1)$ is an ideal in $\\A$. For every $x\\in \\U$ we have \n$$\\tau_2(a_nx)=a_n\\tau_2(x)+\\delta_1(a_n)x+\\delta_2(a_n)x$$\nand\n$$\\tau_2(ax_n)=\\tau_2(x)a_n+x\\delta_1(a_n)+x\\delta_2(a_n).$$\nIf $\\delta_2$ is continuous or $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$, taking limit of the above equations yields \n$$\\tau_2(a_n x)\\rightarrow ax\\quad \\text{and}\\quad \\tau_2(x a_n)\\rightarrow xa.$$\nSince $xa_n, a_nx\\rightarrow 0$, it follows that $ax,xa\\in \\mathfrak{S} (\\tau_2)$. So $\\mathfrak{S} (\\delta_1)\\subseteq(\\mathfrak{S} (\\tau_2):\\U)_\\A$.\n\\\\\n$(iv)$ The proof is similar to part $(iii)$.\n\\end{proof}\nNow we give some results of the preceding theorem.\n\\begin{prop}\\label{au1}\nLet $D$ be a derivation on $\\A\\ltimes\\U$ such that \n$$D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad ((a,x)\\in \\A\\ltimes \\U),$$\nthen \n\\begin{enumerate}\n\\item[(i)]\nif $ann_{\\A}\\U =(0)$, $\\tau_2$ and $\\delta_2$ are continuous, then $D$ is continuous.\n\\item[(ii)]\nif $ann_{\\U}\\U =(0)$, $\\tau_2$ and $\\delta_1$ are continuous, then $\\delta_2$ is continuous. If we also add the assumption of continuity of $\\tau_1$, then $D$ is continuous.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n$(i)$ Since $\\tau_2$ is continuous so $\\mathfrak{S} (\\tau_2)=0$. By Theorem \\ref{joda}-$(i)$ from the continuity of $\\tau_2$ we conclude that $\\mathfrak{S} (\\tau_1)\\subseteq ann_{\\A}\\U=(0)$, thus $\\tau_1$ is continuous. Moreover, since the conditions of Theorem \\ref{joda}-$(iii)$ hold, we have\n$$\\mathfrak{S} (\\delta_1)\\subseteq (\\mathfrak{S} (\\tau_2):\\U)_\\A=((0):\\U)_\\A =ann_{\\A}\\U=(0).$$ Therefore $\\delta_1$ is continuous. So $D$ is continuous.\n\\\\\n$(ii)$ Since the conditions of Theorem \\ref{joda}-$(iv)$ hold and $\\tau_2$ is continuous, it follows that\n $$\\mathfrak{S} (\\delta_2)\\subseteq (\\mathfrak{S} (\\tau_2):\\U)_\\U=((0):\\U)_\\U=ann_{\\U}\\U =(0).$$ So $\\delta_2$ is continuous . If $\\tau_1$ is continuous, then we have the continuity of all the maps and thus $D$ is continuous.\n\\end{proof}\nIf we add the condition $ann_{\\A}\\U=(0)$ to part $(ii)$ of the previous proposition, then as in part $(i)$, this implies that $\\tau_1$ is continuous and in this case $D$ is continuous as well.\n\\par \nNow by the preceding proposition, we obtain some results concerning the automatic continuity of the derivations on $\\A\\ltimes\\U$.\n\\begin{cor}\\label{n1}\nSuppose that for every derivation $D$ on $\\A\\ltimes \\U$ we have \n$x\\tau_1 (y)+\\tau_1(x)y=0$ $ (x,y\\in \\U)$. If $ann_{\\A}\\U=(0)$ and each derivation from $\\U$ to $\\U$ and each derivation from $\\A$ to $\\U$ is continuous, then every derivation on $\\A\\ltimes \\U$ is continuous.\n\\end{cor}\n\\begin{proof}\nBy Theorem \\ref{asll}, for any derivation $D=(\\delta_1 +\\tau_1,\\delta_2+\\tau_2)$ on $\\A\\ltimes \\U$, the maps $\\delta_2:\\A \\rightarrow\\U$ and $\\tau_2:\\U\\rightarrow \\U$ are derivations, since we have \n$x\\tau_1 (y)+\\tau_1(x)y=0$ $ (x,y\\in \\U)$.  So $\\delta_2$ and $\\tau_2$ are continuous by the hypothesis. Now the continuity of $D$ follows from Proposition \\ref{au1}-$(i)$.\n\\end{proof}\n\\begin{cor}\\label{n2}\nSuppose that for any derivation $D$ on $\\A\\ltimes \\U$ we have  $x\\tau_1 (y)+\\tau_1(x)y=0 $ $(x,y\\in \\U)$. If $ann_{\\U}\\U=(0)$, every derivation from $\\U$ to $\\U$ and every derivation from $\\A$ to $\\A$ is continuous and every $\\A$-bimodule homomorphism from $\\U$ to $\\A$ is continuous, then every derivation on $\\A\\ltimes \\U$ is continuous.\n\\end{cor}\n\\begin{proof}\nConsider the derivation $D=(\\delta_1 +\\tau_1,\\delta_2+\\tau_2)$. By Theorem \\ref{asll}, the maps \n $\\delta_1:\\A\\rightarrow \\A$ and  $\\tau_2:\\U\\rightarrow \\U$ are derivation and  $\\tau_1:\\U\\rightarrow \\A$ is an $\\A$-bimodule homomorphism. By the hypothesis, $\\delta_1, \\tau_1$ and $\\tau_2$ are continuous. Therefore  $D$ is continuous by Proposition \\ref{au1}-$(ii)$.\n\\end{proof}\nBy Johnson's and Sinclair's theorem \\cite{john1}, every derivation on a semisimple Banach algebra is continuous, so we have the following corollary.\n\\begin{cor}\\label{semi}\nSuppose that $\\A$ and $\\U$ are semisimple Banach algebras such that $\\U$ has a bounded approximate identity. Then every derivation on $\\A\\ltimes \\U$ is continuous.\n\\end{cor}\n\\begin{proof}\nLet $D=(\\delta_1 +\\tau_1,\\delta_2+\\tau_2)$ be a derivation on $\\A\\ltimes \\U$. By the Cohen's factorization theorem, $\\U ^2 =\\U$ and so $\\tau_1=0$. Also from the hypothesis we conclude that $ann_{\\U}\\U =(0)$. By Johnson's and Sinclair's theorem \\cite{john1}, every derivation on $\\A$ and $\\U$ is continuous. Now by Proposition \\ref{au1}-$(ii)$, every derivation on $\\A\\ltimes \\U$ is continuous.\n\\end{proof}\nAll $C^{*}$-algebras, semigroup algebras, measure algebras and unital simple algebras are semisimple Banach algebras with bounded approximate identity. \n\\par \nIn next results we investigate some conditions under which we can express the derivation $D$ as the sum of two derivations one of which being continuous.\n\\begin{prop}\\label{taj1}\nLet $D$ be a derivation on $\\A\\ltimes\\U$ such that \n\\[D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad ((a,x)\\in \\A\\ltimes \\U),\\]\nthen\n\\begin{enumerate}\n\\item[(i)]\nif $ann_{\\U}\\U =(0)$, $x\\tau_1 (y)+\\tau_1(x)y=0 $ $(x,y\\in \\U)$, $\\delta_1$ and $\\tau_2$ are continuous, then $D=D_1 +D_2$ in which \n$$D_1((a,x))=(\\delta_1(a),\\delta_2(a)+\\tau_2(x))\\quad\\text{and}\\quad D_2((a,x))=(\\tau_1(x),0),$$\nsuch that $D_1$ and $D_2$ are derivations on $\\A\\ltimes\\U$ and $D_1$ is continuous.\n\\item[(ii)]\nif $ann_{\\U}\\U =(0)$, $\\delta_1(\\A)\\subseteq ann_{\\A}\\U$ and $\\tau_1, \\tau_2$ are continuous, then \\[D_1((a,x))=(\\tau_1(x),\\delta_2 (a)+\\tau_2 (x))\\] is a continuous derivation on $\\A\\ltimes \\U$ and $D=D_1 +D_2$ where $D_2((a,x))=(\\delta_1(a),0)$ is a derivation on $\\A\\ltimes \\U$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n$(i)$ By Proposition \\ref{au1}-$(ii)$, $\\delta_2$ is continuous. By Corollary \\ref{tak}, $D=D_1 +D_2$ where \n$$D_1((a,x))=(\\delta_1(a),\\delta_2(a)+\\tau_2(x)) \\quad \\text{and} \\quad D_2((a,x))=(\\tau_1(x),0)\\quad\\quad ((a,x)\\in \\A\\ltimes\\U)$$\nare derivations on $\\A\\ltimes \\U$. By the assumptions and that $\\delta_2$ is continuous, it follows that $D_1$ is a continuous derivation.\n\\\\\n$(ii)$ By Corollary \\ref{tak} \n$$D_1((a,x))=(\\tau_1(x),\\delta_2 (a)+\\tau_2 (x))\\quad\\text{and}\\quad D_2((a,x))=(\\delta_1(a),0)$$\nare derivations on $\\A\\ltimes \\U$. Now by the continuity of $\\tau_2$, $ann_{\\U}\\U =(0)$ and that $\\delta_1(\\A)\\subseteq ann_{\\A}\\U$,  from Theorem \\ref{joda}-$(iv)$, it follows that \n$$\\mathfrak{S} (\\delta_2)\\subseteq (\\mathfrak{S} (\\tau_2):\\U)_\\U =((0):\\U)_\\U =ann_{\\U}\\U=(0).$$ \nTherefore $\\delta_2$ is continuous and hence so is $D_1$.\n\\end{proof}\nTo prove the next proposition we need the following lemma.\n\\begin{lem}(\\cite [Proposition 5.2.2]{da}).\\label{da}\nLet $\\mathcal{X}$, $\\mathcal{Y}$ and $\\mathcal{Z}$ be Banach spaces, and let $T:\\mathcal{X}\\rightarrow \\mathcal{Y}$ be linear.\n\\begin{enumerate}\n\\item[(i)] \nSuppose that $R:\\mathcal{Z}\\rightarrow \\mathcal{X}$ is a continuous surjective linear map. Then $ \\mathfrak{S} (TR)= \\mathfrak{S} (T). $\n\\item[(ii)]\n Suppose that $S:\\mathcal{Y}\\rightarrow \\mathcal{Z}$ is a continuous linear map. Then $ST$ is continuous if and only if $S(\\mathfrak{S} (T))=(0)$.\n\\end{enumerate}\n\\end{lem}\n\\begin{prop}\\label{taj2}\nLet $D$ be a derivation on $\\A\\ltimes \\U$ such that \n$$D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad (a\\in \\A,x\\in \\U)$$\nand $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$. Then under any of the following conditions, \\[D_1((a,x))=(\\delta_1(a)+\\tau_1(x),\\tau_2(x))\\]\nis a continuous derivation on  $\\A\\ltimes \\U$ and $D=D_1+D_2$ where $D_2((a,x))=(0,\\delta_2(a))$ is a derivation on  $\\A\\ltimes \\U$.\n\\begin{enumerate}\n\\item[(i)]\n$ann_{\\A}\\U =(0)$ and $\\tau_2$ is continuous.\n\\item[(ii)]\n$\\A$ possesses a bounded right approximate identity, there is a surjective left $\\A$-module homomorphism  \n$\\phi :\\A\\rightarrow \\U$ and both $\\delta_1$ and $\\tau_1$ are continuous.\n\\item[(iii)]\n $\\A$ possesses a bounded right approximate identity, there is an injective left $\\A$-module homomorphism  \n$\\phi :\\A\\rightarrow \\U$ and both $\\tau_1$ and $\\tau_2$ are continuous.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n From Corollary \\ref{tak}-$(ii)$,\n\\[D_1((a,x))=(\\delta_1(a)+\\tau_1(x),\\tau_2(x)) \\quad \\text{and} \\quad\nD_2((a,x))=(0,\\delta_2(a))\\quad\\quad ((a,x))\\in \\A\\ltimes \\U) \\]\nare derivations on $\\A\\ltimes \\U$.\\\\\n$(i)$ Since $ann_{\\A}\\U =(0)$, as in Proposition \\ref{au1}-$(i)$, it can be proved that $\\tau_1$ is continuous. By continuity of $\\tau_2$, $ann_{\\A}\\U =(0)$, $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$ and Theorem \\ref{joda}-$(iii)$, we have \n$$\\mathfrak{S} (\\delta_1)\\subseteq (\\mathfrak{S} (\\tau_2):\\U)_{\\A}=((0),\\U)_{\\A}=ann_{\\A}\\U =(0).$$\nTherefore $\\delta_1$ is continuous and so is $D_1$.\\par \nBefore proving parts $(ii)$ and $(iii)$ we show that if $\\psi :\\A\\rightarrow \\U$ is any left $\\A$-module homomorphism, then it is continuous. Let $\\{a_k\\}$ be a sequence in $\\A$ such that $a_k\\rightarrow 0$. By the Cohen's factorization theorem there is some $c\\in \\A$ and some sequence $b_k$ in $\\A$ for which $b_k\\rightarrow 0$ and $a_k=b_k c \\, \\, (k\\in \\mathbb{N})$. Thus $\\psi (a_k)=b_k\\psi (c)\\rightarrow 0$ and hence $\\psi$ is continuous.\n\\\\\n$(ii)$ Let $\\psi :\\A\\rightarrow \\U$ be the map $\\psi = \\tau_2 o\\phi - \\phi o\\delta_1$. By the condition $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$ and the properties of $\\delta_1$ and $\\tau_2$ it can be shown that $\\psi$ is a left $\\A$-module homomorphism. Therefore $\\psi$ and $\\phi$ are continuous and hence $\\tau_2 o\\phi$ is continuous. So $\\mathfrak{S} (\\tau_2 o\\phi)=(0)$. On the other hand by Lemma \\ref{da}-$(i)$, since $\\phi$ is surjective, $\\mathfrak{S} (\\tau_2 o\\phi)=\\mathfrak{S} (\\tau_2)$. Thus $\\mathfrak{S} (\\tau_2)=0$. Therefore $\\tau_2$ is continuous. So by the hypothesis $D_1$ is continuous.\n\\\\\n$(iii)$\nAs in the previous part we put $\\psi = \\tau_2 o\\phi - \\phi o\\delta_1$ which is a left $\\A$-module homomorphism. Since $\\tau_2$ and $\\phi$ are continuous, it follows that $\\phi o\\delta_1 $ is continuous as well. By Lemma \\ref{da}-$(ii)$ we have $\\phi(\\mathfrak{S} (\\delta_1))=(0)$. Since $\\phi$ is injective, it follows that $\\mathfrak{S} (\\delta_1)=(0)$. So $\\delta_1$ is continuous and by the hypothesis $D_1$ is continuous as well.\n\\end{proof}\nNote that Propositions \\ref{taj2} also holds if \"bounded right approximate identity\" and \"left $\\A$-module homomorphism\" are replaced respectively by \"bounded left approximate identity\" and \"right $\\A$-module homomorphism\".\n\\begin{rem}\\label{taj11}\nIn Propositions \\ref{taj1} and \\ref{taj2} if we assume that $ann_{\\A}\\U =(0)$, then as in Proposition \\ref{au1}-$(i)$, the continuity of $\\tau_2$ implies the continuity of $\\tau_1$ or if we assume that $\\U$ has a bounded approximate identity, we conclude that $\\tau_1 =0$. Therefore any of the conditions $ann_{\\A}\\U =(0)$ or $\\U$ to have a bounded approximate identity implies that $\\tau_1$ is continuous and therefore so $D_1$ is continuous.\n\\end{rem}\n\\par \nIn the following example we show that the derivation $D_2$ in Propositions \\ref{taj1} and \\ref{taj2}, can be discontinuous.\n\\begin{exm}\nLet $\\mathcal{B}$ be a Banach algebra and $d:\\mathcal{B}\\rightarrow \\mathcal{B}$ be a discontinuous derivation.\n\\begin{enumerate}\n\\item[(i)]\nLet $\\A:=\\mathcal{B}\\times \\mathcal{B}$ be the direct product of Banach algebras and let $\\U :=\\mathcal{B}$ becomes a Banach $\\A$-bimodule with compatible actions, by the following module actions;\n\\[ \n(a,b)x=ax \\quad  \\text{and} \\quad \nx(a,b)= xa\\quad\\quad (a,b\\in \\A , x\\in \\U).\n\\]\nTherefore $(0)\\times \\mathcal{B}\\subseteq ann_{\\A}\\U$.\nDefine the map $\\delta_1: \\A\\rightarrow \\A$ by $\\delta_1((a,b))=(0,d(b))$. Then $\\delta_1$ is a discontinuous derivation on $\\A$ such that \n$$\\delta_1 (\\A)\\subseteq (0)\\times \\mathcal{B}\\subseteq ann_{\\A}\\U. $$\nSo the map $D_2: T(\\A,\\U)\\rightarrow T(\\A,\\U)$ defined by $D_2((a,b),x)=(\\delta_1((a,b)),0)$ is discontinuous.\n\\item[(ii)]\nAssume that $\\U :=\\mathcal{B}\\times \\mathcal{B}$ as the direct product of Banach spaces which becomes a Banach algebra with thr product \n$$(x,y)(x',y')=(xx',0)\\quad \\quad ((x,y),(x',y')\\in \\U).$$\nLet $\\A:=\\mathcal{B}$ and $\\U$ is became a Banach $\\A$-bimodule with the pointwise module actions which are compatible  with its algebraic operations.\\par \nNow consider the map $\\delta_2: \\A\\rightarrow \\U$ defined by $\\delta_2(a)=(0,d(a))$. $\\delta_2$ is a discontinuous derivation such that \n$$\\delta_2(\\A)\\subseteq (0)\\times \\mathcal{B}\\subseteq ann_{\\U}\\U.$$\nHence the map $D_2: \\A \\ltimes \\U \\rightarrow \\A \\ltimes \\U$ defined by \n$$D_2((a,(x,y)))=(0,\\delta_2(a))$$\nis a discontinuous derivation on $\\A\\ltimes \\U$.\n\\item[(iii)]\nSuppose that $\\U$ is a Banach space and $T:\\U\\rightarrow \\mathbb{C}$ is a discontinuous linear functional. \nSet $\\A:=T(\\mathbb{C},\\mathbb{C})$ and we turn $\\U$ into a Banach $\\U$-bimodule by the actions below;\n\\[(a,b)x=ax \\quad \\text{and} \\quad x(a,b)=xb  \\quad\\quad ((a,b)\\in A, x\\in \\U).\\]\nConsider the Banach algebra $T(\\A,\\U)$ and the map $\\tau_1:\\U\\rightarrow \\A$ defined by \n$$\\tau_1(x)=(0,T(x)).$$\nSo $\\tau_1$ is an $\\A$-bimodule homomorphism such that \n$$x\\tau_1(x')+\\tau_1 (x)x'=0\\quad\\quad (x,x'\\in \\U).$$\nThus the map $D_2((a,b),x)=(\\tau_1(x),0)$ is a derivation on $T(\\A,\\U)$ which is discontinuous.\n\\end{enumerate}\n\\end{exm}\nNote that in Proposition \\ref{taj2} if we assume that $\\delta_2$ is continuous, then derivation $D$ is continuous as well.\n\n\\section{The first cohomology group of $\\A\\ltimes\\U$}\nIn this section we determine the first cohomology group of $\\A\\ltimes\\U$ in some special cases. Throughout this section  for two linear spaces $\\mathcal{X}$ and $\\mathcal{Y}$, we shall write $ \\mathcal{X} \\cong \\mathcal{Y}$ to indicate that the spaces are linearly isomorphic. \n\\par \nFrom this point up to the last section we assume that every derivation $D$ on $\\A\\ltimes\\U$ is of the form \n\\[D(a,x)=(\\delta_1 (a),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad (a\\in \\A,x\\in \\U)\\]\nin which $\\delta_1:\\A\\rightarrow\\A$, $\\delta_2:\\A\\rightarrow\\U$ and $\\tau_2:\\U\\rightarrow\\U$\nare derivations such that \n\\[\n\\tau_2 (ax)=a\\tau_2 (x)+\\delta_1 (a)x+\\delta_2 (a)x \\,\\, \\text{and} \\,\\, \n\\tau_2 (xa)=\\tau_2 (x)a+x\\delta_1 (a)+x\\delta_2 (a) \\quad\\quad (a\\in \\A,x\\in \\U).\n\\]\nIndeed, we consider that for every derivation $D$ on $\\A\\ltimes\\U$ we have $\\tau_1=0$, where $\\tau_1$ is as in Theorem \\ref{asll}. \n\\par \nIf for all $\\delta\\in Z^{1}(\\A)$, $\\delta_1(\\A)\\subseteq ann_{\\A}\\U$, then by Corollary \\ref{tak}, the map $D$ given by $D((a,x))=(\\delta(a),0)$ is a continuous derivation on $\\A\\ltimes\\U$ and for every continuous derivation $D$ on $\\A\\ltimes\\U$ the maps $D_1((a,x))=(0,\\delta_2 (a)+\\tau_2 (x))$ and $D_2((a,x))=(\\delta_1(a),0)$ are derivations in $Z^{1}(\\A\\ltimes\\U)$ and $D=D_1+D_2$. By these arguments, in this case,\n\\[Z^{1}(\\A\\ltimes\\U)\\cong \\mathcal{W} \\times Z^{1}(\\A),\\]\nwhere $\\mathcal{W}$ is a linear subspace consisting of all continuous derivations on $\\A\\ltimes\\U$ of the form \n$D(a,x)=(0,\\delta_2 (a)+\\tau_2 (x))$, such that $\\delta_2\\in Z^{1}(\\A,\\U)$, $\\tau_2\\in Z^{1}(\\U)$ and \n\\[\\tau_2 (ax)=a\\tau_2 (x)+\\delta_2 (a)x \\,\\, \\text{and} \\,\\,\\tau_2 (xa)=\\tau_2 (x)a+x\\delta_2 (a)\\quad\\quad (a\\in\\A,x\\in\\U).\\]\n\\begin{thm}\\label{11}\nLet for every derivation $\\delta\\in Z^{1}(\\A)$ we have $\\delta(\\A)\\subseteq ann_{\\A}\\U$. If $H^{1}(\\A,\\U)=(0)$, then \\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Z^{1}(\\A)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]}{\\mathcal{E}},\\]\nwhere $\\mathcal{E}$ is the following linear subspace of $Z^{1}(\\A)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]$;\n\n\\[\\mathcal{E}=\\{(id_{a},r_{a}+id_{\\U,x})\\, \\mid \\, a\\in\\A, \\, id_{\\A , x}=0 \\, \\, (x\\in\\U)\\}.\\]\n\\end{thm}\n\\begin{proof}\nDefine the map \\[\\Phi:Z^{1}(\\A)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]\\rightarrow H^{1}(\\A\\ltimes\\U)\\]\nby $\\Phi ((\\delta,\\tau))=[D_{\\delta,\\tau}]$\nwhere $D_{\\delta,\\tau} ((a,x))=(\\delta (a),\\tau (x))$ and $[D_{\\delta,\\tau}]$ represents the equivalence class of $D_{\\delta,\\tau}$ in $H^{1}(\\A\\ltimes\\U).$ By Corollary \\ref{tak} the maps $D_1:\\A\\ltimes\\U\\rightarrow \\A\\ltimes\\U$ and $D_2:\\A\\ltimes\\U\\rightarrow\\A\\ltimes\\U$ given respectively by $D_1((a,x))=(0,\\tau (x))$ and $D_2((a,x))=(\\delta(a),0)$, are continuous derivations on $\\A\\ltimes\\U$ and so $\\Phi$ is well-defined. Clearly $\\Phi$ is linear. Let $D\\in Z^{1}(\\A\\ltimes\\U)$. By the hypothesis and the discussion before the theorem, $D=D_1+D_2$ where  $D_1((a,x))=(\\delta_1(a),0)$ and $D_2((a,x))=(\\delta_1(a),0)$ are continuous derivations on $\\A\\ltimes\\U$ and $\\delta_1\\in Z^{1}(\\A)$, $\\delta_2\\in Z^{1}(\\A,\\U)$ and $\\tau_2\\in Z^{1}(\\U)$. Since $H^{1}(\\A,\\U)=(0)$, there is some $x_0\\in \\U$ such that $\\delta_2 =id_{\\A,x_0}$. Define the map $\\tau:\\U\\rightarrow \\U $ by $\\tau=\\tau_2 - id_{\\U,x_0}$. By the properties of $\\tau$ and Remark \\ref{inn2}, it follows that $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$. Now we have\n\\[\nD((a,x))-D_{\\delta_1 ,\\tau} ((a,x))=(0,id_{\\A,x_0}(a)+id_{\\U,x_0}(x))= id_{(0,x_0)}(a,x) \\quad \\quad ((a,x)\\in\\A\\ltimes\\U).\\]\nSo $[D]=[D_{\\delta,\\tau}]$ and hence $\\Phi ((\\delta_1,\\tau))=[D_{\\delta_1 ,\\tau}]=[D]$. Thus $\\Phi$ is surjective.\n\\par\nIt can be easily checked that $\\mathcal{E}$ is a linear subspace and if $(\\delta,\\tau)\\in ker \\Phi$, then $D_{\\delta,\\tau}$ is an inner derivation on $\\A\\ltimes\\U$. So for some $(a_0,x_0)\\in \\A\\ltimes\\U$, \n\\[D_{\\delta,\\tau}((a,x))=id_{a_0,x_0}(a,x)=(id_{a_0}(a),id_{\\A,x_0}(a)+r_{a_0}(x)+id_{\\U,x_0}(x)),\\]\nwhich implies that \\[\\delta =id_{a_0}, \\quad \\tau (x)=r_{a_0}(x)+id_{\\U,x_0}(x)\\quad \\text {and} \\quad id_{\\A,x_0}=0 \\quad \\quad (a\\in\\A ,x\\in\\U).\\] Hence $(\\delta,\\tau)\\in \\mathcal{E}$. Conversely, Suppose that $(\\delta,\\tau)\\in \\mathcal{E}$. Then for $a_0\\in\\A$ and $x_0\\in\\U$ we have $\\delta =id_{a_0}$ and $ \\tau =r_{a_0}+id_{\\U,x_0}$ where $id_{\\A,x_0}=0$ for all $a\\in\\A$. So \n\\[\nD_{\\delta,\\tau}((a,x))=(id_{a_0}(a),r_{a_0}(x)+id_{\\U ,x_0}(x))\n= id_{(a_0,x_0)}((a,x)).\\]\nThus $D_{\\delta,\\tau}\\in N^{1}(\\A\\ltimes\\U)$ and hence $(\\delta,\\tau)\\in ker\\Phi$. So $\\mathcal{E}=ker\\Phi$. Therefore by the above arguments we have \n\\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Z^{1}(\\A)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]}{\\mathcal{E}}.\\]\n\\end{proof}\n\\begin{cor}\nSuppose that for every $\\delta\\in Z^{1}(\\A)$, $\\delta (\\A)\\subseteq ann_{\\A}\\U$. If $H^{1}(\\A,\\U)=(0)$ and  $H^{1}(\\A\\ltimes\\U)=(0)$, then $H^{1}(\\A)=(0)$ and $\\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{C_{\\A}(\\U)+I(\\U)}=(0).$\n\\end{cor}\n\\begin{proof}\nLet $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$ and $\\delta\\in Z^{1}(\\A)$. Since $\\delta (\\A)\\subseteq ann_{\\A}\\U$, it follows that \n\\[ (0,\\tau)\\,,\\, (\\delta,0)\\in Z^{1}(\\A)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)].\\]\nBy the hypothesis and according to the preceding theorem, there exist some $a_0\\in\\A$ and $x_0\\in\\U$ such that $id_{\\A,x_0}=0$ and \\[(\\delta,0)=(id_{a_0},r_{a_0}+id_{\\U,x_0})\\quad ,\\quad (0,\\tau)=(id_{a_0},r_{a_0}+id_{\\U,x_0}).\\]\nHence $\\delta=id_{a_0}$ and so $H^{1}(\\A)=(0)$. Moreover, $\\tau=r_{a_0}+id_{\\U,x_0}$ where $id_{a_0}=0$. Thus $\\tau\\in C_{\\A}(\\U)+I(\\U)$.\n \\end{proof}\n If we assume that $\\A$ is a Banach algebra with $ann_{\\A}\\A=(0)$ and $\\delta\\in Z^{1}(\\A)$ where $\\delta\\neq 0$, then $\\delta(\\A)\\not\\subseteq ann_{\\A}\\A =(0)$. So in this case, the condition  $\\delta (\\A)\\subseteq ann_{\\A}\\A$ is not stisfied on $\\A\\ltimes\\A$ for every derivation $\\delta\\in Z^{1}(\\A)$ and this shows that this condition does not hold in general. In the following we give an example of Banach algebras which are satisfying in conditions of Theorem \\ref{11}.\n \\begin{exm}\nLet $\\A$ be a semisimple commutative Banach algebra. By Thomas' theorem \\cite{tho}, we have $Z^{1}(\\A)=(0)$. So in this case, for every $\\delta\\in Z^{1}(\\A)$ and every Banach $\\A$-bimodule $\\U$, $\\delta (\\A)\\subseteq ann_{\\A}\\U$ (in fact $\\delta=0$). In particular for a semisimple commutative Banach algebra $\\A$, if $\\U$ is a Banach $\\A$-bimodule such that $H^{1}(\\A,\\U)=(0)$ and $\\A\\ltimes\\U$ is satisfying in conditions of Theorem \\ref{11}, then\n \\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{R_{\\A}(\\U)+ I(\\U)}.\\]\n \\end{exm}\n Note that if $\\A$ is a commutative Banach algebra with $H^{1}(\\A)=(0)$ and $H^{1}(\\A,\\U)=(0)$, then above example holds again.\n \\par \nWe continue by characterizing the first cohomology group of $\\A\\ltimes\\U$ in another case.\n\\par \nIf for each $\\delta_2\\in Z^{1}(\\A,\\U)$, $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$, then by Corollary \\ref{tak}, the map $D ((a,x))=(0,\\delta_2(a))$ is a continuous derivation on $\\A\\ltimes\\U$ and for each derivation $D\\in Z^{1}(\\A\\ltimes\\U)$ we conclude that the maps $D_1 ((a,x))=(\\delta_1 (a),\\tau_2(x))$ and $D_2((a,x))=(0,\\delta_2(a))$ are derivations in $Z^{1}(\\A\\ltimes\\U)$ and $D=D_1+D_2$. So by this argument in this case,\n \\[Z^{1}(\\A\\ltimes\\U)\\cong Z^{1}(\\A ,\\U)\\times \\mathcal{W}\\]\n where $\\mathcal{W}$ is the linear subspace of all continuous derivations on $\\A\\ltimes\\U$ which are of the  form $D_1((a,x))=(\\delta_1 (a),\\tau_2(x))$ with $\\delta_1\\in Z^{1}(\\A)\\,,\\tau_2\\in Z^{1}(\\U)$ and \n \\[\\tau_2 (ax)=a\\tau_2 (x)+\\delta_1 (a)x\\quad \\text{and} \\quad \\tau_2 (xa)=\\tau_2 (x)a+x\\delta_1 (a)\\quad\\quad ((a,x)\\in \\A\\ltimes\\U).\\]\n  \\begin{thm}\\label{22}\n Suppose that for each derivation $\\delta\\in Z^{1}(\\A,\\U)$ we have $\\delta(\\A)\\subseteq ann_{\\U}\\U$. If $H^{1}(\\A)=(0)$, then \n \\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Z^{1}(\\A,\\U)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]}{\\mathcal{F}},\\]\n where $\\mathcal{F}$ is the following linear subspace of $Z^{1}(\\A,\\U)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]$;\n \\[\\mathcal{F}=\\{ (id_{\\A,x},r_{a}+id_{\\U, x})\\, \\mid \\, x\\in\\U, \\, id_{a}=0 \\, \\, (a\\in\\A) \\}.\\]\n \\end{thm}\n \\begin{proof}\n Define the map \n \\[\\Phi: Z^{1}(\\A,\\U)\\times [Hom_{\\A}(\\U)\\cap Z^{1}(\\U)]\\rightarrow H^{1}(\\A\\ltimes\\U)\\]\n by $\\Phi ((\\delta,\\tau))=[D_{\\delta,\\tau}]$\n where $D_{\\delta,\\tau}((a,x))=(0,\\delta(a)+\\tau(x))$\n is a continuous derivation on $\\A\\ltimes\\U$. Clearly $\\Phi$ is a well-defined linear map. If \n $D\\in Z^{1}(\\A\\ltimes\\U)$, then $D=D_1+D_2$ where $D_1((a,x))=(\\delta_1 (a),\\tau_2(x))$ and $D_2((a,x))=(0,\\delta_2(a))$ are continuous derivations on $\\A\\ltimes\\U$ with $\\delta_1\\in Z^{1}(\\A), \\delta_2\\in Z^{1}(\\A,\\U)$ and $\\tau\\in Z^{1}(\\U)$. Since $H^{1}(\\A)=(0)$, there is some $a_0\\in \\A$ such that $\\delta_1=id_{a_0}$. Consider the map $\\tau=\\tau_2-r_{a_0}$ on $\\U$. We have  \n $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$. Also\n \\[D((a,x))-D_{\\delta_2 ,\\tau}((a,x))=id_{(a_0,0)}((a,x)).\\]\n So $[D]=[D_{\\delta_2 ,\\tau}]$ and hence $\\Phi ((\\delta_2,\\tau))=[D_{\\delta_2,\\tau}]=[D]$.\n Therefore $\\Phi$ is surjective. A straightforward proceeding shows that $\\mathcal{F}$ is a vector subspace and $ker\\Phi =\\mathcal{F}$. This establishes the desired vector space isomorphism.  \n \\end{proof}\n  \\begin{cor}\\label{222}\n  Suppose that for any derivation $\\delta\\in Z^{1}(\\A,\\U)$, $\\delta(\\A)\\subseteq ann_{\\U}\\U$. If \n  $H^{1}(\\A)=(0)$ and $H^{1}(\\A\\ltimes\\U)=(0)$, then $H^{1}(\\A,\\U)=(0)$ and \n  $\\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{C_{\\A}(\\U)+I(\\U)}=(0).$\n  \\end{cor}\n  \\begin{proof}\n  Let $\\delta\\in Z^{1}(\\A,\\U)$ and $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$. By the hypothesis, \n$(\\delta,0),(0,\\tau)\\in  Z^{1}(\\A,\\U)\\times Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$. Again by the hypothesis and the preceding theorem, there are $x_0\\in\\U$ and $a_0\\in \\A$ such that $id_{a_0}=0$ and \n\\[(\\delta,0)=(id_{\\A,x_0},r_{a_0}+id_{\\U,x_0})\\quad , \\quad (0,\\tau)=(id_{\\A,x_0},r_{a_0}+id_{\\U,x_0}).\\]\nNow the result follows from these equalities.\n  \\end{proof}\nIf we assume that $\\A$ is a Banach algebra with $ann_{\\A}\\A =(0)$ and $\\delta\\in Z^{1}(\\A)$ where $\\delta\\neq 0$, then by putting $\\U =\\A$ we have $\\delta(\\A)\\not\\subseteq ann_{\\A}\\A =ann_{\\A}\\U =(0)$. So in this case, the condition $\\delta(\\A)\\subseteq ann_{\\A}\\U$ is not satisfied on $\\A\\ltimes\\U$ for every derivation $\\delta\\in Z^{1}(\\A,\\U)$. This shows that this condition does not hold in general. In the following we give an example of Banach algebras which are satisfying in conditions of Theorem \\ref{22}.\n\\begin{exm}\nIf $\\A$ is a Banach algebra and $\\U$ is a commutative Banach $\\A$-bimodule such that $H^{1}(\\A,\\U)=(0)$, then $Z^{1}(\\A,\\U)=(0)$. So in this case, for every $\\delta\\in Z^{1}(\\A,\\U)$, $\\delta(\\A)\\subseteq ann_{\\U}\\U$ (in fact $\\delta =0$). In this case if $H^{1}(\\A)=(0)$ and $\\A\\ltimes\\U$ satisfies in Theorem \\ref{22}, then\n  \\[ H^{1}(\\A\\ltimes\\U)\\cong \\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{C_{\\A}(\\U)\\cap N^{1}(\\U)}.\\]\n\\end{exm}\n Note that if $\\A$ is a super amenable Banach algebra and $\\U$ is a commutative Banach $\\A$-bimodule, then above example holds.\n \\par \n In the continuation we consider a case on $\\A\\ltimes\\U$ in which for any derivation $\\delta_1\\in Z^{1}(\\A)$ and $\\delta_2\\in Z^{1}(\\A,\\U)$ we have $\\delta_1(\\A)\\subseteq ann_{\\A}\\U$ and $\\delta_2(\\A)\\subseteq ann_{\\U}\\U$.  In fact by these conditions on $\\A\\ltimes\\U$, for every $\\delta_1\\in Z^{1}(\\A)$ and $\\delta_2\\in Z^{1}(\\A,\\U);$\n  \\[\\delta_1(a)x+\\delta_2(a)x=0\\quad \\text {and} \\quad x\\delta_1 (a)+x\\delta_2 (a)=0 \\quad\\quad (a\\in \\A,x\\in \\U).\\]\nIn this case, every derivation $D\\in Z^{1}(A\\ltimes\\U)$ can be written as $D=D_1+D_2$ where \n  \\[D_{1}((a,x))=(\\delta_1 (a),\\delta_2(a))\\quad \\text{and} \\quad D_{2}((a,x))=(0,\\tau_2(x))\\]\nare continuous derivations on $\\A\\ltimes\\U$ and $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$. In this case, we conclude that \n$$R_{\\A}(\\U)+N^{1}(\\U)\\subseteq Hom_{\\A}(\\U)\\cap Z^{1}(\\U).$$  \n\\begin{thm}\\label{33}\n Suppose that for every $\\delta_1\\in Z^{1}(\\A)$ and $\\delta_2\\in Z^{1}(\\A,\\U)$ we have \\[\\delta_1(\\A)\\subseteq ann_{\\A}\\U \\quad \\text{and} \\quad \\delta_{2}(\\A)\\subseteq ann _{\\U}\\U.\\]\nSuppose further that $\\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{R_{\\A}(\\U)+ N^{1}(\\U)}=(0)$. Then \n \\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Z^{1}(\\A)\\times Z^{1}(\\A,\\U)}{\\mathcal{K}},\\]\n where $\\mathcal{K}$ is the following linear subspace of $Z^{1}(\\A)\\times Z^{1}(\\A,\\U)$;\n \\[\\mathcal{K}=\\{(id_{a},id_{\\A,x})\\, \\mid \\,  r_{a}+id_{\\U,x}=0\\,\\,(a\\in \\A,x\\in \\U)\\}.\\]\n \\end{thm}\n  \\begin{proof}\n Define the map \n \\[\\Phi: Z^{1}(\\A)\\times Z^{1}(\\A,\\U)\\rightarrow H^{1}(\\A\\ltimes\\U)\\]\n by $\\Phi ((\\delta_1,\\delta_2))=[D_{\\delta_1 , \\delta_2}]$\n where $D_{\\delta_1 , \\delta_2} ((a,x))=(\\delta_1 (a),\\delta_2(a))$ is a continuous derivation on $\\A\\ltimes\\U$. The map $\\Phi$ is well-defined and linear. If $D\\in Z^{1}(\\A\\ltimes\\U)$, then $D((a,x))=(\\delta_1 (a),\\delta_2(a)+\\tau_2(x))$. By the hypothesis, $\\tau_2=r_{a}+id_{\\U,x}$ where $a\\in\\A$ and $x\\in\\U$. Define the derivations $d_1\\in Z^{1}(\\A)$ and $d_2\\in Z^{1}(\\A,\\U)$ by $d_1=\\delta_1 -id_{a}$ and $d_2=\\delta_2 -id_{\\A,x}$ respectively. Then $D-D_{d_1 ,d_2}=id_{(a ,x )}$. So $\\Phi ((\\delta_1,\\delta_2))=[D_{d_1 , d_2}]=[D]$.\nThus $\\Phi$ is surjective. It can be easily seen that $ker \\Phi =\\mathcal{K}$. So we have the desired vector spaces isomorphism.\n \\end{proof}\n \\begin{cor}\\label{333}\n Suppose that for every $\\delta_1\\in Z^{1}(\\A)$ and $\\delta_2\\in Z^{1}(\\A,\\U)$ we have \\[\\delta_1(\\A)\\subseteq ann_{\\A}\\U \\quad \\text{and} \\quad \\delta_2(\\A)\\subseteq ann _{\\U}\\U.\\] Suppose further that \n  $\\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{{R_{\\A}(\\U)+ N^{1}(\\U)}}=(0)$. If $H^{1}(\\A\\ltimes\\U)=(0)$, then $H^{1}(\\A)=(0)$ and $H^{1}(\\A,\\U)=(0)$.\n  \\end{cor}\n  \\begin{proof}\n  Let $\\delta_1\\in Z^{1}(\\A)$ and $\\delta_2\\in Z^{1}(\\A,\\U)$. By the hypothesis \n  \\[(\\delta_1 , 0)\\,,\\, (0,\\delta_2)\\in Z^{1}(\\A)\\times Z^{1}(\\A,\\U).\\]\n Again by the hypothesis and the preceding theorem, there exists some $(id_{a},id_{\\A,x})\\in \\mathcal{K}$ such that \n  $(\\delta_1 , 0)=(id_{a},id_{\\A,x})$ and $(0,\\delta_2)=(id_{a},id_{\\A,x})$ and so $\\delta_1$ and $\\delta_2$ are inner.\n  \\end{proof} \nIf $H^{1}(\\U)=(0)$, since $R_{\\A}(\\U)+ N^{1}(\\U)\\subseteq Z^{1}(\\U)= N^{1}(\\U)$, It follows that \n $\\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{{R_{\\A}(\\U)+ N^{1}(\\U)}}=(0)$. So if $H^{1}(\\U)=(0)$, Theorem \\ref{33} and Corollary \\ref{333} hold again.  \n \\par \n Let $\\A$ be a Banach algebra with $ann_{\\A}\\A =(0)$ and $\\delta_1 ,\\delta_2\\in Z^{1}(\\A)$ such that $\\delta_1 +\\delta_2 \\neq 0$. If we put $\\tau =\\delta_1+\\delta_2$ and define linear map $D$ on $\\A\\ltimes\\A$ by \n \\[D((a,x))=(\\delta_1 (a),\\delta_2(a)+\\tau(x))\\quad\\quad ((a,x)\\in \\A\\ltimes\\A),\\]\n then $D\\in Z^{1}(\\A\\ltimes\\A )$. But for $a, x\\in\\A$, the equation $\\delta_1 (a)x+\\delta_2 (a)x=0$\n is not necessarily true. This example does not satisfy the conditions of Theorem \\ref{33}. \n \\par \nThe last case we consider is as follows.\n \\begin{thm}\\label{44}\n Let $H^{1}(\\A)=(0)$ and $H^{1}(\\A,\\U)=(0)$. Then \n \\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{C_{\\A}(\\U)+I(\\U)}.\\]\n\\end{thm} \n\\begin{proof}\nDefine the map \n\\[\\Phi : Hom_{\\A}(\\U)\\cap Z^{1}(\\U)\\rightarrow H^{1}(\\A\\ltimes\\U)\\]\nby $\\Phi(\\tau)=[D_{\\tau}]$ where $D_{\\tau}((a,x))=(0,\\tau (x))$ is a continuous derivation on $\\A\\ltimes\\U$. The map $\\Phi$ is well-defined linear. If $D\\in Z^{1}(\\A\\ltimes\\U )$, then $D=(\\delta_1 , \\delta_2 +\\tau_2)$. By hypotheses  $\\delta_1 =id _{a}$ and $\\delta_2 =id_{\\A, x}$ for some  $a\\in \\A$ and $x\\in \\U$. Define the map $\\tau:\\U\\rightarrow \\U$ by $\\tau=\\tau_2-r_{a}-id_{\\U, x}$. Then $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$ and so $D-D_{\\tau}=id_{(a , x)}$. Hence $\\Phi (\\tau)=[D_\\tau]=[D]$. Thus $\\Phi$ is surjective. By Corollary \\ref{tak}-$(iv)$, $ker \\Phi =C_{\\A}(\\U)+I(\\U)$. So the desired vector space isomorphism is established.\n\\end{proof}\nThe following corollary follows immediately from the preceding theorem.\n\\begin{cor}\nIf $H^{1}(\\A)=(0),H^{1}(\\A,\\U)=(0)$ and $H^{1}(\\A\\ltimes\\U)=(0)$, then for every $\\tau\\in Hom_{\\A}(\\U)\\cap Z^{1}(\\U)$ there are some $r_{a}\\in C_{\\A}(\\U)$ and $id_{\\U , x}\\in I(\\U)$ such that $\\tau =r_{a}+id_{x}$.\n\\end{cor}\nIn the following an example of Banach algebras satisfying the conditions of Theorem \\ref{44}, is given.\n\\begin{exm}\nIf $\\A$ is a weakly amenable commutative Banach algebra, then for any commutative Banach $\\A$-bimodule $\\mathcal{X}$ we have $H^{1}(\\A,\\mathcal{X})=(0)$. So in this case, if $\\U$ is a commutative Banach $\\A$-bimodule satisfying the conditions of Theorem \\ref{44}, then $Z^{1}(\\A)=H^{1}(\\A)=(0)$ and $Z^{1}(\\A,\\U)=H^{1}(\\A,\\U)=(0)$ and hence \n\\[H^{1}(\\A\\ltimes\\U)\\cong \\frac{Hom_{\\A}(\\U)\\cap Z^{1}(\\U)}{R_{\\A}(\\U)+ N^{1}(\\U)}.\\]\n\\end{exm}\nThis example could be also derived from Theorem \\ref{22}.\n\\section{Applications}\nIn this section we investigate applications of the previous sections and give some examples.\n\\subsection*{Direct product of Banach algebras}\nLet $\\A$ and $\\U$ be Banach algebras. With trivial module actions $\\A\\U=\\U\\A =(0)$, as we saw in Example \\ref{dp}, $\\A\\ltimes\\U=\\A\\times\\U$ where $\\A\\times\\U$ is $l^{1}$-direct product of Banach algebras. In this case, $ann_{\\A}\\U =\\A$ and so for every derivation $\\delta:\\A\\rightarrow \\A$ we have $\\delta(\\A)\\subseteq ann_{\\A}\\U$. Also in this case, $R_{\\A}(\\U)=(0)$ and $Hom_{\\A}(\\U)=\\mathbb{B}(\\U)$. The following proposition follows from Theorem \\ref{asll}.\n\\begin{prop}\\label{dpd}\nLet $\\A$ and $\\U$ be Banach algebras and $D:\\A\\times\\U\\rightarrow\\A\\times\\U$ be a map. The following are equivalent.\n\\begin{enumerate}\n\\item[(i)]\n$D$ is a derivation.\n\\item[(ii)]\n\\[D((a,x))=(\\delta_1(a)+\\tau_1(x),\\delta_2(a)+\\tau_2(x))\\quad \\quad ((a,x)\\in\\A\\times\\U)\\]\nsuch that $\\delta_1:\\A\\rightarrow\\A,\\tau_2:\\U\\rightarrow\\U$ are derivations and $\\tau_1:\\U\\rightarrow\\A$ and $\\delta_2:\\A\\rightarrow\\U$ are linear maps satisfying the following conditions;\n\\[\\tau_1(\\U)\\subseteq ann_{\\A}\\A , \\,\\, \\delta_2(\\A)\\subseteq ann_{\\U}\\U, \\,\\,  \\tau_1(xy)=0, \\, \\text{and} \\,\\, \\delta_2(ab)=0 \\quad \\quad(a,b\\in\\A , x,y\\in \\U).\\]\n\\end{enumerate}\nMoreover, if $ann_{\\U}\\U=(0)$ or $\\A^{2}=\\A$ and $ann_{\\A}\\A=(0)$ or $\\U ^{2}=\\U$, then $\\delta_2=0$ and $\\tau_1=0$, respectively.\n\\end{prop}\nBy this proposition it is clear if $\\A$ or $\\U$ has a bounded approximate identity, then $\\delta_2=0$ and $\\tau_1=0$. Hence in this case every derivation $D$ on $\\A\\times\\U$ is of the form $D((a,x))=(\\delta(a),\\tau(x))$ where $\\delta$ and $\\tau$ are derivations on $\\A$ and $\\U$, respectively.\n\\begin{rem}\\label{d1}\nIf $\\A$ and $\\U$ are Banach algebras, then for any derivation $\\delta:\\A\\rightarrow\\A$ and $\\tau:\\U\\rightarrow\\U$ the maps $D_1$ and $D_2$ on $\\A\\times\\U$ given by \n\\[D_{1}((a,x))=(\\delta (a),0)\\quad \\text{and} \\quad D_2((a,x))=(0,\\tau (x)),\\]\nare derivations. According to this point, if every derivation on $\\A\\times\\U$ is continuous, then every derivation on $\\A$ and every derivation on $\\U$ is continuous.\n\\par \nAlso let  $\\A$ or $\\U$ have a bounded approximate identity. So by above arguments if every derivation on $\\A$ and every derivation on $\\U$ is continuous, then every derivation on $\\A\\times\\U$ is continuous.\n\\end{rem}\n\\begin{rem}\\label{d2}\nLet $\\A$ and $\\U$ be Banach algebras and $D\\in Z^{1}(\\A\\ltimes\\U)$. If $ann_{\\U}\\U =(0)$ or $\\overline{\\A^{2}}=\\A$ and $ann_{\\A}\\A =(0)$ or $\\overline{\\U ^{2}}=\\U$, then in the representation of  $D$ as in the preceding proposition, $\\delta_2=0$ and $\\tau_1=0$. So $D((a,x))=(\\delta(a),\\tau(x))$ where $\\delta \\in Z^{1}(\\A)$ and $\\tau\\in Z^{1}(\\U)$. In this case we can conclude\n\\[ Z^{1}(\\A\\times\\U)\\cong Z^{1}(\\A)\\times Z^{1}(\\U) \\quad \\text{and} \\quad N^{1}(\\A\\times\\U)\\cong N^{1}(\\A)\\times N^{1}(\\U).\\]\nMoreover\n\\[ H^{1}(\\A\\times\\U)\\cong H^{1}(\\A)\\times H^{1}(\\U).\\]\nIn particular if $\\A$ or $\\U$ have a bounded approximate identity, then above observation holds.\n\\par \nIn the case of isomorphism of the first cohomology group by Theorem \\ref{11}, one can weaken the conditions. Indeed $\\A\\times\\U$ with $\\overline{\\U ^{2}}=\\U$ or $ann_{\\A}\\A =(0)$ satisfies the conditions of Theorem \\ref{11}. Since $Hom_{\\A}(\\U)=\\mathbb{B}(\\U)$ and $R_{\\A}(\\U)=(0)$, in this case  \n\\[H^{1}(\\A\\times\\U)\\cong H^{1}(\\A)\\times H^{1}(\\U)\\]\n(In fact in this case, $\\mathcal{E}=N^{1}(\\A)\\times N^{1}(\\U)$).\n\\par \nIf $\\overline{\\A ^{2}}=\\A$ or $ann_{\\U}\\U =(0)$, by symmetry as above we obtain the same result again.\n\\end{rem}\nAs the next example confirms, it is not necessarily true that for any derivation on $\\A\\times\\U$, $\\tau_1 =0$ or $\\delta_1 =0$ in the decomposition of it. The example is given from \\cite{ess}.\n \\begin{exm}\n Let $\\A$ be a Banach algebra with $ann_{\\A}\\A\\neq 0$. Put $\\U:=ann_{\\A}\\A$. Then $\\U$ is a closed subalgebra of $\\A$. Define the map $D$ on $\\A\\times\\U$ by $D((a,x))=(x,0)$. Then $D$ is a derivation on $\\A\\times\\U$ such that in its representation the map $\\tau_1:\\U\\rightarrow\\A$ is given by $\\tau_1(x)=x$ with $\\tau_1\\neq 0$. \n \\end{exm}\nLet $\\A$ and $\\U$ be Banach algebras and $\\alpha:\\A\\rightarrow \\U$ be a continuous algebra homomorphism with $\\Vert \\alpha\\Vert \\leq 1$. Then the following module actions turn $\\U$ into a Banach $\\A$-bimodule with the compatible actions and norm;\n\\[\nax=\\alpha(a)x \\quad \\text{and} \\quad\nxa=x\\alpha(a)\\quad\\quad (a\\in \\A,x\\in \\U).\\]\nIn this case we can consider $\\A\\ltimes \\U$ with the multiplication given by \n$$(a,x)(b,y)=(ab,\\alpha(a) y+x\\alpha(b)+xy).$$\nWe denote this Banach algebra by $\\A\\ltimes_{\\alpha} \\U$ which is introduced in \\cite{bh}. In the next proposition we see that $\\A\\ltimes_{\\alpha} \\U$ is isomorphic as a Banach algebra to the direct product $\\A\\times \\U$.\n\\begin{prop}\\label{dal}\nLet $\\A$, $\\U$ be Banach algebras and $\\alpha:\\A\\rightarrow \\U$ be a continuous algebra homomorphism with $\\Vert \\alpha\\Vert \\leq 1$. Consider the Banach algebra $\\A\\ltimes_{\\alpha} \\U$ as above. Then $\\A\\ltimes_{\\alpha} \\U$ is isomorphic as a Banach algebra to  $\\A\\times \\U$.\n\\end{prop}\n\\begin{proof}\nDefine $\\theta:\\A\\times \\U\\rightarrow\\A\\ltimes_{\\alpha} \\U$ by $\\theta((a,x))=(a, x-\\alpha(a))$  $(a,x)\\in \\A\\times \\U$. The map $\\theta$ is linear and continuous. Also\n\\[\\theta((a,x)(b,y))=(a,x-\\alpha(a))(b,y-\\alpha(b))=(ab,xy-\\alpha(ab))=\\theta((a,x))\\theta((b,y)),\\]\nfor any $(a,x),(b,y)\\in \\A\\times \\U$. It is obvious that $\\theta$ is bijective. Hence $\\theta$ is a continuous algebra isomorphism.\n\\end{proof}\n\\begin{rem}\nLet $\\A$, $\\U$ and $\\alpha$ are as Proposition \\ref{dal}. By Remarks \\ref{d1}, \\ref{d2} and Proposition \\ref{dal} we have the following.\n\\par \nIf every derivation on $\\A\\ltimes_{\\alpha}\\U$ is continuous, then every derivation on $\\A$ and every derivation on $\\U$ is continuous. If  $\\A$ or $\\U$ have a bounded approximate identity and every derivation on $\\A$ and every derivation on $\\U$ is continuous, then every derivation on $\\A\\ltimes_{\\alpha}\\U$ is continuous. Also if $\\overline{\\U ^{2}}=\\U$ or $ann_{\\A}\\A =(0)$ ($\\overline{\\A ^{2}}=\\A$ or $ann_{\\U}\\U =(0)$), then $H^{1}(\\A\\ltimes_{\\alpha}\\U)\\cong H^{1}(\\A)\\times H^{1}(\\U)$.\n\\end{rem}\nIf $\\U$ is a Banach algebra and $\\A$ is a closed subalgebra of it, then it is clear that the embedding map $i:\\A\\rightarrow\\U$ is continuous algebra homomorphism and hence we can conmsider $\\A\\ltimes_{i} \\U$ which is isomorphic as a Banach algebra to  $\\A\\times \\U$ by Proposition \\ref{dal}. So by above remark we have the following examples.\n\\begin{exm}\n\\begin{enumerate}\n\\item[(i)] If $\\A$ is a semisimle Banach algebra with a bounded approximate identity, then every derivation on $\\A\\ltimes_{i}\\A$ is continuous. \n\\item[(ii)] Consider the group algebra $L^{1}(G)$ as a closed ideal in the measure algebra $M(G)$. Every derivation on $M(G)$ and every derivation on $L^{1}(G)$ is continuous and $M(G)$ is unital. So every derivation on $L^{1}(G)\\ltimes_{i} M(G)$ is continuous.\n\\item[(iii)] Let $\\mathcal{H}$ be a Hilbert space and $\\mathcal{N}$ be a complete nest in $\\mathcal{H}$. The associated nest algebra $Alg\\mathcal{N}$ is a closed subalgebra of $\\mathbb{B}(\\mathcal{H})$ which is unital. By \\cite{chr} every derivation on $Alg\\mathcal{N}$ is  continuous. Also $\\mathbb{B}(\\mathcal{H})$ is a unital $C^{*}$-algebra. Hence every derivation on $Alg\\mathcal{N}\\ltimes_{i} Alg\\mathcal{N}$ and $Alg\\mathcal{N}\\ltimes_{i} \\mathbb{B}(\\mathcal{H})$ is continuous.\n\\end{enumerate}\n\\end{exm}\n\\begin{exm}\n\\begin{enumerate}\n\\item[(i)] Let $\\A$ be a weakly amenable commutative Banach algebra. Since $H^{1}(\\A)=(0)$ and $\\overline{\\A ^{2}}=\\A$, it follows that $H^{1}(\\A\\ltimes_{i}\\A)=(0)$.\n\\item[(ii)]\nSakai showed in \\cite{sa} that every continuous derivation on a $W^{*}$-algebra is inner. Every von Neumann algebra is a $W^{*}$-algebra which is unital. Let $\\A$ be a von Neumann algebra on a Hilbert space $\\mathcal{H}$. Hence $H^{1}(\\A\\ltimes_{i}\\A)=(0)$ and $H^{1}(\\A\\ltimes_{i}\\mathbb{B} (\\mathcal{H}))=(0)$.\n\\item[(iii)]\nLet $\\mathcal{H}$ be a Hilbert space and $\\mathcal{N}$ be a complete nest in $\\mathcal{H}$. In \\cite{chr}, Christensen proved that $H^{1}(Alg\\mathcal{N})=(0)$. Also $\\mathbb{B}(\\mathcal{H})$ is a von Neumann algebra. Hence  \\[ H^{1}(Alg\\mathcal{N}\\ltimes_{i} Alg\\mathcal{N})=(0) \\quad \\text{ and } \\quad H^{1}(Alg\\mathcal{N}\\ltimes_{i} \\mathbb{B}(\\mathcal{H}))=(0).\\]\n\\end{enumerate}\n\\end{exm}\n\\subsection*{Module extension Banach algebras}\nLet $\\A$ be a Banach algebra and $\\U$ be a Banach $\\A$-bimodule. With trivial product $\\U ^{2}=(0)$, as we saw in Example \\ref{me}, $\\A\\ltimes\\U$ is the same as module extension of $\\A$ by $\\U$, namely $T(\\A,\\U)$. In this case, $ann_{\\U}\\U =\\U$ and so for every derivation $\\delta:\\A\\rightarrow\\U$ we have $\\delta(\\A)\\subseteq ann_{\\U}\\U$. By these notes, Theorem \\ref{asll} and Corollary \\ref{tak}, we have the following proposition on derivations on $T(\\A,\\U)$.\n\\begin{prop}\\label{ttd}\n Let $D:T(\\A,\\U)\\rightarrow T(\\A,\\U)$ be a linear map such that \n \\[D((a,x))=(\\delta_1(a)+\\tau_1(x),\\delta_2(a)+\\tau_2(x))\\quad\\quad ((a,x)\\in \\A\\ltimes\\U).\\]\n The following are equivalent.\n \\begin{enumerate}\n \\item[(i)]\n $D$ is a derivation. \n \\item[(ii)]\n $D=D_1+D_2$ where $D_1((a,x))=(\\delta_1(a)+\\tau_1(x),\\tau_2(x))$ and $D_{2}((a,x))=(0,\\delta_2(a))$ are derivations on $T(\\A,\\U)$.\n \\item[(iii)]\n $\\delta_1:\\A\\rightarrow \\A$, $\\delta_2:\\A\\rightarrow \\U$ are derivations, $\\tau_{2}:\\U\\rightarrow \\U$ is a linear map such that \n \\[\\tau_2 (ax)=a\\tau_2 (x)+\\delta_2 (a)x\\quad \\text{and} \\quad \\tau_2 (xa)=\\tau_2 (x)a+x\\delta_2(a)\\quad \\quad (a\\in\\A,x\\in\\U).\\] \n and $\\tau_1:\\U\\rightarrow \\A$ is an $\\A$-bimodule homomorphism such that $x\\tau_1(y)+\\tau_1(x)y=0$ $ (x,y\\in\\U)$. \\\\\n Moreover, $D$ is an inner derivation if and only if $\\delta_1 ,\\delta_2$ are inner derivations, $\\tau_1 =0$ and if $\\delta_1 =ad_{a}$ and $ \\delta_2 =ad_{x}$, then $\\tau_2 =r_{a}$.\n \\end{enumerate}\n \\end{prop}\n  Proposition 2.2 of \\cite{med} is a consequence of this proposition. \n\\begin{rem} \\label{r0} \n By Proposition \\ref{ttd}, the linear map $\\delta:\\A\\rightarrow\\U$ is a derivation if and only if the linear map $D((a,x))=(0,\\delta (a))$ on $T(\\A,\\U)$ is a derivation. So any derivation $\\delta:\\A\\rightarrow\\U$ is continuous, if every derivation on $T(\\A,\\U)$ is continuous.\n \\end{rem} \n \\begin{prop}\\label{tau}\nSuppose that there are $\\A$-bimodule homomorphisms $\\phi:\\A\\rightarrow\\U$ and $\\psi:\\U\\rightarrow\\A$ such that $\\phi o\\psi = I_{\\U}$ ($I_\\U$ is the identity map on $\\U$). If every derivation on $T(\\A,\\U)$ is continuous, then every derivation on $\\A$ is continuous.\n\\end{prop}\n\\begin{proof}\nLet $\\delta$ be a derivation on $\\A$. Define the map $\\tau:\\U\\rightarrow\\U$ by $\\tau =\\phi o\\delta o\\psi$.\nThen for every $a\\in\\A,x\\in\\U$,\n\\[\\tau(ax)=a\\tau(x)+\\delta(a)x\\quad \\text{and} \\quad \\tau(xa)=\\tau(x)a+x\\delta(a).\\]\nSo the map $D:T(\\A,\\U)\\rightarrow T(\\A,\\U)$ defined by $D((a,x))=(\\delta(a),\\tau(x))$ is a derivation which is continuous by the hypothesis. Thus $\\delta$ is continuous.\n\\end{proof}\nIn the previous proposition the assumption of existence of $\\A$-bimodule homomorphisms $\\phi:\\A\\rightarrow\\U$ and $\\psi:\\U\\rightarrow\\A$ with $\\phi o \\psi =I_{\\U}$ is equivalent to that there exists a subbimodule $\\mathcal{V}$ of $\\A$ such that $\\A=\\U\\oplus \\mathcal{V}$ as $\\A$-bimodules direct sum. (In fact in this case, $\\U$ and $\\mathcal{V}$ are ideals of $\\A$). \n \\par \nSince for every derivation $\\delta:\\A\\rightarrow\\U$, $\\delta(\\A)\\subseteq ann_{\\U}\\U$, in the stated cases in Proposition \\ref{taj2}, one can express any derivation $D$ on $T(\\A,\\U)$ as the sum of two derivations one of which being continuous. Also the Proposition \\ref{au1}-$(i)$ holds in the case of module extension Banach algebras. \n\\begin{rem}\\label{r1}\nIf $\\A$ is a Banach algebra and $\\mathcal{I}$ is a closed ideal on it, then $\\frac{\\A}{\\mathcal{I}}$ is a Banach $\\A$-bimodule and so we can consider $T(\\A, \\frac{\\A}{\\mathcal{I}})$. Suppose that $\\A$ possesses a bounded right (or left) approximate identity and every derivation on $\\A$ and every derivation from $\\A$ to $\\frac{\\A}{\\mathcal{I}}$ is continuous. Let $D$ be a derivation on $T(\\A, \\frac{\\A}{\\mathcal{I}})$ which has a structure as in Proposition \\ref{ttd}. Then $\\tau_1:\\frac{\\A}{\\mathcal{I}}\\rightarrow \\A$ is an $\\A$-bimodule homomorphism. Since $\\A$ has a bounded right(left) approximate identity, then so does $\\frac{\\A}{\\mathcal{I}}$. Hence $\\tau_1$ is continuous. Now from Proposition \\ref{taj2}-$(ii)$ it follows that $D$ is continuous. Hence in this case any derivation on $T(\\A, \\frac{\\A}{\\mathcal{I}})$ is continuous.\n\\end{rem}\n\\begin{rem}\\label{r2}\nIf $\\mathcal{I}$ is a closed ideal in a Banach algebra $\\A$ and $\\delta:\\A\\rightarrow\\A$ is a derivation such that $\\delta(\\mathcal{I})\\subseteq \\mathcal{I}$, then the map $\\tau: \\frac{\\A}{\\mathcal{I}}\\rightarrow \\frac{\\A}{\\mathcal{I}}$ defined by $\\tau(a+\\mathcal{I})=\\delta(a)+\\mathcal{I}$ is well-defined and linear and \n\\[\\tau (a(x+\\mathcal{I}))=a\\tau (x+\\mathcal{I})+\\delta (a)(x+\\mathcal{I})\\quad,\\quad \\tau((x+\\mathcal{I})a)=\\tau(x+\\mathcal{I})a+(x+\\mathcal{I})\\delta(a).\\]\n Therefore the map $D$ on  $T(\\A, \\frac{\\A}{\\mathcal{I}})$ defined by $D((a,x))=(\\delta (a),\\tau(x+\\mathcal{I}))$ is a derivation. So if every derivation on  $T(\\A, \\frac{\\A}{\\mathcal{I}})$ is continuous, then every derivation $\\delta:\\A\\rightarrow \\A $ with $\\delta(\\mathcal{I})\\subseteq \\mathcal{I}$ is continuous.\n\\end{rem}\n In Remarks \\ref{r1} and \\ref{r2}, if we let $\\mathcal{I}=(0)$, then we have the following corollary.\n \\begin{cor}\n  Let $\\A$ be a Banach algebra.\n\\begin{enumerate}\n\\item[(i)] If $\\A$ has a right (left) approximate identity and every derivation on $\\A$ is continuous, then any derivation on $T(\\A, \\A)$ is continuous. \n\\item[(ii)] If every derivation on $T(\\A, \\A)$ is continuous, then any derivation on $\\A$ is continuous.\n\\end{enumerate} \n \\end{cor}\nThe part (ii) of this corollary could also be derived from Remark \\ref{r0} or Proposition \\ref{tau}.\n\\par \nIn continue we give some results of Proposition \\ref{taj2} in the case of module extension Banach algebras.\n\\begin{cor}\\label{trs}\nLet $\\A$ be a semisimple Banach algebra which has a bounded approximate identity and $\\U$ be a Banach $\\A$-bimodule with $ann_{\\A}\\U=(0)$. Suppose that there exists a surjective left $\\A$-module homomorphism $\\phi:\\A\\rightarrow\\U$ and every derivation from $\\A$ into $\\U$ is continuous, then every derivation on $T(\\A,\\U)$ is continuous.\n\\end{cor}\n\\begin{proof}\nLet $D$ be a derivation on $T(\\A,\\U)$ which has a structure as in Proposition \\ref{ttd}. Since $\\A$ is semisimple, every derivation from $\\A$ into $\\A$ is continuous. Now by Proposition \\ref{taj2}-$(ii)$ and Remark \\ref{taj11}, it follows that every derivation on $T(\\A,\\U)$ is continuous.\n\\end{proof}\nRingrose in \\cite{rin} proved that every derivation from a $C^{*}$-algebra into a Banach bimodule is continuous. So we have the following example which satisfies the conditions of the above corrolary.\n\\begin{exm}\nLet $\\A$ be a $C^{*}$-algebra and $\\U$ be a Banach $\\A$-bimodule with $ann_{\\A}\\U=(0)$. Suppose that there exists a surjective left $\\A$-module homomorphism $\\phi:\\A\\rightarrow\\U$. Hence $\\A$ is a semisimple Banach algebra with a bounded approximate identity. So by \\cite{rin} and Corrolary \\ref{trs}, any derivation on $T(\\A,\\U)$ is continuous. \n\\end{exm}\nAn element $p$ in an algebra $\\A$ is called an \\textit{idempotent} if $p^{2}=p$.\n\\begin{cor}\nLet $\\A$ be a prime Banach algebra with a non-trivial idempotent\n$p$ (i.e. $p \\neq 0$) such that $\\A p$ is finite dimensional. Then every derivation\non $\\A$ is continuous.\n\\end{cor}\n\\begin{proof}\nLet $\\U:=\\A p$. Then $\\U$ is a closed left ideal in $\\A$. By the following make $\\U$ into a Banach\n$\\A$-bimodule:\n\\[ xa=0 \\quad \\quad (x\\in \\A p, a\\in \\A), \\]\nand the left multiplication is the usual multiplication of $\\A$. So we can consider $T(\\A,\\U)$ in this case. Since $\\A$ is prime, it follows that $ann_{\\A}\\U=(0)$. Let $\\delta:\\A \\rightarrow \\A$ be a derivation. Define the map $\\tau:\\U \\rightarrow \\U$ by $\\tau(ap)=\\delta(ap)p$ $(a\\in \\A)$. The map $\\tau$ is well-defined and linear. Also\n\\[\\tau (ax)=a\\tau (x)+\\delta (a)x\\quad \\text{and} \\quad \\tau (xa)=\\tau (x)a+x\\delta(a)\\quad \\quad (a\\in\\A,x\\in\\U).\\] \nSince $\\U$ is finite dimensional, it follows that $\\tau$ is continuous. By Proposition \\ref{ttd}, the mapping $D:T(\\A,\\U)\\rightarrow T(\\A,\\U)$ defined by $D((a,x))=(\\delta(a),\\tau(x))$ is a derivation. Now for $D$ the conditions of Proposition \\ref{taj2}-$(i)$ hold and hence $D$ is continuous. Therefore $\\delta$ is continuous.\n\\end{proof}\n Now we investigate the first cohomology group of $T(\\A ,\\U)$. \n \\par \n In module extension $T(\\A ,\\U)$ since $\\U ^{2}=(0)$, for every derivation $\\delta:\\A\\rightarrow\\U$ on $T(\\A ,\\U)$ we have $\\delta (\\A)\\subseteq ann_{\\U}\\U$ and also $Z^{1}(\\U)=\\mathbb{B}(\\U)$ and $N^{1}(\\U)=(0)$, we may conclude the following proposition from Theorem \\ref{22}. \n \\begin{prop}\\label{cte}\nConsider the module extension $T(\\A,\\U)$ of a Banach algebra $\\A$ and a Banach $\\A$-bimodule $\\U$. Suppose that $H^{1}(\\A)=(0)$ and the only continuous $\\A$-bimodule homomorphism $T:\\U\\rightarrow\\A$ satisfying $T(x)y+xT(y)=0$ is $T=0$. Then \n\\[H^{1}(T(\\A ,\\U))\\cong H^{1}(\\A ,\\U)\\times \\frac{Hom_{\\A}(\\U)}{C_{\\A}(\\U)}.\\]\n \\end{prop} \nIn fact with the assumptions of the previous proposition, in Theorem \\ref{22} we have $\\mathcal{F}=N^{1}(\\A)\\times C_{\\A}(\\U)$. Proposition \\ref{cte} is the same as Theorem 2.5 of \\cite{med}. So it can be said that Theorem \\ref{22} is a generalization of Theorem 2.5 in \\cite{med}. \n\\begin{rem}\n \\begin{enumerate}\n \\item[(i)]\n If $\\U$ is a closed ideal of a Banach algebra $\\A$ such that $\\overline{\\U^{2}}=\\U$, then for any continuous $\\A$-bimodule homomorphism $T:\\U\\rightarrow\\A$ with  $T(x)y+xT(y)=0\\, (x,y\\in\\U)$ we have $T(xy)=0$. Since $\\overline{\\U^{2}}=\\U$ and $T$ is continuous, it follows that $T=0$. Hence in this case, for any $\\delta\\in Z^{1}(T(\\A ,\\U))$ in its representation, $\\tau_1=0$. Also in this case, for any $r_a \\in C_{\\A}(\\U)$, since $a\\in Z(\\A)$, then $r_{a}=0$. Therefore if $H^{1}(\\A)=(0)$, in this case by Proposition \\ref{22} we have \n \\[H^{1}(T(\\A ,\\U))\\cong H^{1}(\\A,\\U)\\times Hom_{\\A}(\\U).\\]\n \\item[(ii)]\n If Banach algebra $\\A$ has no nonzero nilpotent elements, $\\U$ is a Banach $\\A$-bimodule and $T:\\U\\rightarrow\\A$ is an $\\A$-bimodule homomorphism such that  $T(x)y+xT(y)=0\\, (x,y\\in\\U)$, then $2T(x)T(y)=0\\, (x,y\\in \\U)$. Thus for any $x\\in \\U$, $T(x)^{2}=0$ and by the hypothesis $T(x)=0$. So in this case for any derivation on $T(\\A ,\\U)$, in its structure by Proposition \\ref{ttd} we have $\\tau_1 =0$.\n\\end{enumerate}\n \\end{rem} \n \\begin{exm}\n If $\\A$ is a weakly amenable commutative Banach algebra and $\\U$ is a commutative Banach $\\A$-bimodule such that for any continuous $\\A$-bimodule homomorphism $T:\\U\\rightarrow\\A$ with  $T(x)y+xT(y)=0\\, (x,y\\in\\U)$ implies that $T=0$, then $H^{1}(\\A)=(0)$, $H^{1}(\\A,\\U)=(0)$ and $C_{\\A}(\\U)=(0)$ and thus \n \\[H^{1}(T(\\A ,\\U))\\cong  Hom_{\\A}(\\U).\\]\n \\end{exm}\nVarious examples of the trivial extension of Banach algebras and computing their first cohomology group are given in \\cite{med}. \n\\par \nIf $\\delta\\in Z^{1}(\\A,\\U)$, then the map $D_{\\delta}:T(\\A,\\U)\\rightarrow T(\\A,\\U)$ defined by $D_{\\delta}((a,x))=(0,\\delta(a))$ is a continuous derivation. Now we may define the linear map $\\Phi :Z^{1}(\\A,\\U)\\rightarrow H^{1}(T(\\A ,\\U))$ by $\\Phi (\\delta)=[D_{\\delta}]$. By noting that $\\delta$ is inner if and only if $D_{\\delta}$ is inner (by Corollary \\ref{tak}), it can be seen that $ker\\Phi= N^{1}(T(\\A ,\\U))$. Thus $H^{1}(\\A ,\\U)$ is isomorphic to some subspace of  $H^{1}(T(\\A ,\\U))$. So we have the following corollary. \n \\begin{cor}\n Let $\\A$ be a Banach algebra and $\\U$ be a Banach $\\A$-bimodule. Then there is a linear isomorphism from  \n $H^{1}(\\A ,\\U)$ onto a subspace of $H^{1}(T(\\A ,\\U))$.\n \\end{cor}\n By the above corollary from $H^{1}(T(\\A ,\\U))=(0)$ we conclude that $H^{1}(\\A ,\\U)=(0)$. In particular, if $H^{1}(T(\\A ,\\A))=(0)$, then $H^{1}(\\A)=(0)$. We can also obtain this result from the fact that any derivation $\\delta:\\A\\rightarrow\\A$ gives rise to a derivation $D:T(\\A,\\A)\\rightarrow T(\\A,\\A)$ given by $D((a,x))=(\\delta(a),\\delta(x))$ (by Remark \\ref{r2}). So if $D$ is inner, then $\\delta$ is inner.\n\n \\subsection*{$\\theta$-Lau products of Banach algebras}  \nIn this subsection we assume that $0\\neq \\theta\\in \\Delta (\\A)$ and $\\U$ is a Banach algebra. By the module action given in Example \\ref{la} we turn $\\U$ into a Banach $\\A$-bimodule with comaptible actions and norm and if it is necessary we show this module by $ \\U_{\\theta}$. Note that $ann_{\\A}\\U_{\\theta} =ker \\theta$. Consider $\\A\\ltimes\\U$ and denote it by $\\A\\ltimes_{\\theta}\\U$ which is called $\\theta$-Lau product. In the continuation of this section we always consider $\\A\\ltimes_{\\theta}\\U$ as just mentioned. \n\\par \nThe following proposition characterizes the structure of derivations on $\\A\\ltimes_{\\theta}\\U$, which is obtained from Theorem \\ref{asll}.\n \\begin{prop}\\label{lau-der}\nLet $D:\\A\\ltimes_{\\theta} \\U\\rightarrow \\A \\ltimes_{\\theta}\\U$ be a map. Then the following conditions are equivalent.\n\\begin{enumerate}\n\\item[(i)] $D$ is a derivation.\n\\item[(ii)] \n\\[D((a,x))=(\\delta_1 (a)+\\tau_1 (x),\\delta_2 (a)+\\tau_2 (x))\\quad\\quad (a\\in \\A,x\\in \\U)\\]\nsuch that \n\\begin{enumerate}\n\\item[(a)]\n$\\delta_1 :\\A\\rightarrow\\A, \\delta_2 :\\A\\rightarrow\\U$ are derivations such that \n\\[\\theta (\\delta_1 (a))x+\\delta_2(a)x=0\\quad \\text{and}\\quad \\theta (\\delta_1 (a))x+x\\delta_2(a)=0\\quad (a\\in\\A,x\\in\\U).\\]\n\\item[(b)]\n$\\tau_1:\\U\\rightarrow \\A$ is an $\\A$-bimodule homomorphism such that $\\tau_1(xy)=0\\quad (x,y\\in\\U)$.\n\\item[(c)]\n$\\tau_2:\\U\\rightarrow \\U$ is a linear map such that \n\\[\\tau_2(xy)=\\theta (\\tau_1(y))x+\\theta (\\tau_1(x))y+x\\tau_2(y)+\\tau_2(x)y\\quad \\quad (x,y\\in\\U).\\]\nAlso $D$ is inner if and only if $\\tau_1 =0, \\delta_2 =0, \\delta_1 =id_{a}$ and $\\tau_2 =id_{\\U,x}$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{prop}\nBy the above proposition, for a derivation $D$ on $\\A\\ltimes_{\\theta}\\U$ we have \n\\[\\delta_2 (\\A)\\subseteq Z(\\U), \\quad \\theta (a)\\tau_1 (x) =a\\tau_1(x)=\\tau_1(x)a\\]\nand so $\\tau_1(\\U)\\subseteq Z(\\A)$. Also $x\\tau_1 (y)+\\tau_1 (x)y =0$ for all $x,y\\in \\U$ if and only if $\\tau_1 (\\U)\\subseteq ker \\theta$. Additionally, $\\delta_1 (\\A)\\subseteq ann_{\\A}\\U =ker\\theta$ if and only if $\\delta_2 (\\A)\\subseteq ann_{\\U}\\U$.\n\\begin{rem} \\label{cla}\nIf $\\A$ is a commutative Banach algebra, then by Thomas' theorem \\cite{tho}, for any derivation $\\delta:\\A\\rightarrow\\A$, $\\delta(\\A)\\subseteq rad (\\A)$. So in this case for any derivation $D$ on $\\A\\ltimes_{\\theta}\\U$ we always have $\\delta_1 (\\A)\\subseteq rad (\\A) \\subseteq ker\\theta=ann_{\\A}\\U $ and hence $\\delta_2 (\\A)\\subseteq ann_{\\U}\\U$. Also $D=D_1+D_2+D_{3}$ where $D_1((a,x))=(\\delta_1(a),0)$, $D_{2}((a,x))=(0,\\delta_2(a))$ and $D_3((a,x))=(\\tau_1(x),\\tau_2(x))$ are derivations on $\\A\\ltimes_{\\theta}\\U$. In this case by Corollary \\ref{tak}-$(i)$, for every derivation $\\delta:\\A\\rightarrow\\A$ the map $D$ on $\\A\\ltimes_{\\theta}\\U$ defined by $D((a,x))=(\\delta(a),0)$ is a derivation.\n\\end{rem}\nIn the following we always assume that for any derivation $D\\in \\A\\ltimes_{\\theta}\\U$ we have $\\tau_1 =0$ and study the derivations under this condition. In this case, $\\tau_2$ is then always a derivation on $\\U$. \n\\par \nThe established results about the automatic continuity of derivations in section 3 obviously hold as well as in this special case $\\theta$-Lau product. Now we are ready to state some results concerning the automatic continuity of derivations on $\\A\\ltimes_{\\theta}\\U$.\n\\par \nBy the definition of $\\theta$-Lau product and Corollary \\ref{tak}-$(iv)$, any linear map $\\tau:\\U\\rightarrow\\U$ is a derivation if and only if the linear map $D((a,x))=(0,\\tau(x))$ on $\\A\\ltimes_{\\theta}\\U$ is a derivation. According to this point, the following corollary is clear.\n\\begin{cor}\nIf every derivation on $\\A\\ltimes_{\\theta}\\U$ is continuous, then every derivation on $\\U$ is continuous. In particular, if every derivation on $\\A\\ltimes_{\\theta}\\A$ is continuous, then every derivation on $\\A$ is continuous.\n\\end{cor}\nIf $\\A$ and $\\U$ are semisimple where $\\U$ has a bounded approximate identity, then by Proposition \\ref{semi} every derivatin on $\\A\\ltimes_{\\theta}\\U$ is continuous. In particular if $\\A$ is a semisimple Banach algebra with a bounded approximate identity, then all derivations on $\\A\\ltimes_{\\theta}\\A$ are continuous. In the case of $C^{*}$-algebras we can drop the semisimplicity of $\\U$. In fact by Ringrose's result \\cite{rin}, Proposition \\ref{lau-der} and preceding corollary we have the next corollary.\n\\begin{cor}\nLet $\\A$ be a $C^{*}$-algebra. Then every derivation on $\\A\\ltimes_{\\theta}\\U$ is continuous if and only if every derivation on $\\U$ is continuous.\n\\end{cor}\nFrom Proposition \\ref{taj1}-$(ii)$ and Remark \\ref{cla} we have the following corollary.\n\\begin{cor}\nLet $\\A$ be a commutative Banach algebra and $\\U$ a semisimple Banach algebra. Then every derivation $D$ on $ \\A\\ltimes_{\\theta}\\U $ is of the form $D=D_1+D_2$ where $D_1 ((a,x))=(0,\\delta_2 (a)+\\tau_2(x))$ is a continuous derivation on $\\A\\ltimes_{\\theta}\\U$ and $D_1 ((a,x))=(\\delta_1 (a),0)$ is a derivation on $\\A\\ltimes_{\\theta}\\U$. In particular, in this case, if every derivation on $\\A$ is continuous, then every derivation on $\\A\\ltimes_{\\theta}\\U$ is continuous.\n\\end{cor}\nNote that if $\\A$ is commutative, then $\\U$ is a commutative $\\A$-bimodule and hence the set of all left $\\A$-module homomorphisms, all right $\\A$-module homomorphisms and all $\\A$-module homomorphisms are the same. Now by Proposition \\ref{taj2} and Remark \\ref{cla} we obtain the next corollary.\n\\begin{cor}\nLet $\\A$ be a commutative Baanch algebra which has a bounded approximate identity. Then any derivation $D$ on $\\A\\ltimes_{\\theta}\\U$ is of the form $D=D_1+D_2$ where $D_1 ((a,x))=(\\delta_1 (a),\\tau_2(x))$ and $D_2 ((a,x))=(0,\\delta_2(a))$ are derivations on $\\A\\ltimes_{\\theta}\\U$ and under any of the following conditions, $D_1$ is continuous.\n\\begin{enumerate}\n\\item[(i)]\nThere is a surjective $\\A$-module homomorphism $\\phi:\\A\\rightarrow\\U$ and $\\delta_1$ is continuous.\n\\item[(ii)]\nThere is an injective $\\A$-module homomorphism $\\phi:\\A\\rightarrow\\U$ and $\\tau_2$ is continuous.\n\\end{enumerate}\n\\end{cor}\nIn the continuation we assume that for any continuous derivation $D$ on $\\A\\ltimes_{\\theta}\\U$ we have $\\tau_1 =0$. By the definition of $\\A\\ltimes_{\\theta}\\U$, in this case, $Hom_{\\A}(\\U)=\\mathbb{B}(\\U), N^{1}(\\A,\\U)=(0), Z^{1}(\\A,\\U)=H^{1}(\\A,\\U), R_{\\A}(\\U)=C_{\\A}(\\U)=(0)$ and $N^{1}(\\U)=I(\\U)$. \n\\begin{rem}\\label{pri}\nFor any derivation $\\delta\\in Z^{1}(\\A)$, we have $\\delta(\\A)\\subseteq ker \\theta=ann_{\\A}\\U $, since Sinclair's theorem \\cite{sinc} implies that $\\delta(P)\\subseteq P$ for any primitive ideal $P$ of $\\A$. So in this case for any derivation $D\\in Z^{1}(\\A\\ltimes_{\\theta}\\U)$ we always have $\\delta_1 (\\A)\\subseteq ker \\theta =ann_{\\A}\\U $ and hence $\\delta_2 (\\A)\\subseteq ann_{\\U}\\U$. Also $D=D_1+D_2+D_{3}$ where $D_1((a,x))=(\\delta_1(a),0)$, $D_{2}((a,x))=(0,\\delta_2(a))$ and $D_3((a,x))=(\\tau_1(x),\\tau_2(x))$ are in $Z^{1}(\\A\\ltimes_{\\theta}\\U)$. \n\\end{rem}\nNote that it is not necessarily true that $\\delta\\in Z^{1}(\\A,\\U)$ implies $\\delta(\\A)\\subseteq ann_{\\U}\\U$ as the next example shows this.\n\\begin{exm}\nAssume that $G$ is a non-discrete abelian group. In \\cite{br}, it has been shown that there is a nonzero continuous point derivation $d$ at a nonzero character $\\theta$ on $M(G)$. Now consider $M(G)\\ltimes_{\\theta} \\mathbb{C}$. Every derivation from $M(G)$ into $\\mathbb{C}_\\theta$ is a point derivation at $\\theta$. It is clear that $ann_{\\mathbb{C}}\\mathbb{C} =(0)$. But $d\\in Z^{1}(M(G),\\mathbb{C}_\\theta)$ is a nonzero derivation such that $d(M(G))\\not\\subseteq ann_{\\mathbb{C}}\\mathbb{C} =(0)$.\n\\end{exm}\n\\par \nNow we determine the first cohomology group of $\\A\\ltimes_{\\theta}\\U$ in some other cases.\n\\par \nLinear space $\\mathcal{E}$ in Theorem \\ref{11} in the case of $\\A\\ltimes_{\\theta}\\U$ is $\\mathcal{E}=N^{1}(\\A)\\times N^{1}(\\U)$. Therefore by theorem \\ref{11} and Remark \\ref{pri} we have the next proposition.\n\\begin{prop}\\label{a1}\nIf $H^{1}(\\A,\\U)=(0)$, then \n\\[H^{1}(\\A\\ltimes_{\\theta}\\U)\\cong H^{1}(\\A)\\times H^{1}(\\U).\\]\n\\end{prop}\n\\begin{cor}\nLet $\\A$ be a Banach algebra with $H^{1}(\\A,\\A_{\\theta})=(0)$ and $\\overline{\\A^{2}}=\\A$. Then $H^{1}(\\A \\ltimes_{\\theta}\\A)=(0)$ if and only if $H^{1}(\\A)=(0)$.\n\\end{cor}\n\\par \nLinear space $\\mathcal{F}$ in Theorem \\ref{22} in the case of $\\A\\ltimes_{\\theta}\\U$ is $\\mathcal{F}=(0)\\times N^{1}(\\U)$. So we have the following proposition.\n\\begin{prop}\nLet for any $\\delta\\in Z^{1}(\\A,\\U)$, $\\delta(\\A)\\subseteq ann_{\\U}\\U$. If $H^{1}(\\A)=(0)$, then \n\\[H^{1}(\\A\\ltimes_{\\theta}\\U)\\cong H^{1}(\\A,\\U)\\times H^{1}(\\U).\\]\n\\end{prop}\n The linear space $\\mathcal{K}$ in Theorem \\ref{33} in the case of $\\A\\ltimes_{\\theta}\\U$ is $\\mathcal{K}=N^{1}(\\A)\\times (0)$. Therefore by Theorem \\ref{33} we have the next proposition.\n \\begin{prop}\\label{prop8}\nLet for every $\\delta\\in Z^{1}(\\A,\\U)$, $ \\delta(\\A)\\subseteq ann _{\\U}\\U$.\n If $H^{1}(\\U)=(0)$, then \n \\[H^{1}(\\A\\ltimes_{\\theta}\\U)\\cong H^{1}(\\A)\\times H^{1}(\\A,\\U).\\]\n\\end{prop}\nThe following proposition follows directly from Theorem \\ref{44} and the properties of $\\A\\ltimes_{\\theta}\\U$.\n\\begin{prop}\\label{prop10}\nSuppose that $H^{1}(\\A)=(0)$ and $H^{1}(\\A,\\U)=(0)$. Then\n\\[H^{1}(\\A\\ltimes_{\\theta}\\U)\\cong H^{1}(\\U).\\] \n\\end{prop}\n\\begin{rem}\\label{akh}\nFrom Proposition \\ref{prop10} it follows that if $H^{1}(\\A)=(0), H^{1}(\\A,\\U)=(0)$ and $H^{1}(\\U)=(0)$, then $H^{1}(\\A\\ltimes_{\\theta}\\U)=(0)$. Under some conditions the converse of is also true. That is, if for any $\\delta\\in Z^{1}(\\A,\\U)$ we have $\\delta (\\A)\\subseteq ann_{\\U}\\U$ and $H^{1}(\\A\\ltimes_{\\theta}\\U)=(0)$, then $H^{1}(\\A)=(0), H^{1}(\\A,\\U)=(0)$ and $H^{1}(\\U)=(0)$. It does not yield from above propositions. We prove it directly as follows.  By the hypotheses, for any $\\delta_1\\in Z^{1}(\\A)$, $\\delta_2\\in Z^{1}(\\A,\\U)$ and $\\tau\\in Z^{1}(\\U)$ the map $D:\\A\\ltimes_{\\theta}\\U\\rightarrow \\A\\ltimes_{\\theta}\\U$ given by $D((a,x))=(\\delta_1(a),0)$, $D((a,x))=(0,\\delta_2(a))$ or $D((a,x))=(0,\\tau_2(x))$ is a continuous derivation. If $H^{1}(\\A\\ltimes_{\\theta}\\U)=(0)$, then $\\delta_2=0$ and $\\delta_1, \\tau_2$ are inner.\n\\end{rem}\n \\begin{exm}\nLet $\\A$ be a weakly amenable commutative Banach algebra and $\\overline{\\U^{2}}=\\U$. So $Z^{1}(\\A)=(0)$, $Z^{1}(\\A,\\U)=(0)$ and from Remark \\ref{akh} we have $H^{1}(\\A\\ltimes_{\\theta}\\U)=(0)$ if and only if $H^{1}(\\U)=(0)$.\n \\end{exm}\n In particular if $\\A$ is a weakly amenable commutative Banach algebra, above example implies that $H^{1}(\\A\\ltimes_{\\theta}\\A)=(0)$. Note that for a weakly amenable Banach algebra $\\A$ the equality $\\overline{\\A^{2}}=\\A$ always holds.\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n \\label{Sec_intro}\n\n  In this paper we present results on the longitudinal double spin\nasymmetry $A_1^{\\rho }$ for exclusive incoherent  $\\rho ^0$ production in the\nscattering of high energy muons on nucleons.\nThe experiment was carried out at CERN by the COMPASS collaboration\nusing the 160~GeV muon beam and the large $^{6}$LiD polarised target.\n\nThe studied reaction is \n\\begin{equation} \n \\mu  +  N  \\rightarrow  \\mu^{\\prime}  +  \\rho ^0  +  N^{\\prime},       \\label{murho}\n\\end{equation}\nwhere $N$ is a quasi-free nucleon from the polarised deuterons.\nThe reaction (\\ref{murho}) can be described in terms of the virtual photoproduction\nprocess\n\\begin{equation}\n \\gamma^{\\ast}  +  N  \\rightarrow  \\rho^0  +  N^{\\prime}.   \\label{phorho} \n\\end{equation}\nThe reaction (\\ref{phorho}) can be regarded as a fluctuation of the\nvirtual photon into a quark-antiquark pair (in partonic language),\nor an off-shell vector meson (in Vector Meson Dominance model), which then scatters\noff the target nucleon resulting in the production of an on-shell vector meson.\nAt high energies this is predominantly a diffractive process and plays\nan important role in the investigation of Pomeron exchange and its\ninterpretation in terms of multiple gluon exchange.\n\nMost of the presently available information on the spin structure \nof reaction (\\ref{phorho})\nstems from the $\\rho ^0$ spin density matrix elements,\nwhich are obtained from the analysis of angular distributions\nof  $\\rho ^0$ production and decay \\cite{Schilling73}. Experimental results\non $\\rho ^0$ spin density matrix elements \ncome from various experiments \\cite{NMC,E665-1,ZEUS,H1,HERMES00} including\nthe preliminary results from COMPASS \\cite{sandacz}.\n \nThe emerging picture of the spin structure of the considered process \nis the following. At low photon virtuality $Q^2$ the cross section by transverse virtual photons\n$\\sigma _T$\ndominates, while the relative contribution of the cross section by longitudinal\nphotons $\\sigma _L$ \nrapidly increases with $Q^2$. At $Q^2$ of about 2~(GeV\/{\\it c})$^2$ both\ncomponents become comparable and at a larger $Q^2$ the contribution of\n$\\sigma _L$ becomes dominant and continues to grow, although\nat lower rate than at low $Q^2$. \nApproximately, the so called $s$-channel helicity\nconservation (SCHC) is valid, i.e.\\ the helicity of the vector meson is the same\nas the helicity of the parent virtual photon. The data indicate that \nthe process can be described approximately by the \nexchange in the $t$-channel of an object with natural \nparity $P$.\nSmall deviations from SCHC are observed, also at the highest energies, whose\norigin is still to be understood. \nAn interesting suggestion was made in Ref.~\\cite{igivanov} that at high energies\nthe magnitudes of various helicity amplitudes for the reaction (\\ref{phorho})\nmay shed a light on the spin-orbital momentum structure of the vector meson. \n\nA complementary information can be obtained from measurements of the double\nspin cross section asymmetry, when the information on both the beam\nand target polarisation is used.\nThe asymmetry is defined as\n\\begin{equation}\n A_{1}^{\\rho}  = \n   \\frac{\n    \\sigma_{1\/2}  -  \\sigma_{3\/2}\n   }{\n    \\sigma_{1\/2}  +  \\sigma_{3\/2}}\n   ,\n   \\label{A1def}\n\\end{equation}\nwhere $\\sigma_{1\/2  (3\/2)}$ stands for the cross sections of the reaction (\\ref{phorho})\nand the subscripts denote the total virtual photon--nucleon angular momentum\ncomponent along the virtual photon\ndirection.\nIn the following we will also use the asymmetry\n\\ALL\\\nwhich is defined for reaction (\\ref{murho}) as the asymmetry of\nmuon--nucleon cross sections for antiparallel and parallel beam and\ntarget longitudinal spin orientations.\n\n\nIn the Regge approach \\cite{manaenkov} the longitudinal double spin asymmetry\n$A_1^{\\rho }$ can arise due \nto the interference of amplitudes for exchange in the $t$-channel of Reggeons with natural parity\n(Pomeron, $\\rho $, $\\omega $, $f$, $A_2$ ) with amplitudes for Reggeons with \nunnatural parity \n($\\pi , A_1$).\nNo significant asymmetry is expected \nwhen only a non-perturbative \nPomeron is exchanged because it has small spin-dependent couplings as found from \nhadron-nucleon data for cross sections and polarisations.\n\nSimilarly, in the approach of Fraas \\cite{fraas76}, \nassuming approximate validity of \nSCHC, the spin asymmetry $A_1^{\\rho }$ arises from the interference between\nparts of the helicity amplitudes for transverse photons corresponding\nto the natural and unnatural parity exchanges\nin the $t$ channel. While a measurable asymmetry can arise even from a small\ncontribution of the unnatural parity exchange, the latter may remain\nunmeasurable in the cross sections.\nA significant unnatural-parity contribution may indicate an exchange \nof certain Reggeons like $\\pi$, $A_{1}$ or in\npartonic terms an exchange of $q\\bar{q}$ pairs.\n\nIn the same reference a theoretical prediction for $A_1^{\\rho}$ was\npresented, which is based on the description of forward exclusive $\\rho^{0}$\nleptoproduction and inclusive inelastic lepton-nucleon\nscattering by the off-diagonal Generalised Vector Meson Dominance (GVMD) model,\napplied to the case of polarised lepton--nucleon scattering.\nAt the values of Bjorken variable $x < 0.2$, with additional assumptions\n\\cite{HERMES01}, \\Aor\\ can be related \nto the $A_{1}$ asymmetry for inclusive inelastic lepton scattering at the same\n\\xBj\\ as\n\\begin{equation}\n A_1^{\\rho }  =  \\frac{2 A_1}{1  +  (A_1)^2} .    \\label{A1-Fraas}\n\\end{equation} \nThis prediction is consistent with the HERMES results for both the proton\nand deuteron targets, although with rather large errors.\n\nIn perturbative QCD, there exists a general proof of factorisation \\cite{fact}\nfor exclusive vector meson production by longitudinal photons. \nIt allows a decomposition of the full amplitude for reaction (\\ref{phorho})\ninto three components:\na hard scattering amplitude for the exchange of quarks or gluons,\na distribution amplitude for the meson and \nthe non-perturbative description of the target nucleon in terms of\nthe generalised parton distributions (GPDs), which are related to the internal\nstructure\nof the nucleon.\nNo similar proof of factorisation exists for transverse virtual photons, and as a consequence\nthe interpretation of $A_{1}^{\\rho}$ in perturbative QCD is not possible\nat leading twist. However, a model including higher twist effects proposed\nby Martin et al. \\cite{mrt} describes the behaviour of both $\\sigma_{L}$\nas well as of $\\sigma _T$ reasonably well. An extension of this model by Ryskin\n\\cite{Ryskin} for the\nspin dependent cross sections allows to relate\n\\Aor\\ to the spin dependent GPDs of gluons and quarks in the nucleon.\nThe applicability of this model is limited to the range $Q^2 \\geq 4~(\\mathrm{GeV}\/c)^2$.\nMore recently another pQCD-inspired model involving GPDs has been proposed by Goloskokov and Kroll \n\\cite{krgo,gokr}. The non-leading twist asymmetry $A_{LL}$ results from the \ninterference between the dominant GPD $H_g$ and the helicity-dependent GPD\n$\\tilde{H}_g$. The asymmetry is estimated to be of the order \n$ k_T^2  \\tilde{H}_g \/ (Q^2  H_g  )$,\nwhere $k_T$ is the transverse momentum of the quark and the antiquark.\n\nUp to now little experimental information has been available on the\ndouble spin asymmetries for exclusive leptoproduction of vector mesons. \nThe first observation of a non-zero asymmetry $A_1^{\\rho }$ in polarised\nelectron--proton deep-inelastic scattering was reported by the HERMES experiment \n\\cite{HERMES01}.\nIn the deep inelastic region $(0.8 < Q^2 < 3~(\\mathrm{GeV}\/c)^2)$\nthe measured asymmetry is equal to 0.23 $\\pm$ 0.14 (stat) $\\pm$ 0.02 (syst)\n\\cite{HERMES03},\nwith little dependence on the kinematical variables. In contrast, for the `quasi-real photoproduction' data, with \n$\\langle Q^2\\rangle = 0.13~(\\mathrm{GeV}\/c)^2$, the asymmetry for the proton target is consistent with zero.\nOn the other hand the measured asymmetry $A_1^{\\rho }$ for the polarised deuteron target and the asymmetry $A_1^{\\phi }$ for exclusive production of $\\phi $\nmeson \neither on polarised protons or deuterons\n are consistent with zero both in the deep inelastic and in the \nquasi-real photoproduction regions\n \\cite{HERMES03}.\n\nThe HERMES result indicating a non-zero $A_1^{\\rho }$ for the proton\ntarget differs from the unpublished result of similar measurements by \nthe SMC experiment \\cite{ATrip-1} at comparable values of $Q^2$\\ but at\nabout three times higher values of the photon-nucleon centre of mass energy $W$, i.e.\\ at smaller \\xBj. The SMC measurements of \n\\ALL\\ \nin several bins of $Q^2$ are consistent with zero\nfor both proton and deuteron targets. \n\n\\section{The experimental set-up}\n \\label{Sec_exper}\n\nThe experiment \\cite{setup} was performed with\nthe high intensity positive muon beam from the CERN M2 beam line.\nThe $\\mu ^{+}$ beam intensity is $2 \\cdot 10^8$ per spill of 4.8 s with a cycle time of\n16.8~s. The average beam energy is 160~GeV and the momentum spread is $\\sigma _{p}\/p = 0.05$.\nThe momentum of each beam muon is measured upstream of the experimental area in a beam\nmomentum station consisting of several planes of scintillator strips or scintillating fibres\nwith a dipole magnet in between. The precision of the momentum determination is typically\n$\\Delta p\/p \\leq 0.003$. The $\\mu ^{+}$ beam is naturally polarised by the weak decays \nof the parent\nhadrons. The polarisation of the muon varies with its energy and the \naverage polarisation is $-0.76$.\n\nThe beam traverses the two cells of the polarised target, \neach 60~cm long, 3~cm in diameter and separated by 10~cm, which are placed one after the other. \nThe target cells are filled with $^{6}$LiD which is used as polarised deuteron target material\nand is longitudinally polarised by dynamic nuclear polarisation (DNP).\nThe two cells are polarised in opposite directions so that data from both spin directions\nare recorded at the same time. The typical values of polarisation are about 0.50.\nA mixture of liquid $^3$He\nand $^4$He, used to refrigerate the target, and a small amount of heavier nuclei are also\npresent in the target.  \nThe spin directions in the two target cells are reversed every 8 hours by rotating\nthe direction of the magnetic field in the target. \nIn this way fluxes and acceptances cancel in the calculation of spin asymmetries, provided that\nthe ratio of acceptances of the two cells remains unchanged after the reversal. \n\nThe COMPASS spectrometer is designed to reconstruct the scattered muons\nand the produced hadrons in wide momentum and angular ranges.\nIt is divided in two stages with two dipole magnets, SM1 and SM2. The first magnet, \nSM1, accepts charged particles of momenta larger than 0.4~GeV\/{\\it c}, and the second one, SM2, those\nlarger than 4~GeV\/{\\it c}. The angular acceptance of the spectrometer is limited by\nthe aperture of the polarised target magnet. For the upstream\nend of the target it is $\\pm 70$~mrad.\n\nTo match the expected particle flux at various locations in the spectrometer, COMPASS\nuses various tracking detectors. Small-angle tracking is provided by stations of\nscintillating fibres, silicon detectors, micromesh gaseous chambers and gas electron\nmultiplier chambers. Large-angle tracking devices are multiwire proportional chambers,\ndrift chambers and straw detectors. Muons are identified in large-area mini drift \ntubes and drift tubes placed downstream of hadron absorbers. Hadrons are detected by\ntwo large iron-scintillator sampling calorimeters installed in front of the absorbers\nand shielded to avoid electromagnetic contamination.\nThe identification of charged particles is possible with a RICH detector, although\nin this paper we have not utilised the information from the RICH.\n\nThe data recording system is activated by various triggers indicating the presence\nof a scattered muon and\/or an energy deposited by hadrons\nin the calorimeters. In addition to the inclusive trigger, in which the scattered muon\nis identified by coincidence signals in the trigger hodoscopes, several semi-inclusive\ntriggers were used. They select events fulfilling the requirement to detect\nthe scattered muon together with the energy deposited in the hadron calorimeters exceeding\na given threshold. In 2003 the acceptance was further extended towards high $Q^2$\nvalues by the addition of a standalone calorimetric trigger in which no condition \nis set for the scattered muon. \nThe COMPASS trigger system allows us to cover a wide range of  $Q^2$, from quasi-real\nphotoproduction to deep inelastic interactions.\n \nA more detailed description of the COMPASS apparatus can be found in Ref.~\\cite{setup}\n\n\\section{Event sample}\n \\label{Sec_sample}\n\nFor the present analysis the whole data sample taken in 2002 and 2003 with the \nlongitudinally polarised target is used. For an event to be accepted for further analysis it is required to originate in the target, have a reconstructed \nbeam track, a\nscattered  muon track, and only two additional tracks \nof oppositely charged hadrons associated to the primary vertex. \nThe fluxes of beam muons passing through each target cell are equalised using\nappropriate cuts on the position and angle of the beam tracks.\n\nThe charged\npion mass hypothesis is assigned to each hadron track and the invariant mass of two\npions, $m_{\\pi \\pi}$, calculated. A cut on the invariant mass of two pions, $0.5 < m_{\\pi \\pi} < 1~\\mathrm{GeV}\/c^2$, is applied to select\nthe $\\rho ^0$.\nAs slow recoil target particles are not detected, in order to select\nexclusive events we use the cut on the missing energy, $-2.5 < E_{miss} < 2.5~\\mathrm{GeV}$, and on the transverse momentum of $\\rho^0$ with respect to the direction of\nvirtual photon, $p_t^2 < 0.5~(\\mathrm{GeV}\/c)^2$. Here $E_{miss} = (M^{2}_{X} - M^{2}_{p})\/2M_{p}$, where $M_X$ is the missing mass of the unobserved recoiling\nsystem and $M_p$ is the proton mass.\nCoherent interactions on the\ntarget nuclei are removed by a cut $p_t^2 > 0.15~(\\rm{GeV}\/c)^2$. \nTo avoid large corrections for acceptance and misidentification of\nevents, additional cuts \n$\\nu > 30~\\mathrm{GeV}$ and $E_{\\mu'} > 20~\\mathrm{GeV}$ are applied.\n\n\nThe distributions of $m_{\\pi \\pi}$, $E_{miss}$ and $p_t^2$ are shown in \nFig.~\\ref{hadplots}. Each plot is obtained applying all cuts except\nthose corresponding to the displayed variable. \nOn the left top panel of Fig.~\\ref{hadplots} a clear peak of the \\rn\\ resonance, centred at 770~MeV\/$c^2$, is visible on the top of the small contribution of\nbackground of the non-resonant \\pip\\pin\\ pairs. Also the skewing of the resonance peak towards smaller values\nof \\mpipi, due to an interference with the non-resonant background, is noticeable. A small bump below 0.4~GeV\/$c^2$\nis due to assignment of the charged pion mass to the kaons from decays of $\\phi$ mesons.\nThe mass cuts eliminate the non-resonant background outside of the \\rn\\ peak, as well as the contribution of $\\phi$\nmesons.\n\n\\begin{figure}[t]\n \\begin{center}\n \\epsfig{figure=mpipi_0203.eps,height=5cm,width=7.5cm}\n \\epsfig{figure=Emiss_0203.eps,height=5cm,width=7.5cm}\n \\epsfig{figure=ptsq_0203.eps,height=5cm,width=7.5cm}\n \\caption{Distributions of \\mpipi\\ (top left), \\Emi\\ (top right) \nand \\pts\\ (bottom) for the exclusive sample. \nThe arrows show cuts imposed on each variable to define the final sample.}\n \\label{hadplots}\n \\end{center}\n\\end{figure}\n \nOn the right top panel of the figure the peak at $E_{miss} \\approx 0$ is the signal of \nexclusive \\rn\\ production. The width of the peak, $\\sigma \\approx 1.1~\\mathrm{GeV}$,\nis due to the spectrometer resolution. Non-exclusive events, where in addition to the recoil nucleon other undetected hadrons are produced, appear at $E_{miss} > 0$.\nDue to the finite resolution, however, they are not resolved from the exclusive peak.\nThis background\nconsists of two components: the double-diffractive events where additionally to \\rn\\ an excited nucleon state is produced in the\nnucleon vertex of reaction (\\ref{phorho}), and events with semi-inclusive \\rn\\ production, in which other\nhadrons are produced but escape detection. \n\nThe $p_t^2$ distribution shown on the bottom panel of the figure\nindicates a contribution from coherent production on target \nnuclei at small $p_t^2$ values. A three-exponential fit to this distribution \nwas performed, which indicates also \na contribution of non-exclusive background increasing\nwith $p_t^2$.\nTherefore to select the sample of exclusive incoherent \\rn\\ production, \nthe aforementioned $p_t^2$ cuts, indicated by arrows, were applied. \n\nAfter all selections the final sample consists of about 2.44 million events.\nThe distributions of $Q^2$, \\xBj\\ and $W$ are shown in Fig.~\\ref{kinplots}.\nThe data cover a wide range in $Q^2$\\ and \\xBj\\ which extends towards the small values \nby almost two orders of magnitude compared to the similar studies reported in \nRef.~\\cite{HERMES03}. The sharp edge of the $W$ distribution at the low $W$ values\nis a consequence of the cut applied on $\\nu $. For this sample \n$\\langle W \\rangle$ is equal to 10.2~GeV and $\\langle p_{t}^{2}\n\\rangle = 0.27(\\mathrm{GeV}\/c)^2$.\n\n\\begin{figure}[t]\n \\begin{center}\n \\epsfig{figure=Qsq_0203_incoh_liny.eps,height=5cm,width=7.5cm}\n \\epsfig{figure=Qsq_0203_incoh_logy.eps,height=5cm,width=7.5cm}\n \\epsfig{figure=xBj_0203_incoh.eps,height=5cm,width=7.5cm}\n \\epsfig{figure=Wene_0203_incoh.eps,height=5cm,width=7.5cm}\n \n \\caption{Distributions of the kinematical variables for the final sample: $Q^2$\\ with linear and\n           logarithmic vertical axis scale (top left and right panels respectively), \\xBj\\ (bottom left), and the energy\n           $W$ (bottom right).}\n \\label{kinplots}\n \\end{center}\n\\end{figure}\n\n\\section{Extraction of asymmetry \\Aor}\n \\label{Sec_extract} \n\nThe cross section asymmetry $A_{LL}  =  \n(\\sigma_{\\uparrow \\downarrow}  -  \\sigma_{\\uparrow \\uparrow})\/\n(\\sigma_{\\uparrow \\downarrow}  +  \\sigma_{\\uparrow \\uparrow})$ for reaction\n(\\ref{murho}) , for antiparallel ($\\uparrow \\downarrow $) and parallel \n($\\uparrow \\uparrow $) spins of\nthe incoming muon and the target nucleon, is related to the virtual-photon\nnucleon asymmetry $A_{1}^{\\rho}$ by\n\\begin{equation}\n A_{LL}  =  D  \\left( A_{1}^{\\rho}  +  \\eta  A_{2}^{\\rho}  \\right) ,      \\label{A1rALL}\n\\end{equation}\nwhere the factors $D$ and $\\eta $ depend on the event kinematics and \n$A_{2}^{\\rho}$ is related to\nthe interference cross section for exclusive production by \nlongitudinal and transverse virtual photons. \nAs the presented results extend into the range of very small $Q^2$, \nthe exact formulae for the depolarisation factor $D$ and kinematical \nfactor $\\eta$ \\cite{JKir}\nare used without neglecting terms proportional to the lepton mass squared $m^2$.\nThe depolarisation factor is given by\n\\begin{equation}\n D(y, Q^{2}) = \n   \\frac{\n    y \\left[ (1 + \\gamma^{2} y\/2) (2  -  y) - \n    2 y^{2} m^{2} \/ Q^{2}  \\right]\n   }{\n    y^{2}  (1  -  2 m^{2} \/ Q^{2})  (1  +  \\gamma^{2})  + \n    2  (1  +  R)  (1  -  y  -  \\gamma^{2} y^{2}\/4)\n   },    \\label{depf}\n\\end{equation}\nwhere $R  =  \\sigma_{L} \/ \\sigma_{T}$, $\\sigma_{L(T)}$ is the cross section for reaction (\\ref{phorho})\ninitiated by longitudinally (transversely) polarised virtual photons, the fraction of the muon energy lost $y = \\nu \/E_{\\mu }$ and $\\gamma^{2}  =  Q^{2} \/ \\nu^{2}$.\nThe kinematical factor $\\eta (y, Q^{2})$  is the same as for the inclusive \nasymmetry.\n\nThe asymmetry $A_{2}^{\\rho}$  obeys the positivity limit $A_{2}^{\\rho} < \\sqrt{R}$, analogous to the one for the inclusive case.\nFor $Q^2 \\leq  0.1~(\\mathrm{GeV}\/c)^2$ the ratio $R$ for the reaction (\\ref{phorho}) is\nsmall, cf.\\ Fig.~\\ref{R-pict}, and the positivity limit constrains $A_{2}^{\\rho}$ to small values. Although for larger $Q^2$\\ the ratio $R$ for the process (\\ref{phorho}) increases\nwith $Q^2$, because of small values of $\\eta $ the product $\\eta \\sqrt{R}$ is small in the  whole $Q^2$\\ range of our sample.\nTherefore the second term in Eq.~\\ref{A1rALL} can be neglected, so that\n\\begin{equation}\n A_{1}^{\\rho} \\simeq \\frac{1}{D}  A_{LL},\n\\end{equation}\nand the effect of this approximation is included in the systematic uncertainty of \\Aor.\n\\begin{figure}[thb]\n\\begin{center}\n  \\epsfig{file=RLT.eps,height=5cm,width=7.4cm}\n  \\caption{The ratio $R= \\sigma_L\/\\sigma_T$ as a function of $Q^2$\\ measured in the E665 experiment.\n           The curve is a fit to the data described in the text.}\n  \\label{R-pict}\n \\end{center}\n\\end{figure}\n\nThe number of events $N_i$ collected from a given target cell in a given time interval\nis related to the spin-independent cross section $\\bar{\\sigma}$ for reaction (\\ref{phorho})\nand to the asymmetry $A_1^{\\rho}$ by\n\\begin{equation}\nN_i =  a_i \\phi _i n_i \\bar{\\sigma } (1+P_B P_T f D A_1^{\\rho }  ),\n\\label{nevents}\n\\end{equation}\nwhere $P_B$ and $P_T$ are the beam and target polarisations, $\\phi _i$ is the incoming\nmuon flux, $a_i$ the acceptance for the target cell, $n_i$ corresponding number of target nucleons, and $f$ the target dilution factor.\nThe asymmetry is extracted from the data sets taken before and after a reversal of the\ntarget spin directions. The four relations of Eq.~\\ref{nevents}, corresponding to the two \ncells ($u$ and $d$) and the two spin orientations (1 and 2) lead to a second-order\nequation in \\Aor\\ for the ratio $(N_{u,1}N_{d,2}\/N_{d,1}N_{u,2})$. Here fluxes cancel out\nas well as acceptances, if the ratio of acceptances for the two cells is the same before\nand after the reversal \\cite{SMClong}. In order to minimise the statistical error\nall quantities used in the asymmetry calculation are evaluated event by event with the\nweight factor $w=P_BfD$. The polarisation of the beam muon, $P_B$, is obtained from\na simulation of the beam line and parameterised as a function of the beam momentum. The target polarisation is not \nincluded in the event weight $w$ because it may vary in time and generate false\nasymmetries. An average $P_T$ is used for each target cell and each spin orientation.\n\n The ratio $R$, which enters the formula for $D$ and strongly depends on $Q^2$\\ for reaction (\\ref{phorho}), was calculated on an event-by-event basis\nusing the parameterisation \n\\begin{equation}\n R(Q^{2})  = \n   a_{0}  (Q^{2})^{a_{1}} ,\n\\end{equation}\nwith $a_{0} = 0.66 \\pm 0.05$, and $a_{1} = 0.61 \\pm 0.09$.\nThe parameterisation was obtained by the Fermilab E665 experiment from a fit\nto their $R$ measurements for exclusive $\\rho ^0$ muoproduction on protons \\cite{E665-1}.\nThese are shown in Fig.~\\ref{R-pict} together with the fitted $Q^2$-dependence.\nThe preliminary COMPASS results on $R$ for the incoherent exclusive $\\rho ^0$\nproduction on the nucleon \\cite{sandacz}, which cover a broader kinematic region in $Q^2$\n, agree reasonably well with this parameterisation.\nThe uncertainty of $a_{0}$ and $a_{1}$ is included in the \nsystematic error of $A_{1}^{\\rho}$.\n\nThe dilution factor $f$ gives the fraction of events of reaction (\\ref{phorho}) \noriginating from nucleons in polarised deuterons inside the target material.\nIt is calculated event-by-event using the formula\n\\begin{equation}\n f  =  C_1  \\cdot f_{0}  =  C_1  \\cdot \n   \\frac{\n    n_{\\rm D}\n   }{\n    n_{\\rm D} +  \\Sigma_{A}  n_{\\rm A} \n    (\\tilde{\\sigma}_{\\rm A} \/ \\tilde{\\sigma}_{\\rm D})\n   }.   \n   \\vspace*{3mm}\n\\end{equation}\nHere $n_{\\rm D}$ and $n_{\\rm A}$ denote numbers of nucleons in deuteron and nucleus of atomic mass $A$\nin the target, and \n$\\tilde{\\sigma}_{\\rm D}$ and $\\tilde{\\sigma}_{\\rm A}$ are the cross sections\n per nucleon for reaction (\\ref{phorho}) occurring on the deuteron and on the nucleus of atomic mass\n$A$, respectively. The sum runs over all nuclei present in the COMPASS target.\nThe factor $C_1$ takes into account that there are two polarised deuterons in\nthe $^6$LiD molecule,\nas the $^6$Li nucleus is in a first approximation composed of a deuteron and\nan $\\alpha $ particle.\n\nThe measurements of the $\\tilde{\\sigma}_{\\rm A} \/ \\tilde{\\sigma}_{\\rm D}$ \nfor incoherent exclusive $\\rho ^0$ production come from the NMC \\cite{NMC}, E665 \\cite{E665-2} and early experiments on \\rn\\ photoproduction \\cite{BPSY}. They were fitted in Ref.~\\cite{ATrip-2}\nwith the formula:\n\\begin{equation}\n \\tilde{\\sigma}_{\\rm A}  = \n   \\sigma_{\\rm p} \\cdot A^{\\alpha(Q^{2})  -  1} , \\hspace*{1cm}\n    {\\rm with}\\ \\ \\alpha(Q^{2})  -  1  =  -\\frac{1}{3}  \\exp\\{-Q^{2}\/Q^{2}_{0}\\},\n    \\label{alpha}\n\\end{equation}\nwhere $\\sigma_{\\rm p}$ is the cross section for reaction (\\ref{phorho})\non the free proton.\nThe value of the fitted parameter $Q^{2}_{0}$ is  equal to $9 \\pm 3~(\\mathrm{GeV}\/c)^2$.\nThe measured values of the parameter $\\alpha $ and the fitted curve $\\alpha (Q^2)$\nare shown on the left panel of \nFig.~\\ref{alpha-fig} taken from Ref.~\\cite{ATrip-2}.\nOn the right panel of the figure the average value of $f$ is plotted for\nthe various $Q^2$ bins used in the present analysis. The values of $f$\nare equal to about 0.36 in most of the $Q^2$ range, rising to about 0.38 at the\nhighest $Q^2$.\n\n\\begin{figure}[t]\n\\begin{center}\n  \\epsfig{file=alpha.eps,height=5cm,width=7.5cm}\n  \\epsfig{file=f_Qsq_0203.eps,height=5cm,width=7.5cm}\n  \\caption{(Left) Parameter $\\alpha$ of Eq.~\\ref{alpha} as a function of $Q^2$\\ (from Ref.~\\cite{ATrip-2}). The experimental points and the fitted curve are shown. See text for details. (Right) The dilution factor $f$ as a function of $Q^2$.}\n  \\label{alpha-fig}\n \\end{center}\n\\end{figure}\n\n  The radiative corrections (RC) have been neglected in the present analysis, in particular in the calculation of $f$, because\nthey are expected to be small for reaction (\\ref{murho}). They were evaluated \\cite{KKurek} to be\nof the order of 6\\% for the NMC exclusive \\rn\\ production analysis.\nThe small values of RC are mainly due to the requirement of event exclusivity via cuts on \\Emi\\ and $p_t^2$,\nwhich largely suppress the dominant external photon radiation.\nThe internal (infrared and virtual) RC were\nestimated in Ref.~\\cite{KKurek} to be of the order of 2\\%.  \n\n\\section{Systematic errors}\n \\label{Sec_syst}\n\nThe main systematic uncertainty of \\Aor\\ comes from an estimate of possible false asymmetries. \nIn order to improve the accuracy of this estimate,\nin addition to the standard sample of incoherent events, a second sample\nwas selected by changing the $p_t^2$ cuts to\n\\begin{equation}\n 0 < p_{t}^{2} < 0.5~(\\mathrm{GeV}\/c)^{2},    \\label{ptsext}\n\\end{equation}  and keeping all the remaining selections and cuts the same\nas for the `incoherent sample'.\nIn the following it will be\n referred to as the `extended $p_t^2$ sample'.\nSuch an extension of the \\pts\\ range allows one to obtain a sample which is\nabout\nfive times larger than the incoherent sample. \nHowever, in addition to incoherent events such a sample contains a large fraction of events originating from coherent \\rn\\ production.\nTherefore, for the estimate of the dilution factor $f$ a different nuclear dependence of the\nexclusive cross section was used, applicable for the sum of coherent and incoherent\ncross sections \\cite{NMC}.\nThe physics asymmetries \\Aor\\ for both samples are consistent within statistical errors.\n\nPossible, false experimental asymmetries were searched for by modifying the selection\nof data sets used for the asymmetry calculation. The grouping of the data into\nconfigurations with opposite target-polarisation was varied from large samples,\ncovering at most two weeks of data taking, into about 100 small samples, taken in time\nintervals of the order of 16 hours.\nA statistical test was performed on the distributions of asymmetries obtained\nfrom these small samples. In each of the $Q^2$\\ and \\xBj\\ bins the dispersion of the values of\n$A_{1}^{\\rho}$ around their mean agrees with the statistical error.\nTime-dependent effects which would lead to a broadening of these distributions\nwere thus not observed. Allowing the dispersion of $A_{1}^{\\rho}$ to vary within its\ntwo standard deviations we \nobtain for each bin an upper bound for the systematic error arising from\ntime-dependent effects\n\\begin{equation}\n \\sigma_{\\rm falseA,tdep}  < \n   0.56~\\sigma_{\\rm stat}.\n\\end{equation}\nHere $\\sigma _{\\rm stat}$ is the statistical error on $A_{1}^{\\rho}$ for the extended\n$p_t^2$ sample.\nThe uncertainty on the estimates of possible false asymmetries due to the time-dependent \neffects is the dominant contribution to the total systematic error in most of the kinematical\nregion.\n\nAsymmetries for configurations where spin effects cancel out were calculated to\ncheck the cancellation of effects due to fluxes and acceptances. They were found compatible\nwith zero within the statistical errors. Asymmetries obtained with different\nsettings of the microwave (MW) frequency, used for DNP, were compared in order to test possible\neffects related to the orientation of the target magnetic field. \nThe results for the extended $p_t^2$ sample tend to show that there is a small\ndifference between asymmetries for the two MW configurations.\nHowever, because the numbers of events of the data samples taken with each MW setting \nare approximately balanced, the effect of this difference on \\Aor\\ is\nnegligible for the total sample.\n \nThe systematic error on $A_{1}^{\\rho}$ also contains an overall scale uncertainty of\n6.5\\% due to uncertainties on $P_B$ and $P_T$. The uncertainty of the \nparameterisation of $R(Q^2)$ \naffects the depolarisation factor $D$. The uncertainty of the dilution factor\n$f$ is mostly due to uncertainty of the parameter $\\alpha (Q^2)$ which takes into account\nnuclear effects in the incoherent $\\rho ^0$ production. \nThe neglect of the $A_2^{\\rho }$ term mainly affects the highest bins of $Q^2$\\ and \\xBj.\n\nAnother source of systematic errors is due to the contribution of\nthe non-exclusive background to our sample.\nThis background originates from two sources.\nFirst one is due to the  production of\n\\rn\\ accompanied by the dissociation of the target nucleon,\nthe second one is the production of \\rn\\ in inclusive scattering.\nIn order to evaluate the amount of background in the sample of exclusive\nevents it is necessary to\ndetermine the $E_{miss}$ dependence for the non-exclusive background\nin the region under the exclusive peak (cf.\\ Fig.~\\ref{hadplots} ).\nFor this purpose complete Monte Carlo simulations of the experiment were \nused, with events \ngenerated by either the PYTHIA 6.2 or LEPTO 6.5.1 generators.\nEvents generated with LEPTO come only from deep inelastic scattering and cover the range of $Q^2 > 0.5~(\\mathrm{GeV}\/c)^2$. \nThose generated with PYTHIA cover the whole kinematical range of the experiment and include exclusive production of vector mesons and processes with diffractive excitation of the target nucleon or the vector meson, in addition to inelastic\nproduction.\n\nThe generated MC events were reconstructed\nand selected for the analysis using the same procedure as for the data.\nIn each bin of $Q^2$\\ the $E_{miss}$ distribution for the MC was \nnormalised to the corresponding one for the data\nin the range of large $E_{miss} > 7.5~\\mathrm{GeV}$. Then the normalised MC\ndistribution was used to estimate the number of background events under the exclusive peak in the data.  \nThe fraction of background events in the sample of\nincoherent exclusive $\\rho ^0$ production was estimated to be about $0.12 \\pm 0.06$\nin most of the kinematical range, except in the largest $Q^2$\\ region, where it is about\n$0.24 \\pm 0.12$. \nThe large uncertainties of these fractions reflect the differences between \nestimates from LEPTO and PYTHIA in the region where they overlap. In \nthe case of PYTHIA the\nuncertainties on the cross sections for diffractive photo- and\nelectroproduction of vector mesons also contribute.\nFor events generated with PYTHIA the $E_{miss}$  \ndistributions for various physics processes could be studied separately. \nIt was found\nthat events of \\rn\\ production with an\nexcitation of the target nucleon into $N^*$ resonances\nof small mass, $M < 2~\\mathrm{GeV}\/c^2$, cannot be resolved from the exclusive\npeak and therefore were not included                 \nin the estimates of number of background events.\n\nAn estimate of the asymmetry \\Aor\\ for the background was obtained using a non-exclusive sample,\nwhich was selected with the standard cuts used in this analysis,\nexcept the cut on $E_{miss}$ which was modified to $E_{miss} > 2.5$~GeV. In different high-$E_{miss}$ bins \\Aor\\ for this sample was\nfound compatible with zero.\n\nBecause no indication of a non-zero \\Aor\\ for the background was found,\nand also due to a large uncertainty of the estimated amount of background \nin the exclusive sample, no background corrections were made.\nInstead, the effect of background was treated as a source\nof systematic error. Its contribution \nto the total systematic error was not significant in most of the kinematical\nrange, except for the highest $Q^2$ and $x$.\n\nThe total systematic error on \\Aor\\ was obtained as a quadratic sum of the errors from\nall discussed sources. \nIts values for each $Q^2$\\ and \\xBj\\ bin are given\nin Tables~\\ref{binsq2-1}  and \\ref{binsx-1}. The total systematic error \namounts to about 40\\% of the \nstatistical error for most of the kinematical range. Both errors become\ncomparable in the highest bin of $Q^2$. \n\n\\section{Results}\n \\label{Sec_resu}\n\nThe COMPASS results on $A^{\\rho}_1$ are shown as a function of $Q^2$\\  and \\xBj\\ \nin Fig.~\\ref{A1-Com} and listed in \nTables~\\ref{binsq2-1} and \\ref{binsx-1}.\nThe statistical errors are represented by vertical bars and the total systematic\nerrors by shaded bands.\n\n\\begin{figure}[ht]\n\\begin{center}\n  \\epsfig{file=A1_Qsq_incoh_cons_0203.eps,height=5cm,width=7.5cm}\n  \\epsfig{file=A1_xBj_incoh_cons_0203.eps,height=5cm,width=7.5cm}\n  \\caption{\\Aor\\ as a function of $Q^2$\\ (left) and \\xBj\\ (right) from the present analysis.\nError bars correspond to statistical errors, while bands at the bottom represent the systematical errors.}\n  \\label{A1-Com}\n \\end{center}\n\\end{figure}\n\\begin{table}[ht]\n \\caption{Asymmetry $A_1^{\\rho }$ as a function of $Q^2$. Both the statistical errors (first) and the total systematic errors (second) are listed.}\n \\label{binsq2-1}\n \\small\n \\begin{center}\n  \\begin{tabular}{||c|c|c|c|c||}\n   \\hline\n   \\hline\n   \\raisebox{0mm}[4mm][2mm]{$Q^2$\\ range}\\ \\  &\n                            $\\langle  Q^{2}  \\rangle$ [$(\\mathrm{GeV}\/c)^2$]  &  $\\langle  x  \\rangle$  &\n                            $\\langle  \\nu  \\rangle$ [GeV] & $A_1^{\\rho }$ \\\\   \\hline\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.0004 - 0.005~$  & \\ \\ \\ 0.0031\\ \\ \\  & $4.0 \\cdot 10^{-5}$ & 42.8  & $-0.030 \\pm 0.045 \\pm 0.014$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.005 - 0.010$  &  ~0.0074 & $8.4 \\cdot 10^{-5}$ & 49.9             & $~~0.048 \\pm 0.038 \\pm 0.013$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.010  - 0.025$ &  0.017  & $1.8 \\cdot 10^{-4}$ & 55.6             & $~~0.063 \\pm 0.026 \\pm 0.014$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.025 - 0.050$  &  0.036  & $3.7 \\cdot 10^{-4}$ & 59.9             & $-0.035 \\pm 0.027 \\pm 0.009$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.05  - 0.10$   &  0.072  & $7.1 \\cdot 10^{-4}$ & 62.0             & $-0.010 \\pm 0.028 \\pm 0.008$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.10   - 0.25$  &  0.16~  & 0.0016 & 62.3                           & $-0.019 \\pm 0.029 \\pm 0.009$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.25  - 0.50$   &  0.35~  & 0.0036 & 60.3                           & $~~0.016 \\pm 0.045 \\pm 0.014$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.5   - 1~~$     &  0.69~  & 0.0074 & 58.6                           & $~~0.141 \\pm 0.069 \\pm 0.030$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $1     - 4$     &  1.7~~  & 0.018~~  & 59.7                           & $~~0.000 \\pm 0.098 \\pm 0.035$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $~4    - 50$  & 6.8~~  & 0.075~~  & 55.9                               & $-0.85  \\pm 0.50  \\pm 0.39 $\\\\\n   \\hline\n   \\hline\n  \\end{tabular}\n \\end{center}\n  \\normalsize\n\\end{table}\n\\begin{table}[ht]\n \\caption{Asymmetry $A_1^{\\rho }$ as a function of \\xBj. Both the statistical errors (first) and the total systematic errors (second) are listed.}\n \\label{binsx-1}\n \\small\n \\begin{center}\n  \\begin{tabular}{||c|c|c|c|c||}\n   \\hline\n   \\hline\n   \\raisebox{0mm}[4mm][2mm]{\\xBj\\ range}\\ \\  &   $\\langle  x  \\rangle$  &  $\\langle  Q^{2}  \\rangle$ [$(\\mathrm{GeV}\/c)^2$]  &\n                           $\\langle  \\nu  \\rangle$ [GeV] & $A_1^{\\rho }$  \\\\   \\hline\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $8 \\cdot 10^{-6} - 1 \\cdot 10^{-4}$   & $5.8 \\cdot 10^{-5}$ & ~0.0058 & 51.7 & $~~0.035 \\pm 0.026 \\pm 0.011$ \\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $1 \\cdot 10^{-4} - 2.5 \\cdot 10^{-4}$  & $1.7 \\cdot 10^{-4}$ & 0.019 &  59.7  & $~~0.036 \\pm 0.024 \\pm 0.010$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $2.5 \\cdot 10^{-4} - 5 \\cdot 10^{-4}$   & $3.6 \\cdot 10^{-4}$ & 0.041 &  61.3  & $-0.039 \\pm 0.027 \\pm 0.012$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $5 \\cdot 10^{-4} - 0.001$ &  $7.1 \\cdot 10^{-4}$ & 0.082 & 60.8  & $-0.010 \\pm 0.030 \\pm 0.010$\\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.001 - 0.002$ &  0.0014 & 0.16~~ & 58.6 & $-0.005 \\pm 0.036 \\pm 0.013$ \\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.002 - 0.004$ &  0.0028 & 0.29~~ & 54.8 & $ ~~0.032 \\pm 0.050 \\pm 0.019$ \\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.004 - 0.01~~$  &  0.0062 & 0.59~~ & 50.7 & $ ~~0.019 \\pm 0.069 \\pm 0.026$ \\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.01  - 0.025$ &  0.015~~  & 1.3~~~~  & 47.5 & $-0.03 \\pm 0.14 \\pm 0.06$ \\\\\n   \\hline\n   \\raisebox{0mm}[4mm][2mm] \\ \\ $0.025 - 0.8~~~~$  &   0.049~~  & 3.9~~~~  & 43.8 & $-0.27 \\pm 0.38 \\pm 0.19$  \\\\\n   \\hline\n   \\hline\n  \\end{tabular}\n \\end{center}\n \\normalsize\n\\end{table}\n\nThe wide range in $Q^2$\ncovers four orders of magnitude from $3 \\cdot 10^{-3}$ to 7~$(\\mathrm{GeV}\/c)^2$. The domain in \\xBj\\ which is strongly correlated\nwith $Q^2$, varies from $5 \\cdot 10^{-5}$ to about 0.05 (see Tables for more details).\nFor the whole kinematical range the \\Aor\\\nasymmetry measured by COMPASS is consistent with zero. As discussed in the introduction,\nthis indicates that the role of unnatural parity exchanges,\nlike $\\pi$- or $A_{1}$-Reggeon exchange, \nis small in that kinematical domain, which is to be expected if diffraction\nis the dominant process for reaction (\\ref{phorho}). \n\nIn Fig.~\\ref{A1-Com-Her} the COMPASS results are compared to the HERMES results on $A^{\\rho}_1$ obtained\non a deuteron target \\cite{HERMES03}.\nNote that the lowest $Q^2$\\ and \\xBj\\ HERMES points, referred to as `quasi-photoproduction', come from measurements where the kinematics of the small-angle scattered \nelectron was not measured but estimated from a MC simulation. This is in contrast to COMPASS,\nwhere scattered muon kinematics is measured even at the smallest $Q^2$.\n\\begin{figure}[ht]\n\\begin{center}\n  \\epsfig{file=A1_Qsq_incoh_cons_0203_wHermes.eps,height=5cm,width=7.5cm}\n  \\epsfig{file=A1_xBj_incoh_cons_0203_wHermes.eps,height=5cm,width=7.5cm}\n  \\caption{\\Aor\\ as a function of $Q^2$\\ (left) and \\xBj\\ (right) from the present analysis (circles)\ncompared to HERMES results on the deuteron target (triangles). For the COMPASS results inner bars represent statistical errors, while the outer bars correspond to the total error.\n For the HERMES results vertical bars represent the quadratic sum of statistical and systematic errors. The curve represents the prediction explained in the text.}\n   \\label{A1-Com-Her}\n\\end{center}\n\\end{figure}\n\nThe results from both experiments are consistent within errors. \nThe kinematical range covered by the present analysis extends further towards small\nvalues of \\xBj\\ and $Q^2$\\ by almost two orders of \nmagnitude. In each of the two experiments $A^{\\rho}_1$ is measured\nat different average  $W$, which is equal to about 10~GeV for\nCOMPASS and 5~GeV for HERMES. Thus, no significant $W$ dependence\nis observed for $A^{\\rho}_1$ on an isoscalar nucleon target.\n\nThe \\xBj\\ dependence of the measured \\Aor\\ is compared in Fig.~\\ref{A1-Com-Her} to the prediction given by Eq.~\\ref{A1-Fraas}, which\nrelates $A_1^{\\rho}$ to the asymmetry $A_{1}$ for the inclusive inelastic\nlepton-nucleon scattering. To produce the curve\nthe inclusive asymmetry $A_{1}$ was parameterised as\n$A_1(x) = (x^{\\alpha } - \\gamma^{\\alpha }) \\cdot (1- e^{-\\beta x})$ , where\n$\\alpha = 1.158 \\pm 0.024$, $\\beta = 125.1 \\pm 115.7$ and $\\gamma = 0.0180 \\pm 0.0038$.\nThe values of the parameters have been obtained\nfrom a fit of $A_{1}(x)$ to the world data\nfrom polarised deuteron targets \\cite{smc,e143,e155_d,smc_lowx,hermes_new,compass_a1_recent} including COMPASS measurements at very \nlow $Q^2$ and \\xBj\\ \\cite{compass_a1_lowq2}. Within the present accuracy the results on \\Aor\\ are consistent with this prediction.  \n\n\nIn the highest $Q^2$\\ bin,  $\\langle  Q^{2} \\rangle = 6.8~(\\mathrm{GeV}\/c)^2$, in the kinematical domain of applicability of pQCD-inspired models which relate\nthe asymmetry to the spin-dependent\nGPDs for gluons and quarks (cf.\\ Introduction), \none can observe a hint of a possible nonzero asymmetry, although with a\nlarge error.\nIt should be noted that in Ref.~\\cite{ATrip-1} a negative value of $A_{LL}$ different from zero by about 2 standard deviations\nwas reported at $\\langle  Q^{2}  \\rangle  = 7.7~(\\mathrm{GeV}\/c)^2$.\nAt COMPASS, including the data \ntaken with the longitudinally polarised deuteron target in 2004 and 2006 will result in an\nincrease of\nstatistics by a factor of about three compared to the present paper,\nand thus may help to clarify the issue.\n \nFor the whole $Q^2$ range future COMPASS data, to be taken with the polarised proton target, would be very valuable for checking if the role of the flavour-blind\nexchanges is indeed dominant, as expected for the Pomeron-mediated process.\n\n \n\\section{Summary}\n \\label{Sec_summ}\n\nThe longitudinal double spin asymmetry \\Aor\\ for the diffractive muoproduction of \\rn\\ meson, $\\mu + N \\rightarrow \n\\mu + N + \\rho$, has been measured by scattering longitudinally polarised muons off longitudinally polarised deuterons from the\n$^6$LiD target and selecting incoherent exclusive $\\rho^0$ production.\nThe presented results for the COMPASS 2002 and 2003 data cover a range of energy $W$ from about 7 to 15~GeV.\n\nThe $Q^2$\\ and \\xBj\\ dependence of \\Aor\\ is presented in a wide kinematical range\n$3 \\cdot 10^{-3} \\leq Q^{2} \\leq 7~(\\mathrm{GeV}\/c)^2$ and $5 \\cdot 10^{-5} \\leq x \\leq 0.05$.\nThese results extend the range in $Q^2$\\ and \\xBj\\ by two orders of magnitude down\nwith respect to the existing data from HERMES.\n\nThe asymmetry $A^{\\rho}_1$ is compatible with zero in the whole \\xBj\\ and\n$Q^2$\\ range.\nThis may indicate that the role of unnatural parity exchanges like $\\pi$- or $A_{1}$-Reggeon exchange is\nsmall in that kinematical domain.\n\nThe \\xBj\\ dependence of measured \\Aor\\ is consistent with the prediction of Ref.~\\cite{HERMES01} which relates $A^{\\rho}_1$\nto the asymmetry $A_{1}$ for the inclusive inelastic lepton--nucleon scattering.\n\n\\section{Acknowledgements}\n \\label{Sec_ack}\n\nWe gratefully acknowledge the support of the CERN management and staff and\nthe skill and effort of the technicians of our collaborating institutes.\nSpecial thanks are due to V. Anosov and V. Pesaro for their support during the\ninstallation and the running of the experiment. This work was made possible by \nthe financial support of our funding agencies.\n\n\n\\noindent\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\n\n\nMany tasks in natural language processing, computational biology, reinforcement learning, and time series analysis rely on learning\nwith sequential data, i.e.\\  estimating functions defined over sequences of observations from training data.  \nWeighted  finite automata~(WFAs) and recurrent neural networks~(RNNs) are two powerful and flexible classes of models which can efficiently represent such functions.\nOn the one hand, WFAs are tractable, they encompass a wide range of machine learning models~(they can for example compute any probability distribution defined by a hidden Markov \nmodel~(HMM)~\\cite{denis2008rational} and can model the transition and observation behavior of\npartially observable Markov decision processes~\\cite{thon2015links}) and they offer appealing theoretical guarantees. In particular, \nthe so-called \n\\emph{spectral\nmethods} for learning HMMs~\\cite{hsu2009spectral}, WFAs~\\cite{bailly2009grammatical,balle2014spectral} and related models~\\cite{glaude2016pac,boots2011closing}, \nprovide an alternative to Expectation-Maximization based algorithms that is both computationally efficient and \nconsistent. \nOn the other hand, RNNs are \nremarkably expressive models --- they can represent any computable function~\\cite{siegelmann1992computational} --- and they have successfully\ntackled many practical problems in speech and audio recognition~\\cite{graves2013speech,mikolov2011extensions,gers2000learning}, but\ntheir theoretical  analysis is difficult. Even though recent work provides interesting results on their\nexpressive power~\\cite{khrulkov2018expressive,yu2017long} as well as alternative training algorithms coming with learning guarantees~\\cite{sedghi2016training},\nthe theoretical understanding of RNNs is still limited.\n\n\\footnotetext{\\footnotemark[1] Mila \\footnotemark[2] Universit\u00e9 de Montr\u00e9al \\footnotemark[3] McGill University}\n\n\\renewcommand*{\\thefootnote}{\\arabic{footnote}}\n\nIn this work, we bridge a gap between these two classes of models by unraveling a fundamental connection between WFAs and second-order RNNs~(2-RNNs): \n\\textit{when considering input sequences of discrete symbols, 2-RNNs with linear activation functions and WFAs are one and the same}, i.e.\\  they are expressively\nequivalent and there exists a one-to-one mapping between the two classes~(moreover, this mapping conserves model sizes).  While connections between\nfinite state machines~(e.g.\\  deterministic finite automata)  and recurrent neural networks have been noticed and investigated in the past~(see e.g.\\  \\cite{giles1992learning,omlin1996constructing}), to the\nbest of our knowledge this is the first time that such a \\rev{rigorous} equivalence between linear 2-RNNs and \\emph{weighted} automata is explicitly formalized. \n\\rev{More precisely, we pinpoint exactly the class of recurrent neural architectures to which weighted automata are equivalent, namely second-order RNNs with\nlinear activation functions.}\nThis result naturally leads to the observation that linear 2-RNNs are a natural generalization of WFAs~(which take sequences of \\emph{discrete} observations as\ninputs) to sequences of \\emph{continuous vectors}, and raises the question of whether the spectral learning algorithm for WFAs can be extended to linear 2-RNNs. \nThe second contribution of this paper is to show that the answer is in the positive: building upon the spectral learning algorithm for vector-valued WFAs introduced\nrecently in~\\cite{rabusseau2017multitask}, \\emph{we propose the first provable learning algorithm for second-order RNNs with linear activation functions}.\nOur learning algorithm relies on estimating  sub-blocks of the so-called Hankel tensor, from which the parameters of a 2-linear RNN can be recovered\nusing basic linear algebra operations. One of the key technical difficulties in designing this algorithm resides in estimating\nthese sub-blocks from training data where the inputs are sequences of \\emph{continuous} vectors. \nWe leverage multilinear properties of linear 2-RNNs and the fact that the Hankel sub-blocks can be reshaped into higher-order tensors of low tensor train rank~(a\nresult we believe is of independent interest) to perform this estimation efficiently using matrix sensing and tensor recovery techniques.\nAs a proof of concept, we validate our theoretical findings in a simulation study on toy examples where we experimentally compare  \ndifferent recovery methods and investigate the robustness of our algorithm to noise and rank mis-specification. \\rev{We also show that refining the estimator returned\nby our algorithm using stochastic gradient descent can lead to significant improvements.}\n\n\\rev{\n\\paragraph{Summary of contributions.} We formalize a \\emph{strict equivalence between weighted automata and second-order RNNs with linear activation\nfunctions}~(Section~\\ref{sec:WFAs.and.2RNNs}), showing that linear 2-RNNs can be seen as a natural extension of (vector-valued) weighted automata for input sequences of \\emph{continuous} vectors. We then\npropose a \\emph{consistent learning algorithm for linear 2-RNNs}~(Section~\\ref{sec:Spectral.learning.of.2RNNs}).\nThe relevance of our contributions can be seen from two perspectives.\nFirst, while learning feed-forward neural networks with linear activation functions is a trivial task (it reduces to linear or reduced-rank regression), this\nis not at all the case for recurrent architectures with linear activation functions; to the best of our knowledge, our algorithm is the \\emph{first consistent learning algorithm\nfor the class of functions computed by linear second-order recurrent networks}. Second, from the perspective of learning weighted automata, we propose a  natural extension of WFAs to continuous inputs and \\emph{our learning algorithm addresses the long-standing limitation of the spectral learning method to discrete inputs}.\n}\n\n\\paragraph{Related work.}\nCombining the spectral learning algorithm for WFAs with matrix completion techniques~(a problem which is closely related to matrix sensing) has\nbeen theoretically investigated in~\\cite{balle2012spectral}. An extension of probabilistic transducers to continuous inputs~(along with a spectral learning algorithm) has been proposed in~\\cite{recasens2013spectral}.\nThe connections between tensors  and RNNs have been previously leveraged to study the expressive power of RNNs in~\\cite{khrulkov2018expressive}\nand to achieve model compression in~\\cite{yu2017long,yang2017tensor,tjandra2017compressing}. \nExploring relationships between RNNs and automata has recently received a renewed interest~\\cite{peng2018rational,chen2018recurrent,li2018nonlinear}. In particular, such connections have been explored for interpretability purposes~\\cite{weiss2018extracting,ayache2018explaining} and the ability of RNNs to learn classes of formal languages\nhas been investigated in~\\cite{avcu2017subregular}.  \\rev{Connections between the tensor train decomposition and WFAs have been previously noticed in~\\cite{critch2013algebraic,critch2014algebraic,rabusseau2016thesis}.} \nThe predictive state RNN model introduced in~\\cite{downey2017predictive} is closely related to 2-RNNs and the authors propose\nto use the spectral learning algorithm for predictive state representations to initialize a gradient based algorithm; their approach however comes without\ntheoretical guarantees. Lastly, a provable algorithm for RNNs relying on the tensor method of moments has been proposed in~\\cite{sedghi2016training} but\nit is limited to first-order RNNs with quadratic activation functions~(which do not encompass linear 2-RNNs).\n\n\\emph{The proofs of the results given in the paper can be found in the supplementary material.}\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec:prelim}\nIn this section, we first present basic notions of tensor algebra before introducing  second-order recurrent neural \nnetwork, \nweighted finite automata and the spectral learning algorithm.\nWe start by introducing some notation.\nFor any integer $k$ we use $[k]$ to denote the set of integers from $1$ to $k$. We use\n$\\lceil l \\rceil$ to denote the smallest integer greater or equal to $l$.\nFor any set $\\Scal$, we denote by $\\Scal^*=\\bigcup_{k\\in\\Nbb}\\Scal^k$ the set of all\nfinite-length sequences of elements of $\\Scal$~(in particular, \n$\\Sigma^*$ will denote the set of strings on a finite alphabet $\\Sigma$). \nWe use lower case bold letters  for vectors (e.g.\\  $\\vec{v} \\in \\Rbb^{d_1}$),\nupper case bold letters for matrices (e.g.\\  $\\mat{M} \\in \\Rbb^{d_1 \\times d_2}$) and\nbold calligraphic letters for higher order tensors (e.g.\\  $\\ten{T} \\in \\Rbb^{d_1\n\\times d_2 \\times d_3}$). We use $\\ten_i$ to denote the $i$th canonical basis \nvector of $\\Rbb^d$~(where the dimension $d$ will always appear clearly from context).\nThe $d\\times d$ identity matrix will be written as $\\mat{I}_d$.\nThe $i$th row (resp. column) of a matrix $\\mat{M}$ will be denoted by\n$\\mat{M}_{i,:}$ (resp. $\\mat{M}_{:,i}$). This notation is extended to\nslices of a tensor in the straightforward way.\nIf $\\vec{v} \\in \\Rbb^{d_1}$ and $\\vec{v}' \\in \\Rbb^{d_2}$, we use $\\vec{v} \\otimes \\vec{v}' \\in \\Rbb^{d_1\n\\cdot d_2}$ to denote the Kronecker product between vectors, and its\nstraightforward extension to matrices and tensors.\nGiven a matrix $\\mat{M} \\in \\Rbb^{d_1 \\times d_2}$, we use $\\vectorize{\\mat{M}} \\in \\Rbb^{d_1\n\\cdot d_2}$ to denote the column vector obtained by concatenating the columns of\n$\\mat{M}$. The inverse of $\\mat{M}$ is denoted by $\\mat{M}^{-1}$, its Moore-Penrose pseudo-inverse\nby $\\mat{M}^\\dagger$, and the transpose of its inverse by $\\mat{M}^{-\\top}$; the Frobenius norm\nis denoted by $\\norm{\\mat{M}}_F$ and the nuclear norm by $\\norm{\\mat{M}}_*$.\n\n\\paragraph{Tensors.}\nWe first recall basic definitions of tensor algebra; more details can be found\nin~\\cite{Kolda09}. \nA \\emph{tensor} $\\ten{T}\\in \\Rbb^{d_1\\times\\cdots \\times d_p}$ can simply be seen\nas a multidimensional array $(\\ten{T}_{i_1,\\cdots,i_p}\\ : \\ i_n\\in [d_n], n\\in [p])$. The\n\\emph{mode-$n$} fibers of $\\ten{T}$ are the vectors obtained by fixing all\nindices except  the $n$th one, e.g.\\  $\\ten{T}_{:,i_2,\\cdots,i_p}\\in\\Rbb^{d_1}$.\nThe \\emph{$n$th mode matricization} of $\\ten{T}$ is the matrix having the\nmode-$n$ fibers of $\\ten{T}$ for columns and is denoted by\n$\\tenmat{T}{n}\\in \\Rbb^{d_n\\times d_1\\cdots d_{n-1}d_{n+1}\\cdots d_p}$.\nThe vectorization of a tensor is defined by $\\vectorize{\\ten{T}}=\\vectorize{\\tenmat{T}{1}}$.\nIn the following $\\ten{T}$ always denotes a tensor of size $d_1\\times\\cdots \\times d_p$.\n\nThe \\emph{mode-$n$ matrix product} of the tensor $\\ten{T}$ and a matrix\n$\\mat{X}\\in\\Rbb^{m\\times d_n}$ is a tensor  denoted by $\\ten{T}\\ttm{n}\\mat{X}$. It is \nof size $d_1\\times\\cdots \\times d_{n-1}\\times m \\times d_{n+1}\\times\n\\cdots \\times d_p$ and is defined by the relation \n$\\ten{Y} = \\ten{T}\\ttm{n}\\mat{X} \\Leftrightarrow \\tenmat{Y}{n} = \\mat{X}\\tenmat{T}{n}$.\nThe \\emph{mode-$n$ vector product} of the tensor $\\ten{T}$ and a vector\n$\\vec{v}\\in\\Rbb^{d_n}$ is a tensor defined by $\\ten{T}\\ttv{n}\\vec{v} = \\ten{T}\\ttm{n}\\vec{v}^\\top\n\\in \\Rbb^{d_1\\times\\cdots \\times d_{n-1}\\times d_{n+1}\\times\n\\cdots \\times d_p}$.\nIt is easy to check that the $n$-mode product satisfies $(\\ten{T}\\ttm{n}\\mat{A})\\ttm{n}\\mat{B} = \\ten{T}\\ttm{n}\\mat{BA}$\nwhere we assume compatible dimensions of the tensor $\\ten{T}$ and\nthe matrices $\\mat{A}$ and $\\mat{B}$.\n\n\nGiven strictly positive integers $n_1,\\cdots, n_k$ satisfying\n$\\sum_i n_i = p$, we use the notation $\\tenmatgen{\\ten{T}}{n_1,n_2,\\cdots,n_k}$ to denote the $k$th order tensor \nobtained by reshaping $\\ten{T}$ into a tensor\\footnote{Note that the specific ordering used to perform matricization, vectorization\nand such a reshaping is not relevant as long as it is consistent across all operations.} of size \n$(\\prod_{i_1=1}^{n_1} d_{i_1}) \\times (\\prod_{i_2=1}^{n_2} d_{n_1 + i_2}) \\times \\cdots \\times (\\prod_{i_k=1}^{n_k} d_{n_1+\\cdots+n_{k-1} + i_k})$.\nIn particular we have $\\tenmatgen{\\ten{T}}{p} = \\vectorize{\\ten{T}}$ and $\\tenmatgen{\\ten{T}}{1,p-1} = \\tenmat{\\ten{T}}{1}$.\n\n\nA rank $R$ \\emph{tensor train (TT) decomposition}~\\cite{oseledets2011tensor} of a tensor \n$\\ten{T}\\in\\Rbb^{d_1\\times \\cdots\\times d_p}$ consists in factorizing $\\ten{T}$ into the product of $p$ core tensors\n$\\ten{G}_1\\in\\Rbb^{d_1\\times R},\\ten{G}_2\\in\\Rbb^{R\\times d_2\\times R},\n\\cdots, \\ten{G}_{p-1}\\in\\Rbb^{R\\times d_{p-1} \\times R},\n\\ten{G}_p \\in \\Rbb^{R\\times d_p}$, and is defined\\footnote{The classical definition of the TT-decomposition allows the rank $R$ to be different\nfor each mode, but this definition is sufficient for the purpose of this paper.} by\n\\begin{align*}\n\\MoveEqLeft\\ten{T}_{i_1,\\cdots,i_p} =  \n&(\\ten{G}_1)_{i_1,:}(\\ten{G}_2)_{:,i_2,:}\\cdots \n (\\ten{G}_{p-1})_{:,i_{p-1},:}(\\ten{G}_p)_{:,i_p}\n\\end{align*}\n %\nfor all indices $i_1\\in[d_1],\\cdots,i_p\\in[d_p]$; we will use the notation $\\ten{T} = \\TT{\\ten{G}_1,\\cdots,\\ten{G}_p}$\nto denote such a decomposition. A tensor network representation of this decomposition is shown in Figure~\\ref{fig:tn.TT}.\n While the problem of finding the best approximation of  TT-rank $R$\nof a given tensor is NP-hard~\\cite{hillar2013most}, \na quasi-optimal SVD based compression algorithm~(TT-SVD) has been proposed \nin~\\cite{oseledets2011tensor}.\nIt is worth mentioning that the TT decomposition is invariant under change of basis: \nfor any invertible matrix $\\mat{M}$ and any core tensors $\\ten{G}_1,\\ten{G}_2,\\cdots,\\ten{G}_p$, we have\n$\\TT{\\ten{G}_1,\\cdots,\\ten{G}_p} = \\TT{\\ten{G}_1\\ttm{2}\\mat{M}^{-\\top},\\ten{G}_2\\ttm{1}\\mat{M}\\ttm{3}\\mat{M}^{-\\top},\\cdots,\n\\ten{G}_{p-1}\\ttm{1}\\mat{M}\\ttm{3}\\mat{M}^{-\\top},\\ten{G}_p\\ttm{1}\\mat{M}}$.\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.45\\textwidth}{0.08\\textwidth}{%\n\\begin{tikzpicture}\n\t\\input{tikz_tensor_networks}\n\t\\node[tensor](G1){$\\ten{G}_1$};\n\t\\node[draw = none,below=0.8cm of G1](G11){};\n\t\n\t\\node[tensor,right = 1cm of G1](G2){$\\ten{G}_2$};\n\t\\node[draw=none,below=0.8cm of G2](G22){};\n\t\n\t\\node[tensor,right = 1cm of G2](G3){$\\ten{G}_3$};\n\t\\node[draw=none,below=0.8cm of G3](G32){};\n\t\n\t\\node[tensor,right = 1cm of G3](G4){$\\ten{G}_4$};\n\t\\node[draw=none,below=0.8cm of G4](G41){};\n\t\n\t\\node[tensor,left=3cm of G1](T){$\\ten{T}$};\n\t\\node[draw=none,below left = 0.2cm and 1cm of T](T1){};\n\t\\node[draw=none,below left = 0.8cm and 0.2cm of T](T2){};\n\t\\node[draw=none,below right = 0.8cm and 0.2cm of T](T3){};\n\t\\node[draw=none,below right = 0.2cm and 1cm of T](T4){};\n\t\\node[draw=none,right=1.5cm of T](eq){$=$};\t\n\t\t\n\t\\edgeports{T}{1}{above left}{T1}{}{}{$d_1$};\n\t\\edgeports{T}{2}{below right}{T2}{}{}{$d_2$};\n\t\\edgeports{T}{3}{below right=-0.1cm and 0.1cm}{T3}{}{}{$d_3$};\n\t\\edgeports{T}{4}{above right}{T4}{}{}{$d_4$};\n\t\n\t\\edgeports{G1}{1}{below left = -0.1cm and 0.01cm }{G11}{}{}{$d_1$};\n\t\n\t\\edgeports{G2}{1}{below left}{G1}{2}{below right}{$R$};\t\n\t\\edgeports{G2}{2}{below left = -0.1cm and 0.01cm }{G22}{}{}{$d_2$};\n\t\n\t\\edgeports{G3}{1}{below left}{G2}{3}{below right}{$R$};\t\n\t\\edgeports{G3}{2}{below left = -0.1cm and 0.01cm }{G32}{}{}{$d_3$};\n\t\n\t\\edgeports{G4}{1}{below left}{G3}{3}{below right}{$R$};\t\n\t\\edgeports{G4}{2}{below left = -0.1cm and 0.01cm }{G41}{}{}{$d_4$};\n\n\t\n\\end{tikzpicture}\n}%\n\\end{center}\n\\caption{Tensor network representation of a rank $R$ tensor train decomposition~(nodes represent tensors and an edge between two nodes\nrepresents a contraction between the corresponding modes of the two tensors).}\n\\label{fig:tn.TT}\n\\end{figure}\n\n\\paragraph{Second-order RNNs.}\nA \\emph{second-order recurrent neural network} (2-RNN)~\\cite{giles1990higher,pollack1991induction,lee1986machine}\\footnote{Second-order reccurrent architectures have also been successfully used more recently, see e.g.\\  \\cite{sutskever2011generating} and \\cite{wu2016multiplicative}.} with $n$ hidden units can be defined as a tuple \n$M=(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$ where $\\vec{h}_0\\in\\Rbb^n$ is the initial state, $\\Aten\\in\\Rbb^{ n\\times d \\times n }$ is the transition tensor, and\n$\\vecs{\\vvsinfsymbol}\\in \\Rbb^{p\\times n}$ is the output matrix,\nwith $d$ and $p$ being the input and output dimensions respectively. \nA 2-RNN maps any sequence of inputs $\\vec{x}_1,\\cdots,\\vec{x}_k\\in\\Rbb^d$ to\na sequence of outputs $\\vec{y}_1,\\cdots,\\vec{y}_k\\in\\Rbb^p$ defined for any $ t=1,\\cdots,k$ by\n\\begin{equation}\\label{eq:2RNN.definition}\n\\vec{y}_t = z_2(\\vecs{\\vvsinfsymbol}\\vec{h}_t) \\text{ with }\\vec{h}_t = z_1(\\Aten\\ttv{1}\\vec{x}_t\\ttv{2}\\vec{h}_{t-1})\n\\end{equation}\nwhere $z_1:\\Rbb^n\\to\\Rbb^n$ and $z_2:\\Rbb^p\\to\\Rbb^p$ are activation functions.\nAlternatively, one can think of a 2-RNN as computing\na function $f_M:(\\Rbb^d)^*\\to\\Rbb^p$ mapping each input sequence $\\vec{x}_1,\\cdots,\\vec{x}_k$ to the corresponding final output $\\vec{y}_k$.\nWhile $z_1$ and $z_2$ are usually non-linear\ncomponent-wise functions, we consider in this paper the case where both $z_1$ and $z_2$ are the identity, and we refer to\nthe resulting model as a \\emph{linear 2-RNN}.\nFor a linear 2-RNN $M$, the function $f_M$ is multilinear in the sense that, for any integer $l$, its restriction to the domain $(\\Rbb^d)^l$ is\nmultilinear. Another useful observation is that linear 2-RNNs are invariant under change of basis: for any invertible matrix\n$\\P$, the linear 2-RNN $\\tilde{M}=(\\P^{-\\top}\\vec{h}_0,\\Aten\\ttm{1}\\P\\ttm{3}\\P^{-\\top},\\P\\vecs{\\vvsinfsymbol})$ is such that $f_{\\tilde{M}}=f_M$. A linear 2-RNN $M$ with $n$ states is called \\emph{minimal} if its number of hidden units is minimal~(i.e.\\  any linear 2-RNN computing $f_M$ has\nat least $n$ hidden units).\n\n\n\\paragraph{Weighted automata and spectral learning.} \\emph{Vector-valued weighted finite automaton}~(vv-WFA) have\nbeen introduced in~\\cite{rabusseau2017multitask} as a natural generalization of weighted automata from scalar-valued functions\nto vector-valued ones. A $p$-dimensional vv-WFA with $n$ states is a tuple $A=\\vvwa$\nwhere $\\vecs{\\szerosymbol}\\in\\Rbb^n$ is the initial weights vector, $\\vecs{\\vvsinfsymbol}\\in\\Rbb^{p\\times n}$ is the matrix of final weights, \nand $\\mat{A}^\\sigma\\in\\Rbb^{n\\times n}$ is the transition matrix for each symbol $\\sigma$ in a finite alphabet $\\Sigma$.\nA vv-WFA $A$ computes a function $f_A:\\Sigma^*\\to\\Rbb^p$ defined by \n$$f_A(x) =\\vecs{\\vvsinfsymbol} (\\mat{A}^{x_1}\\mat{A}^{x_2}\\cdots\\mat{A}^{x_k})^\\top\\vecs{\\szerosymbol} $$\nfor each word $x=x_1x_2\\cdots x_k\\in\\Sigma^*$. We call a vv-WFA \\emph{minimal} if its number of states\nis minimal. Given a function $f:\\Sigma^*\\to \\Rbb^p$ we denote by $\\rank(f)$ the number of states of a minimal vv-WFA computing $f$~(which is\nset to $\\infty$ if $f$ cannot be computed by a vv-WFA).\n\nThe  spectral learning algorithm for  vv-WFAs relies on the following fundamental theorem relating\nthe rank of a function $f:\\Sigma^*\\to\\Rbb^d$ to its Hankel tensor $\\ten{H} \\in \\Rbb^{\\Sigma^*\\times\\Sigma^*\\times p}$, which is defined\nby $\\ten{H}_{u,v,:} = f(uv)$ for all $u,v\\in\\Sigma^*$.\n\\begin{theorem}[\\cite{rabusseau2017multitask}]\n\\label{thm:fliess-vvWFA}\nLet $f:\\Sigma^*\\to\\Rbb^d$ and let $\\ten{H}$ be its Hankel tensor. Then $\\rank(f) = \\rank(\\tenmat{H}{1})$.\n\\end{theorem}\nThe vv-WFA learning algorithm leverages the fact that the proof of this theorem is constructive: one can recover a vv-WFA computing $f$\nfrom any low rank factorization of $\\tenmat{H}{1}$. In practice, a finite sub-block $\\ten{H}_{\\Pcal,\\Scal} \\in \\Rbb^{\\Pcal\\times \\Scal\\times p}$ of the Hankel tensor  is used\nto recover the vv-WFA, where $\\Pcal,\\Scal\\subset\\Sigma^*$ are finite sets of prefixes and suffixes forming a \\emph{complete basis} for $f$, i.e.\\  such that\n$\\rank(\\tenmatpar{\\ten{H}_{\\Pcal,\\Scal}}{1}) = \\rank(\\tenmat{H}{1})$. More details can be found \nin~\\cite{rabusseau2017multitask}.\n\n\n\\section{A Fundamental Relation between WFAs and Linear 2-RNNs}\\label{sec:WFAs.and.2RNNs}\n\nWe start by unraveling a fundamental connection between vv-WFAs and linear 2-RNNs: vv-WFAs and\nlinear 2-RNNs are expressively equivalent for representing functions defined over sequences of\ndiscrete symbols. \\rev{Moreover, both models have the same capacity in the sense that there is a direct\ncorrespondence between the hidden units of a linear 2-RNN and the states of a vv-WFA computing the same function}. More formally, we have the following theorem. \n\\begin{theorem}\\label{thm:2RNN-vvWFA}\nAny function that can be computed by a vv-WFA with $n$ states can be computed by a linear 2-RNN with $n$ hidden units.\nConversely, any function that can be computed by a linear 2-RNN with $n$ hidden units on sequences of one-hot vectors~(i.e.\\  canonical basis \nvectors) can be computed by a WFA with $n$ states.\n\n\nMore precisely, the WFA $A=\\vvwa$ with $n$ states and the linear 2-RNN $M=(\\vecs{\\szerosymbol},\\Aten,\\vecs{\\vvsinfsymbol})$ with\n$n$ hidden units, where $\\Aten\\in\\Rbb^{n\\times \\Sigma \\times n}$ is defined by $\\Aten_{:,\\sigma,:}=\\mat{A}^\\sigma$ for all $\\sigma\\in\\Sigma$, are\nsuch that\n$f_A(\\sigma_1\\sigma_2\\cdots\\sigma_k) = f_M(\\vec{x}_1,\\vec{x}_2,\\cdots,\\vec{x}_k)$ for all sequences of input symbols $\\sigma_1,\\cdots,\\sigma_k\\in\\Sigma$,\nwhere for each $i\\in[k]$ the input vector $\\vec{x}_i\\in\\Rbb^\\Sigma$ is\nthe one-hot encoding of the symbol $\\sigma_i$.\n\n\\end{theorem}\n\nThis result first implies that linear 2-RNNs defined over sequence of discrete symbols~(using one-hot encoding) \\emph{can be provably learned using the spectral\nlearning algorithm for WFAs\/vv-WFAs}; indeed, these  algorithms have been proved to return consistent estimators.\n\\rev{Let us stress again that, contrary to the case of feed-forward architectures, learning recurrent networks with linear activation functions is not a trivial task.}\nFurthermore, Theorem~\\ref{thm:2RNN-vvWFA} reveals that linear 2-RNNs are a natural generalization of classical weighted automata  to functions\ndefined over sequences of continuous vectors~(instead of discrete symbols). This spontaneously raises the question of whether the spectral learning algorithms\nfor WFAs and vv-WFAs can be extended to the general setting of linear 2-RNNs; we show that the answer is in the positive in the next section.\n\n\n\n \n\n\n\n\\section{Spectral Learning of Linear 2-RNNs}\\label{sec:Spectral.learning.of.2RNNs}\nIn this section, we extend the learning algorithm for vv-WFAs to linear 2-RNNs, \\rev{thus at the same time addressing the limitation of the spectral learning algorithm to discrete inputs and  providing the first consistent learning algorithm for linear second-order RNNs.}\n\n\n\n\\subsection{Recovering 2-RNNs from Hankel Tensors}\\label{subsec:SL-2RNN}\nWe first present an identifiability result showing how one can recover a linear 2-RNN computing a function $f:(\\Rbb^d)^*\\to \\Rbb^p$ from observable tensors extracted from some Hankel tensor\nassociated with $f$. Intuitively, we obtain this result by reducing the problem to the one of learning a vv-WFA. This is done by considering the restriction of\n$f$ to canonical basis vectors; loosely speaking, since the domain of this restricted function is isomorphic to $[d]^*$, this allows us to fall back onto the \nsetting of sequences of discrete symbols.\n\nGiven a function $f:(\\Rbb^d)^*\\to\\Rbb^p$, we define its Hankel tensor $\\ten{H}_f\\in \\Rbb^{[d]^* \\times [d]^* \\times p}$ by\n$$(\\ten{H}_f)_{i_1\\cdots i_s,  j_1\\cdots j_t,:} = f(\\ten_{i_1},\\cdots,\\ten_{i_s},\\ten_{j_1},\\cdots,\\ten_{j_t}),$$ \nfor all $i_1,\\cdots,i_s,j_1,\\cdots,j_t\\in [d]$, which is infinite in two\nof its modes. It is easy to see that $\\ten{H}_f$ is also the Hankel tensor associated with the function $\\tilde{f}:[d]^* \\to \\Rbb^p$ mapping any\nsequence $i_1i_2\\cdots i_k\\in[d]^*$ to $f(\\ten_{i_1},\\cdots,\\ten_{i_k})$. Moreover, in the special case where $f$ can be computed by a linear 2-RNN, one\ncan use the multilinearity of $f$ to show that $f(\\vec{x}_1,\\cdots,\\vec{x}_k) = \\sum_{i_1,\\cdots,i_k = 1}^d (\\vec{x}_1)_{i_1}\\cdots(\\vec{x}_l)_{i_k} \\tilde{f}(i_1\\cdots i_k)$,\n\\rev{giving us some intuition on how one could} learn $f$ by learning a vv-WFA computing $\\tilde{f}$ using the spectral learning algorithm. \nThat is, given a large enough sub-block $\\ten{H}_{\\Pcal,\\Scal}\\in \\Rbb^{\\Pcal\\times \\Scal\\times p}$ of $\\ten{H}_f$ for some prefix and suffix sets $\\Pcal,\\Scal\\subseteq [d]^*$, \none should be able to recover a vv-WFA computing $\\tilde{f}$ and consequently a linear 2-RNN computing $f$ using Theorem~\\ref{thm:2RNN-vvWFA}. \n\\rev{Before devoting the remaining of this section to formalize this intuition~(leading to Theorem~\\ref{thm:2RNN-SL}), it is worth \nobserving that while this approach is sound, it is not realistic since it requires observing entries of the Hankel tensor $\\ten{H}_f$, which implies having access to input\/output examples where the inputs are  \\emph{sequences of canonical basis vectors}; This issue will be discussed in more details and addressed in the \nnext section.} \n\n\n\n\\emph{For the sake of clarity, we present the learning algorithm for the particular case where there exists an $L$ such that \nthe prefix and suffix sets consisting of all sequences of length $L$, that is $\\Pcal = \\Scal = \n[d]^L$, forms a complete basis for $\\tilde{f}$}~\\rev{(i.e.\\  the sub-block $\\ten{H}_{\\Pcal,\\Scal}\\in\\Rbb^{[d]^L\\times [d]^L\\times p}$ of the Hankel tensor $\\ten{H}_f$ is\nsuch that $\\rank(\\tenmatpar{\\ten{H}_{\\Pcal,\\Scal}}{1}) = \\rank(\\tenmatpar{\\ten{H}_f}{1})$)}. This assumption allows us to present all the key elements of the algorithm  in a simpler way, the technical details\nneeded to lift this assumption are given in the supplementary material.\n\nFor any integer $l$, we define the finite tensor \n$\\ten{H}^{(l)}_f\\in \\Rbb^{ d\\times \\cdots \\times d\\times p}$ of order $l+1$ by\n$$  (\\ten{H}^{(l)}_f)_{i_1,\\cdots,i_l,:} = f(\\ten_{i_1},\\cdots,\\ten_{i_l}) \\ \\ \\ \\text{for all } i_1,\\cdots,i_l\\in [d].$$ \nObserve that for any integer $l$, the tensor $\\ten{H}^{(l)}_f$ can be obtained by reshaping a finite sub-block of the Hankel tensor $\\ten{H}_f$. \nWhen $f$ is computed by a linear $2$-RNN, we have the useful property that, for any integer $l$,\n\\begin{equation}\\label{eq:}\n f(\\vec{x}_1,\\cdots,\\vec{x}_l)  = \\ten{H}^{(l)}_f \\ttv{1} \\vec{x}_1 \\ttv{2} \\cdots \\ttv{l} \\vec{x}_l\n\\end{equation}\nfor any sequence of inputs $\\vec{x}_1,\\cdots,\\vec{x}_l\\in\\Rbb^d$~(which can be shown using the\nmultilinearity of $f$). Another fundamental property of the tensors $\\ten{H}^{(l)}_f$ is that they are of low tensor train rank. Indeed, for any $l$, one can check that\n$\\ten{H}^{(l)}_f = \\TT{\\Aten\\ttv{1}\\vecs{\\szerosymbol}, \\underbrace{\\Aten, \\cdots, \\Aten}_{l-1\\text{ times}}, \\vecs{\\vvsinfsymbol}^\\top}$~(the tensor network representation of this\ndecomposition is shown in Figure~\\ref{fig:Hl.TT.rank}).\nThis property will be particularly relevant to the learning algorithm we design in the following section, but it is also a fundamental relation\nthat deserves some attention on its own: it implies in particular that, beyond the classical relation between the rank of the Hankel matrix $\\H_f$ and the number states of\na minimal WFA computing $f$, the Hankel matrix possesses a deeper structure intrinsically connecting weighted automata to the tensor\ntrain decomposition. \n\\rev{We now state the main result of this section, showing that a (minimal) linear 2-RNN computing a function $f$ can be exactly recovered from sub-blocks of the Hankel tensor $\\ten{H}_f$.}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{0.45\\textwidth}{0.08\\textwidth}{%\n\\begin{tikzpicture}\n\t\\input{tikz_tensor_networks}\n\t\\node[tensor](G1){$\\vecs{\\szerosymbol}$};\n\t\\node[draw=none,left of = G1](H){$\\ten{H}^{(4)}_f=\\ \\ \\ $};\n\t\\node[tensor,right = 1cm of G1](G2){$\\ten{A}$};\n\t\\node[draw=none,below=0.8cm of G2](G22){};\n\t\n\t\\node[tensor,right = 1cm of G2](G3){$\\ten{A}$};\n\t\\node[draw=none,below=0.8cm of G3](G32){};\n\t\n\n\t\\node[tensor,right = 1cm of G3](G4){$\\ten{A}$};\n\t\\node[draw=none,below=0.8cm of G4](G42){};\t\n\t\n\t\\node[tensor,right = 1cm of G4](G4-){$\\ten{A}$};\n\t\\node[draw=none,below=0.8cm of G4-](G42-){};\t\n\t\n\t\\node[tensor,right = 1cm of G4-](G5){$\\vecs{\\vvsinfsymbol}$};\n\t\\node[draw=none,below=0.8cm of G5](G51){};\n\t\n\n\t\n\t\\edgeports{G2}{1}{below left = 0.01cm }{G1}{1}{below right}{$n$};\t\n\t\\edgeports{G2}{2}{below left = -0.1cm and 0.01cm }{G22}{}{}{$d$};\n\t\n\t\\edgeports{G3}{1}{below left}{G2}{3}{below right}{$n$};\t\n\t\\edgeports{G3}{2}{below left = -0.1cm and 0.01cm }{G32}{}{}{$d$};\n\t\n\t\\edgeports{G4}{1}{below left}{G3}{3}{below right}{$n$};\t\n\t\\edgeports{G4}{2}{below left = -0.1cm and 0.01cm }{G42}{}{}{$d$};\n\t\n\t\\edgeports{G4-}{1}{below left}{G4}{3}{below right}{$n$};\t\n\t\\edgeports{G4-}{2}{below left = -0.1cm and 0.01cm }{G42-}{}{}{$d$};\n\t\n\t\\edgeports{G5}{1}{below left}{G4-}{3}{below right}{$n$};\t\n\t\\edgeports{G5}{2}{below left = -0.1cm and 0.01cm }{G51}{}{}{$p$};\n\n\t\n\\end{tikzpicture}\n}%\n\\end{center}\n\\caption{Tensor network representation of the TT decomposition of the Hankel tensor $\\ten{H}^{(4)}_f$ induced\nby a linear $2$-RNN $(\\vecs{\\szerosymbol},\\Aten,\\vecs{\\vvsinfsymbol})$.}\n\\label{fig:Hl.TT.rank}\n\\end{figure}\n\n\n\n\\begin{theorem}\\label{thm:2RNN-SL}\nLet $f:(\\Rbb^d)^*\\to \\Rbb^p$ be a function computed by a minimal linear $2$-RNN  with $n$ hidden units and let\n$L$ be an integer such that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$.\n\nThen, for any $\\P\\in\\Rbb^{d^L\\times n}$ and $\\S\\in\\Rbb^{n\\times d^Lp}$ such that $\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1} = \\P\\S$, the\nlinear 2-RNN $M=(\\vecs{\\szerosymbol},\\Aten,\\vecs{\\vvsinfsymbol})$ defined by\n\n$$\\vecs{\\szerosymbol} = (\\S^\\dagger)^\\top\\tenmatgen{\\ten{H}^{(L)}_f}{L+1}, \\ \\ \\ \\ \\vecs{\\vvsinfsymbol}^\\top = \\P^\\dagger\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}\n$$\n$$\\Aten = (\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1})\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top$$\nis a minimal linear $2$-RNN computing $f$.\n\\end{theorem}\nFirst observe that such an integer $L$ exists under the assumption that $\\Pcal = \\Scal = \n[d]^L$ forms a complete basis for $\\tilde{f}$.\nIt is also worth mentioning that a necessary condition for $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$ is that\n$d^L\\geq n$, i.e.\\  $L$ must be of the order $\\log_d(n)$.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Hankel Tensors Recovery from Linear Measurements}\\label{subsec:Hankel.tensor.recovery}\n\n\n\nWe showed in the previous section that, given the Hankel tensors $\\ten{H}^{(L)}_f$, $\\ten{H}^{(2L)}_f$ and $\\ten{H}^{(2L+1)}_f$, one can recover \na linear 2-RNN computing $f$ if it exists. This first implies that the class of functions that can be computed by linear 2-RNNs is learnable  in Angluin's\nexact learning model~\\cite{angluin1988queries} where one has access to an oracle that can answer membership queries~(e.g.\\  \\textit{what is the value computed by the target $f$ \non~$(\\vec{x}_1,\\cdots,\\vec{x}_k)$?}) and  equivalence queries~(e.g.\\  \\textit{is my current hypothesis $h$ equal to the target $f$?}). While this fundamental result is \nof significant theoretical interest, assuming access to such an oracle is unrealistic. In this section, we show that a stronger learnability result can be obtained in a more realistic setting,\nwhere we  only\nassume access to randomly generated input\/output examples $((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)})\\in(\\Rbb^d)^*\\times\\Rbb^p$ where $\\vec{y}^{(i)} = f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)})$.\n\n\nThe key observation is that such an input\/output example $((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)})$ can be seen as a \\emph{linear measurement} of the \nHankel tensor $\\ten{H}^{(l)}$. Indeed, we have\n\\begin{align*}\n\\vec{y}^{(i)} &= f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}) = \\ten{H}^{(l)}_f \\ttv{1} \\vec{x}_1 \\ttv{2} \\cdots \\ttv{l} \\vec{x}_l \\\\\n&= \n\\tenmatgen{\\ten{H}^{(l)}}{l,1}^\\top \\vec{x}^{(i)}\n\\end{align*}\nwhere $\\vec{x}^{(i)} = \\vec{x}^{(i)}_1\\otimes\\cdots \\otimes \\vec{x}_{l}^{(i)}\\in\\Rbb^{d^l}$. Hence,  by regrouping $N$ output examples $\\vec{y}^{(i)}$ into\nthe matrix $\\Ymat\\in\\Rbb^{N\\times p}$ and the corresponding input vectors $\\vec{x}^{(i)}$ into the matrix $\\mat{X}\\in\\Rbb^{N\\times d^l}$,\none can recover $\\ten{H}^{(l)}$ by solving the linear system $\\Ymat = \\mat{X}\\tenmatgen{\\ten{H}^{(l)}}{l,1}$, which has a unique\nsolution whenever $\\mat{X}$ is of full column rank. \nThis  naturally leads to the following theorem, whose proof relies on the fact that\n$\\mat{X}$ will be of full column rank whenever $N\\geq d^l$ and  the components of each $\\vec{x}^{(i)}_j$ for $j\\in[l],i\\in[N]$\nare drawn independently from a continuous distribution over $\\Rbb^{d}$~(w.r.t. the Lebesgue measure).\n\n\n\n\n\n\n\n\\begin{theorem}\\label{thm:learning-2RNN}\nLet $(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$ be a minimal linear 2-RNN with $n$ hidden units computing a function $f:(\\Rbb^d)^*\\to \\Rbb^p$, and let $L$ be an integer\\footnote{Note that the theorem can be adapted if such an integer $L$ does not exists~(see supplementary material).}\nsuch that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$.\nSuppose we have access to $3$ datasets\n$D_l = \\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)}) \\}_{i=1}^{N_l}\\subset(\\Rbb^d)^l\\times \\Rbb^p$ for $l\\in\\{L,2L,2L+1\\}$ \nwhere the entries of each $\\vec{x}^{(i)}_j$ are drawn independently from the standard normal distribution and where each\n$\\vec{y}^{(i)} = f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)})$.\n\nThen, if $N_l \\geq d^l$ for $l =L,\\ 2L,\\ 2L+1$,\nthe linear 2-RNN $M$ returned by Algorithm~\\ref{alg:2RNN-SL} with the least-squares method satisfies $f_M = f$ with probability one.\n\\end{theorem}\n\n\n\\begin{algorithm}[tb]\n   \\caption{\\texttt{2RNN-SL}: Spectral Learning of linear 2-RNNs }\n   \\label{alg:2RNN-SL}\n\\begin{algorithmic}[1]\n   \\REQUIRE Three training datasets $D_L,D_{2L},D_{2L+1}$ with input sequences of length $L$, $2L$ and $2L+1$ respectively, a \\texttt{recovery\\_method}, rank $R$ and learning rate $\\gamma$~(for IHT\/TIHT).\n  \n   \\FOR{$l\\in\\{L,2L,2L+1\\}$}\n   \\STATE\\label{alg.firstline.forloop} Use $D_l = \\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)}) \\}_{i=1}^{N_l}\\subset(\\Rbb^d)^l\\times \\Rbb^p$ to build $\\mat{X}\\in\\Rbb^{N_l\\times d^l}$ with rows $\\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)}$ for $i\\in[N_l]$ and  $\\Ymat\\in\\Rbb^{N_l\\times p}$ with rows $\\vec{y}^{(i)}$ for $i\\in[N_l]$.\n   \\IF{\\texttt{recovery\\_method} = \"Least-Squares\"}\n   \\STATE\\label{alg.line.lst-sq} $\\ten{H}^{(l)} = \\displaystyle\\argmin_{\\ten{T}\\in \\Rbb^{d\\times\\cdots\\times d\\times p}} \\norm{\\mat{X}\\tenmatgen{\\ten{T}}{l,1} - \\Ymat}_F^2$.\n   \\ELSIF{\\texttt{recovery\\_method} = \"Nuclear Norm\"}\n   \\STATE $\\ten{H}^{(l)} = \\displaystyle\\argmin_{\\ten{T}\\in \\Rbb^{d\\times\\cdots\\times d\\times p}} \\norm{\\tenmatgen{\\ten{T}}{\\ceil{l\/2},l-\\ceil{l\/2} + 1}}_*$ subject to $\\mat{X} \\tenmatgen{\\ten{T}}{l,1} = \\Ymat$.\n   \\label{alg.line.nucnorm}\n   \\ELSIF{\\texttt{recovery\\_method} = \"(T)IHT\"}\n   \\STATE Initialize $\\ten{H}^{(l)} \\in \\Rbb^{d\\times\\cdots\\times d\\times p}$ to $\\mat{0}$.\n   \\REPEAT\\label{alg.line.iht.start}\n   \\STATE\\label{alg.line.iht.gradient} $\\tenmatgen{\\ten{H}^{(l)}}{l,1} = \\tenmatgen{\\ten{H}^{(l)} }{l,1} + \\gamma\\mat{X}^\\top(\\Ymat - \\mat{X}\\tenmatgen{\\ten{H}^{(l)} }{l,1})$\n  \n   \\STATE $\\ten{H}^{(l)} = \\texttt{project}(\\ten{H}^{(l)},R)$~(using either SVD for IHT or TT-SVD for TIHT)\n  \n   \\UNTIL{convergence}\\label{alg.line.iht.end}\n   \\ENDIF\\label{alg.lastline.forloop} \n  \n   \\ENDFOR\n   \\STATE\\label{alg.line.svd} Let $\\tenmatgen{\\ten{H}^{(2L)}}{L,L+1} = \\P\\S$ be a rank $R$ factorization.\n   \\STATE Return the linear 2-RNN $(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$ where \n   \\begin{align*}\n    \\vecs{\\szerosymbol}\\ &= (\\S^\\dagger)^\\top\\tenmatgen{\\ten{H}^{(L)}_f}{L+1},\\ \\ \\ \\ \\vecs{\\vvsinfsymbol}^\\top = \\P^\\dagger\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}\\\\\n    \\Aten\\ &= (\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1})\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top\n\\end{align*}  \n\\end{algorithmic}\n\\end{algorithm}\n\nA few remarks on this theorem are in order.   The first observation is that the $3$\ndatasets $D_L$, $D_{2L}$ and $D_{2L+1}$ can either be drawn independently or not~(e.g.\\  the sequences in $D_{L}$ can\nbe prefixes of the sequences in $D_{2L}$ but it is not necessary). In particular, the result still holds when  the datasets $D_l$ are constructed from a unique dataset \n$S =\\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_T^{(i)}),(\\vec{y}^{(i)}_1,\\vec{y}^{(i)}_2,\\cdots,\\vec{y}^{(i)}_T)) \\}_{i=1}^{N}$\nof input\/output sequences with $T\\geq 2L+1$, where $\\vec{y}^{(i)}_t = f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_t^{(i)})$ for any $t\\in[T]$.\nObserve that having access to such input\/output training sequences is not an unrealistic assumption: for example when training RNNs for\nlanguage modeling the output $\\vec{y}_t$ is the  conditional probability vector of the next symbol, and  for classification tasks the output is\nthe one-hot encoded label for all time steps. Lastly, when the outputs $\\vec{y}^{(i)}$ are noisy, one can solve the least-squares problem\n$\\norm{\\Ymat - \\mat{X}\\tenmatgen{\\ten{H}^{(l)}}{l,1}}^2_F$ to approximate the Hankel tensors; we will empirically evaluate this approach \nin Section~\\ref{sec:xp} and we defer its theoretical analysis  in the noisy setting to future work.\n\n\n\n\\subsection{\\rev{Leveraging the low rank structure of the Hankel tensors}}\nWhile the least-squares method is sufficient to obtain the theoretical guarantees of Theorem~\\ref{thm:learning-2RNN}, it does not leverage\nthe low rank structure of the Hankel tensors $\\ten{H}^{(L)}$, $\\ten{H}^{(2L)}$ and $\\ten{H}^{(2L+1)}$. We now  propose three alternative recovery\nmethods to leverage this structure, whose  sample efficiency will be assessed in a simulation study in Section~\\ref{sec:xp}~(deriving improved sample\ncomplexity guarantees using these methods is left for future work). In the noiseless setting, we first propose to replace solving the linear \nsystem $\\Ymat = \\mat{X}\\tenmatgen{\\ten{H}^{(l)}}{l,1}$ with a nuclear norm minimization problem~(see line~\\ref{alg.line.nucnorm} of Algorithm~\\ref{alg:2RNN-SL}), thus leveraging the\nfact that $\\tenmatgen{\\ten{H}^{(l)}}{\\ceil{l\/2},l-\\ceil{l\/2} + 1}$ is potentially of low matrix rank. We also propose to use iterative hard thresholding~(IHT)~\\cite{jain2010guaranteed}\nand its tensor counterpart TIHT~\\cite{rauhut2017low}, which are based on the classical projected gradient descent algorithm and have shown to be\nrobust to noise in practice. These two methods are implemented in lines~\\ref{alg.line.iht.start}-\\ref{alg.line.iht.end} of Algorithm~\\ref{alg:2RNN-SL}. There,\nthe \\texttt{project} method either projects $\\tenmatgen{\\ten{H}^{(l)}}{\\ceil{l\/2},l-\\ceil{l\/2} + 1}$ onto the manifold of low rank matrices \nusing SVD~(IHT) or projects  $\\ten{H}^{(l)}$ onto the manifold of tensors with TT-rank $R$~(TIHT). \n\n\\input{xp_figures.tex}\n\n\n\\rev{\nThe low rank structure of the Hankel tensors can also be leveraged to improve the scalability of the learning algorithm.\nOne can check that the computational complexity of Algorithm~\\ref{alg:2RNN-SL} is exponential in the maximum sequence length: indeed,\nbuilding the matrix $\\mat{X}$ in line~2 is already in $\\bigo{N_ld^l}$, where $l$ is in turn equal to $L,\\ 2L$ and $2L+1$.\nFocusing on the TIHT recovery method, a careful analysis shows that the computational complexity of the algorithm\nis in \n$$\\bigo{d^{2L+1}\\left(p(TN+R) +R^2\\right) + TL\\max(p,d)^{2L+3}}, $$\nwhere $N=\\max(N_L,N_{2L},N_{2L+1})$ and $T$ is the number of iterations of the loop on line~\\ref{alg.line.iht.start}.\nThus, in its present form, our approach cannot scale to high dimensional inputs and long sequences. However, one can  leverage\nthe low tensor train rank structure of the Hankel tensors to circumvent this issue:\nby storing \nboth the estimates of the Hankel tensors $\\ten{H}^{(l)}$ and the matrices $\\mat{X}$ in TT format~(with decompositions of ranks $R$ and $N$ respectively),\nall the operations needed to implement Algorithm~\\ref{alg:2RNN-SL} with the TIHT recovery method can be performed in time $\\bigo{T(N+R)^3(Ld + p)}$~(more details can be found in the supplementary\nmaterial). By leveraging the tensor train structure, one can thus lift the dependency on $d^{2L+1}$ by paying the price of an increased cubic complexity \nin the number of examples $N$ and the number of states $R$. While the\ndependency on the number of states is not a major issue~($R$ should be negligible w.r.t. $N$), the dependency on $N^3$ can quickly become prohibitive for realistic application scenario. \nFortunately, this issue can  be\ndealt with by using mini-batches of training data for the gradient updates on line~\\ref{alg.line.iht.gradient} instead of the whole dataset $D_l$, in which case the overall complexity\nof Algorithm~\\ref{alg:2RNN-SL} becomes $\\bigo{T(M+R)^3(Ld + p)}$ where $M$ is the mini-batch size~(the overall algorithm in TT format is summarized in Algorithm~\\ref{alg:2RNN-SL-TT} in the supplementary material).\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\input{xp}\n\n\n\n\n\n\\section{Conclusion and Future Directions}\n\nWe proposed the first provable learning algorithm for second-order RNNs with linear activation functions:\nwe showed that linear 2-RNNs are a natural extension of vv-WFAs to the setting of input sequences of \\emph{continuous vectors}~(rather than\ndiscrete symbol) and we extended the vv-WFA spectral learning\nalgorithm to this setting. We believe that the results presented in this paper open a number of exciting and promising research directions on both the\ntheoretical and practical perspectives. We first plan to use the spectral learning estimate as a starting point for\ngradient based methods to train non-linear 2-RNNs. More precisely, linear 2-RNNs can be thought of as 2-RNNs using LeakyRelu activation functions with negative slope $1$, therefore one could use \na linear 2-RNN as initialization before gradually reducing the negative slope parameter during training. The extension of the spectral method to linear 2-RNNs also\nopens the door to scaling up the classical spectral algorithm to problems with large discrete alphabets~(which is a known caveat of the spectral algorithm for WFAs) since\nit allows one to use low dimensional embeddings of large vocabularies~(using e.g.\\  word2vec or latent semantic analysis). From the theoretical perspective, we plan on \nderiving learning guarantees for  linear 2-RNNs in the noisy setting~(e.g.\\  using the PAC learnability framework). Even though it is intuitive that such guarantees should hold~(given\nthe continuity of all operations used in our algorithm), we believe that such an analysis may entail results of independent interest. In particular, analogously to the\nmatrix case  studied in~\\cite{cai2015rop}, obtaining rate optimal convergence rates for the recovery of the low TT-rank Hankel tensors from rank one measurements is an interesting\ndirection; such a result could for example allow one to improve the generalization bounds provided in~\\cite{balle2012spectral} for spectral learning of \ngeneral WFAs. \n\n\n\n\\newpage\n\\subsubsection*{Acknowledgements} This work was done while G. Rabusseau was an IVADO postdoctoral scholar at McGill University. \n{\n\\bibliographystyle{plain}\n\n\n\\section*{\\centering \\LARGE Connecting Weighted Automata and Recurrent Neural Networks through Spectral Learning \\\\ \\vspace*{0.3cm}(Supplementary Material)}\n\n\n\\section{Proofs}\n\\subsection{Proof of Theorem~\\ref{thm:2RNN-vvWFA}}\n\\begin{theorem*}\nAny function that can be computed by a vv-WFA with $n$ states can be computed by a linear 2-RNN with $n$ hidden units.\nConversely, any function that can be computed by a linear 2-RNN with $n$ hidden units on sequences of one-hot vectors~(i.e.\\  canonical basis \nvectors) can be computed by a WFA with $n$ states.\n\n\nMore precisely, the WFA $A=\\vvwa$ with $n$ states and the linear 2-RNN $M=(\\vecs{\\szerosymbol},\\Aten,\\vecs{\\vvsinfsymbol})$ with\n$n$ hidden units, where $\\Aten\\in\\Rbb^{n\\times \\Sigma \\times n}$ is defined by $\\Aten_{:,\\sigma,:}=\\mat{A}^\\sigma$ for all $\\sigma\\in\\Sigma$, are\nsuch that\n$f_A(\\sigma_1\\sigma_2\\cdots\\sigma_k) = f_M(\\vec{x}_1,\\vec{x}_2,\\cdots,\\vec{x}_k)$ for all sequences of input symbols $\\sigma_1,\\cdots,\\sigma_k\\in\\Sigma$,\nwhere for each $i\\in[k]$ the input vector $\\vec{x}_i\\in\\Rbb^\\Sigma$ is\nthe one-hot encoding of the symbol $\\sigma_i$.\n\\end{theorem*}\n\\begin{proof}\nWe first show by induction on $k$ that, for any sequence $\\sigma_1\\cdots\\sigma_k\\in\\Sigma^*$, the hidden state $\\vec{h}_k$ computed by $M$~(see\nEq.~\\eqref{eq:2RNN.definition})\non the corresponding one-hot encoded sequence $\\vec{x}_1,\\cdots,\\vec{x}_k\\in\\Rbb^d$\nsatisfies $\\vec{h}_k = (\\mat{A}^{\\sigma_1}\\cdots\\mat{A}^{\\sigma_k})^\\top\\vecs{\\szerosymbol}$. The case $k=0$\nis immediate. Suppose the result true for sequences of length up to $k$. One can check easily check that $\\Aten\\ttv{2}\\vec{x}_i = \\mat{A}^{\\sigma_i}$\nfor any index $i$. Using the induction hypothesis it then follows that\n\\begin{align*}\n\\vec{h}_{k+1} &= \\Aten \\ttv{1}\\vec{h}_k \\ttv{2} \\vec{x}_{k+1} = \\mat{A}^{\\sigma_{k+1}}\\ttv{1} \\vec{h}_k = (\\mat{A}^{\\sigma_{k+1}})^\\top \\vec{h}_k\\\\\n&= (\\mat{A}^{\\sigma_{k+1}})^\\top (\\mat{A}^{\\sigma_1}\\cdots\\mat{A}^{\\sigma_k})^\\top\\vecs{\\szerosymbol} = (\\mat{A}^{\\sigma_1}\\cdots\\mat{A}^{\\sigma_{k+1}})^\\top\\vecs{\\szerosymbol} .\n\\end{align*} \nTo conclude, we thus have\n\\begin{equation*}\nf_M(\\vec{x}_1,\\vec{x}_2,\\cdots,\\vec{x}_k) = \\vecs{\\vvsinfsymbol}\\vec{h}_{k} = \\vecs{\\vvsinfsymbol}(\\mat{A}^{\\sigma_1}\\cdots\\mat{A}^{\\sigma_{k}})^\\top\\vecs{\\szerosymbol} = f_A(\\sigma_1\\sigma_2\\cdots\\sigma_k).\\qedhere\n\\end{equation*}\n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{thm:2RNN-SL}}\n\n\n\\begin{theorem*}\nLet $f:(\\Rbb^d)^*\\to \\Rbb^p$ be a function computed by a minimal linear $2$-RNN  with $n$ hidden units and let\n$L$ be an integer such that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$.\n\nThen, for any $\\P\\in\\Rbb^{d^L\\times n}$ and $\\S\\in\\Rbb^{n\\times d^Lp}$ such that $\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1} = \\P\\S$, the\nlinear 2-RNN $M=(\\vecs{\\szerosymbol},\\Aten,\\vecs{\\vvsinfsymbol})$ defined by\n$$\\vecs{\\szerosymbol} = (\\S^\\dagger)^\\top\\tenmatgen{\\ten{H}^{(L)}_f}{L+1},\\ \\ \\ \\ \\Aten = (\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1})\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top,\\ \\ \\ \\ \n\\vecs{\\vvsinfsymbol}^\\top = \\P^\\dagger\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}$$\nis a minimal linear $2$-RNN computing $f$.\n\\end{theorem*}\n\\begin{proof}\nLet $\\P\\in\\Rbb^{d^L\\times n}$ and $\\S\\in\\Rbb^{n\\times d^Lp}$ be such that $\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1} = \\P\\S$\nDefine the tensors \n$$\\Pten^* = \\TT{\\Aten^\\star\\ttv{1}\\vecs{\\szerosymbol}^\\star, \\underbrace{\\Aten^\\star, \\cdots, \\Aten^\\star}_{L-1\\text{ times}}, \\mat{I}_n}\\in\\Rbb^{d\\times\\cdots\\times d\\times n}\\ \\ \\ \\  \n\\text{ and }\\ \\ \\ \\ \n\\Sten^* = \\TT{\\mat{I}_n,\\underbrace{\\Aten^\\star, \\cdots, \\Aten^\\star}_{L\\text{ times}}, \\vecs{\\vvsinfsymbol}^\\star}\\in\\Rbb^{n\\times d\\times\\cdots\\times d\\times p}$$ \nof order $L+1$ and $L+2$ respectively, and let $\\P^\\star = \\tenmatgen{\\Pten^*}{l,1} \\in\\Rbb^{d^l\\times n}$ \nand $\\S = \\tenmatgen{\\Sten^*}{1,L+1}  \\in\\Rbb^{n\\times d^lp}$. Using the identity $\\ten{H}^{(j)}_f = \\TT{\\Aten\\ttv{1}\\vecs{\\szerosymbol}, \\underbrace{\\Aten, \\cdots, \\Aten}_{j-1\\text{ times}}, \\vecs{\\vvsinfsymbol}^\\top}$ for\nany $j$, one can easily check the following identities:\n\\begin{gather*}\n\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1} = \\P^\\star\\S^\\star,\\ \\  \\ \\ \\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1}= \\Aten^\\star \\ttm{1} \\P^\\star \\ttm{3} (\\S^\\star)^\\top,\\\\\n\\tenmatgen{\\ten{H}^{(L)}_f}{L,1} = \\P^\\star(\\vecs{\\vvsinfsymbol}^\\star)^\\top, \\ \\ \\ \\ \\ \\ \\\n\\tenmatgen{\\ten{H}^{(L)}_f}{L+1} = (\\S^\\star)^\\top\\vecs{\\szerosymbol}.\n\\end{gather*}\n\nLet $\\mat{M} = \\P^\\dagger\\P^\\star$. We will show that $\\vecs{\\szerosymbol} = \\mat{M}^{-\\top}\\vecs{\\szerosymbol}^\\star$, $\\Aten = \\Aten^\\star \\ttm{1}\\mat{M}\\ttm{3}\\mat{M}^{-\\top}$ and\n$\\vecs{\\vvsinfsymbol} = \\mat{M}\\vecs{\\vvsinfsymbol}^\\star$, which will entail the results since linear 2-RNN are invariant under change of basis~(see Section~\\ref{sec:prelim}). First observe that $\\mat{M}^{-1} = \\S^\\star\\S^\\dagger$. Indeed,\nwe have\n$\\P^\\dagger\\P^\\star\\S^\\star\\S^\\dagger = \\P^\\dagger\\tenmatgen{\\ten{H}^{(2l)}_f}{l,l+1}\\S^\\dagger = \\P^\\dagger\\P\\S\\S^\\dagger = \\mat{I}$\nwhere we used the fact that $\\P$~(resp. $\\S$) is of full column rank~(resp. row rank) for the last equality. \n\nThe following derivations then follow from basic tensor algebra:\n\\begin{align*}\n\\vecs{\\szerosymbol} \n&= \n(\\S^\\dagger)^\\top\\tenmatgen{\\ten{H}^{(L)}_f}{L+1} \n=\n(\\S^\\dagger)^\\top (\\S^\\star)^\\top\\vecs{\\szerosymbol}\n=\n(\\S^\\star\\S^\\dagger)^\\top\n=\n\\mat{M}^{-\\top}\\vecs{\\szerosymbol}^\\star,\\\\\n\\ \\\\\n\\Aten \n&= \n(\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1})\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top\\\\\n&=\n(\\Aten^\\star \\ttm{1} \\P^\\star \\ttm{3} (\\S^\\star)^\\top)\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top\\\\\n&=\n\\Aten^\\star \\ttm{1} \\P^\\dagger\\P^\\star \\ttm{3} (\\S^\\star\\S^\\dagger)^\\top =  \\Aten^\\star \\ttm{1}\\mat{M}\\ttm{3}\\mat{M}^{-\\top},\\\\\n\\ \\\\\n\\vecs{\\vvsinfsymbol}^\\top \n&= \n\\P^\\dagger\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}\n=\n\\P^\\dagger\\P^\\star(\\vecs{\\vvsinfsymbol}^\\star)^\\top\n=\n\\mat{M}\\vecs{\\vvsinfsymbol}^\\star,\n\\end{align*}\nwhich concludes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:learning-2RNN}}\n\n\n\\begin{theorem*}\nLet $(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$ be a minimal linear 2-RNN with $n$ hidden units computing a function $f:(\\Rbb^d)^*\\to \\Rbb^p$, and let $L$ be an integer\\footnote{Note that the theorem can be adapted if such an integer $L$ does not exists~(see supplementary material).}\nsuch that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$.\n\nSuppose we have access to $3$ datasets\n$D_l = \\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)}) \\}_{i=1}^{N_l}\\subset(\\Rbb^d)^l\\times \\Rbb^p$ for $l\\in\\{L,2L,2L+1\\}$ \nwhere the entries of each $\\vec{x}^{(i)}_j$ are drawn independently from the standard normal distribution and where each\n$\\vec{y}^{(i)} = f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)})$.\n\nThen, whenever $N_l \\geq d^l$ for each $l\\in\\{L,2L,2L+1\\}$,\nthe linear 2-RNN $M$ returned by Algorithm~\\ref{alg:2RNN-SL} with the least-squares method satisfies $f_M = f$ with probability one.\n\\end{theorem*}\n\\begin{proof}\nWe just need to show for each $l\\in \\{L,2L,2L+1\\}$ that, under the hypothesis of the Theorem, the Hankel tensors $\\hat{\\ten{H}}^{(l)}$ computed in line~\\ref{alg.line.lst-sq} of\nAlgorithm~\\ref{alg:2RNN-SL} are equal to the true Hankel tensors $\\ten{H}^{(l)}$ with probability one. Recall that these tensors are computed by solving the least-squares\nproblem\n$$\\hat{\\ten{H}}^{(l)} = \\argmin_{T\\in \\Rbb^{d\\times\\cdots\\times d\\times p}} \\norm{\\mat{X}\\tenmatgen{\\ten{T}}{l,1} - \\Ymat}_F^2$$\nwhere $\\mat{X}\\in\\Rbb^{N_l\\times d_l}$ is the matrix  with rows $\\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)}$ for each $i\\in[N_l]$. Since  $\\mat{X}\\tenmatgen{\\ten{H}^{(l)}}{l,1} = \\Ymat$ and since the solution\nof the least-squares problem is unique as soon as $\\mat{X}$ is of full column rank, we just need to show that this is the case with probability one\nwhen the entries of the vectors $\\vec{x}^{(i)}_j$ are drawn at random from a standard normal distribution. The result will  then directly follow\nby applying Theorem~\\ref{thm:2RNN-SL}.\n\nWe will show that the set \n$$\\Scal = \\{ (\\vec{x}_1^{(i)},\\cdots, \\vec{x}_l^{(i)}) \\mid \\ i\\in[N_l],\\ dim(span(\\{ \\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)} \\})) < d^l\\} $$\nhas Lebesgue measure $0$ in $((\\Rbb^d)^{l})^{N_l}\\simeq \\Rbb^{dlN_l}$ as soon as $N_l \\geq d^l$, which will imply that it has probability $0$ under any continuous probability, hence\nthe result. For any  $S=\\{(\\vec{x}_1^{(i)},\\cdots, \\vec{x}_l^{(i)})\\}_{i=1}^{N_l}$, we denote  by $\\mat{X}_S\\in\\Rbb^{N_l\\times d^l}$  the matrix  with rows $\\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)}$.\nOne can easily check that $S\\in\\Scal$ if and only if $\\mat{X}_S$ is of rank strictly less than $d^l$, which is equivalent to the determinant of \n$\\mat{X}_S^\\top\\mat{X}_S$ being equal to $0$. Since this determinant is a polynomial in the entries of the vectors $\\vec{x}_j^{(i)}$, $\\Scal$ is an algebraic\nsubvariety of $\\Rbb^{dlN_l}$.\nIt is then easy to check that the polynomial $det(\\mat{X}_S^\\top\\mat{X}_S)$ is not uniformly 0 when $N_l \\geq d^l$. Indeed, \nit suffices to choose the vectors $\\vec{x}_j^{(i)}$ such that the family  $(\\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)})_{n=1}^{N_l}$ spans the whole space \n$\\Rbb^{d^l}$~(which is possible since we can choose arbitrarily any of the $N_l\\geq d^l$ elements of this family),  hence the result. \nIn conclusion, $\\Scal$ is a proper algebraic subvariety of $\\Rbb^{dlN_l}$ and hence has Lebesgue\nmeasure zero~\\cite[Section 2.6.5]{federer2014geometric}.\n\n\n\n\\end{proof}\n\n\n\\section{Lifting the simplifying assumption}\\label{app:lift}\nWe now show how all our results still hold when there does not exist an $L$ such that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$.\nRecall that this simplifying assumption followed from assuming that the sets $\\Pcal=\\Scal=[d]^L$ form a complete basis for the function\n$\\tilde{f}:[d]^*\\to \\Rbb^p$ defined by $\\tilde{f}(i_1i_2\\cdots i_k) = f(\\ten_{i_1},\\ten_{i_2},\\cdots,\\ten_{i_k})$. \nWe first show that there always exists an integer $L$ such that $\\Pcal=\\Scal=\\cup_{i\\leq L} [d]^i$ forms a complete basis for $\\tilde{f}$.\n Let $M = (\\vecs{\\szerosymbol}^\\star,\\Aten^\\star,\\vecs{\\vvsinfsymbol}^\\star)$  be a linear 2-RNN with $n$ hidden units computing\n$f$~(i.e.\\  such that $f_M=f$). It follows from Theorem~\\ref{thm:2RNN-vvWFA} and from the discussion  at the beginning of Section~\\ref{subsec:SL-2RNN}\nthat there exists a vv-WFA computing $\\tilde{f}$ and it is easy to check that $\\rank(\\tilde{f}) = n$.  This implies $\\rank(\\tenmatpar{\\ten{H}_f}{1}) = n$ by\nTheorem~\\ref{thm:fliess-vvWFA}. Since  $\\Pcal=\\Scal=\\cup_{i\\leq l} [d]^i$ converges to $[d]^*$ as $l$ grows to infinity, there exists an $L$ such that \nthe finite sub-block $\\tilde{\\ten{H}}_f \\in \\Rbb^{\\Pcal\\times\\Scal\\times p}$ of $\\ten{H}_f\\in \\Rbb^{[d]^*\\times[d]^*\\times p}$\nsatisfies $\\rank(\\tenmatpar{\\tilde{\\ten{H}}_f}{1}) = n$, i.e.\\  such that\n$\\Pcal=\\Scal=\\cup_{i\\leq L} [d]^i$ forms a complete basis for $\\tilde{f}$. \n\nNow consider the finite sub-blocks $\\tilde{\\ten{H}}^{+}_f\\in \\Rbb^{\\Pcal \\times [d] \\times \\Scal \\times p}$ and  $\\tilde{\\H}^{-}_f\\in \\Rbb^{\\Pcal  \\times p}$ of $\\ten{H}_f$  defined by\n$$(\\tilde{\\ten{H}}^{+}_f)_{u,i,v,:}=\\tilde{f}(uiv),\\ \\ \\ \\text{and} (\\tilde{\\H}^{-}_f)_{u,:}=f(u)$$\nfor any $u\\in \\Pcal=\\Scal$ and any $i\\in [d]$. One can check that Theorem~\\ref{thm:2RNN-SL} holds by replacing \\emph{mutatis mutandi}\n$\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}$ by $\\tenmatpar{\\tilde{\\ten{H}}_f}{1}$, $\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1}$ by  $\\tilde{\\ten{H}}^{+}_f$, $\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}$ by $\\tilde{\\H}^{-}_f$\nand $\\tenmatgen{\\ten{H}^{(L)}_f}{L+1}$ by $\\vectorize{\\tilde{\\H}^{-}_f}$.\n\nTo conclude, it suffices to observe that both $\\tilde{\\ten{H}}^{+}_f$ and  $\\tilde{\\H}^{-}_f$ can be constructed\nfrom the entries for the tensors $\\ten{H}^{(l)}$ for $1\\leq l \\leq 2L+1$, which can be recovered~(or estimated in the noisy setting) using\nthe techniques described in Section~\\ref{subsec:Hankel.tensor.recovery}~(corresponding to lines \\ref{alg.firstline.forloop}-\\ref{alg.lastline.forloop} of\nAlgorithm~\\ref{alg:2RNN-SL}).\n\nWe thus showed that linear 2-RNNs can be provably learned even when  there does not exist an $L$ such that $\\rank(\\tenmatgen{\\ten{H}^{(2L)}_f}{L,L+1}) = n$. In this\nsetting, one needs to estimate enough of the tensors $\\ten{H}^{(l)}$ to reconstruct a complete sub-block $\\tilde{\\ten{H}}_f$ of the Hankel tensor \n$\\ten{H}$~(along with the corresponding tensor $\\tilde{\\ten{H}}^{+}_f$ and matrix $\\tilde{\\H}^{-}_f$)\nand recover the linear 2-RNN by applying Theorem~\\ref{thm:2RNN-SL}. In addition, one needs to have access to sufficiently large datasets \n$D_l$ for each $l\\in [2L+1]$ rather than only the three datasets mentioned in Theorem~\\ref{thm:learning-2RNN}.  However the data requirement remains the\nsame in the case where we assume that each of the datasets $D_l$ is constructed from a unique training  dataset \n$S =\\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_T^{(i)}),(\\vec{y}^{(i)}_1,\\vec{y}^{(i)}_2,\\cdots,\\vec{y}^{(i)}_T)) \\}_{i=1}^{N}$\nof input\/output sequences.\n\n\\section{Leveraging the tensor train structure for computational efficiency}\nThe overall learning algorithm using the TIHT recovery method in TT format is summarized in Algorithm~\\ref{alg:2RNN-SL-TT}. The key ingredients to improve the complexity of Algorithm~\\ref{alg:2RNN-SL} are (i) to estimate the gradient using mini-batches of data and  (ii) to directly use the TT format to represent and perform operations on the tensors $\\ten{H}^{(l)}$ and the tensors $\\Xten^{(l)}\\in\\Rbb^{M\\times d\\times \\cdots \\times d}$ defined by\n\\begin{equation}\\label{eq:Xten}\n \\Xten_{i,:,\\cdots,:}= \\vec{x}^{(i)}_1\\otimes\\vec{x}_2^{(i)}\\otimes\\cdots\\otimes\\vec{x}_l^{(i)}\\ \\  \\ \\text{for }i\\in[M]\n\\end{equation}\nwhere $M$ is the size of a mini-batch of training data~($\\ten{H}^{(l)}$ is of TT-rank $R$ by design and it can easily be shown that $\\Xten^{(l)}$ is of TT-rank at most $M$, cf. Eq.~\\eqref{eq:XtenTT}). \nThen, all the operations of the algorithm can be expressed in terms of these tensors and performed efficiently in TT format.\nMore precisely, the products and sums needed to compute the gradient update on line~\\ref{algtt.line.iht.gradient} can be performed in~$\\bigo{(R+M)^2(ld+p)+(R+M)^3d}$. After the gradient update, the tensor $\\ten{H}^{(l)}$ has TT-rank at most $(M+R)$ but can be efficiently projected back to a tensor of TT-rank $R$ using the tensor train rounding operation~\\cite{oseledets2011tensor} in $\\bigo{(R+M)^3(ld+p)}$~(which is the operation dominating the complexity of the whole algorithm).\nThe subsequent operations on line~\\ref{line.algtt.return} can be performed efficiently in the TT format in~$\\bigo{R^3d+R^2p}$~(using the method described in~\\cite{klus2018tensor} to compute the pseudo-inverses of the matrices $\\P$ and $\\S$). \nThe overall complexity of Algorithm~2 is thus in~$\\bigo{T(R+M)^3(Ld+p)}$ where $T$ is the number of iterations of the inner loop.\n\n\n\\begin{algorithm}\n   \\caption{\\texttt{2RNN-SL-TT}: Spectral Learning of linear 2-RNNs  \\textbf{in tensor train format}}\n   \\label{alg:2RNN-SL-TT}\n\\begin{algorithmic}[1]\n   \\REQUIRE Three training datasets $D_L,D_{2L},D_{2L+1}$ with input sequences of length $L$, $2L$ and $2L+1$ respectively, rank $R$, learning rate $\\gamma$ and\n   mini-batch size $M$.\n  \n   \\FOR{$l\\in\\{L,2L,2L+1\\}$}\n   \\STATE Initialize all cores of the rank $R$ TT-decomposition $\\ten{H}^{(l)} = \\TT{\\Gten^{(l)}_1,\\cdots,\\Gten^{(l)}_{l+1}} \\in \\Rbb^{d\\times\\cdots\\times d\\times p}$ to $\\mat{0}$.\\\\\n   \/\/ \\emph{Note that all the updates of  $\\ten{H}^{(l)}$ stated below are in effect applied directly to the core tensors $\\Gten^{(l)}_k$, i.e. the tensor $\\ten{H}^{(l)}$ is never \n   explicitely constructed.}\n   \n   \\REPEAT\\label{algtt.line.iht.start}\n   \\STATE Subsample a minibatch $$\\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)}) \\}_{i=1}^{M}\\subset(\\Rbb^d)^l\\times \\Rbb^p$$ of size $M$ from $D_l$.\n   \\STATE Compute the rank $M$ TT-decomposition of the tensor $\\Xten=\\Xten^{(l)}$~(defined in Eq.~\\eqref{eq:Xten}), which is given by\n   \\begin{equation}\\label{eq:XtenTT}\n       \\Xten = \\TT{\\mat{I}_M, \\Aten_1,\\cdots,\\Aten_l} \\text{ where the cores are defined by } (\\Aten_k)_{i,:,j}=\\delta_{ij}\\vec{x}^{(i)}_k \\ \\ \\text{ and } \\ \\  (\\Aten_l)_{i,:} = \\vec{x}^{(i)}_k\n   \\end{equation}\n   for all $1\\leq k <l$, $i,j\\in[M]$, where $\\delta$ is the Kroencker symbol.\n   \\STATE\\label{algtt.line.iht.gradient} Perform the gradient update using efficient addition and product operations in TT format~(see~\\cite{oseledets2011tensor}):\n   $$\\tenmatgen{\\ten{H}^{(l)}}{l,1} = \\tenmatgen{\\ten{H}^{(l)} }{l,1} + \\gamma\\tenmatgen{\\Xten}{1,l}^\\top(\\Ymat - \\tenmatgen{\\Xten}{1,l}\\tenmatgen{\\ten{H}^{(l)} }{l,1})$$\n   \\STATE Project the Hankel tensor $\\ten{H}^{(l)}$~(which is now of rank at most $R+M$) back onto the manifold of tensor of TT-rank $R$ using the TT rounding\n   operation~(see again~\\cite{oseledets2011tensor}):\n   $$\\ten{H}^{(l)} = \\texttt{TT-rounding}(\\ten{H}^{(l)},R)$$\n  \n   \\UNTIL{convergence}\\label{algtt.line.iht.end}\n  \n   \\ENDFOR\n   \\STATE\\label{algtt.line.svd} Let $\\P = \\tenmatgen{\\TT{\\Gten^{(2L)}_1,\\cdots,\\Gten^{(2L)}_L,\\mat{I}_R}}{L,1}$ and\n   $\\S = \\tenmatgen{\\TT{\\mat{I}_R,\\Gten^{(2L)}_{L+1},\\cdots,\\Gten^{(2L)}_{2L+1}}}{1,L+1}$~(observe that\n   $\\tenmatgen{\\ten{H}^{(2L)}}{L,L+1} = \\P\\S$ is a rank $R$ factorization).\n   \\STATE\\label{line.algtt.return} Return the linear 2-RNN $(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$ where \n   \\begin{align*}\n    \\vecs{\\szerosymbol}\\ &= (\\S^\\dagger)^\\top\\tenmatgen{\\ten{H}^{(L)}_f}{L+1}\\\\\n    \\Aten\\ &= (\\tenmatgen{\\ten{H}^{(2L+1)}_f}{L,1,L+1})\\ttm{1}\\P^\\dagger\\ttm{3}(\\S^\\dagger)^\\top\\\\\n\\vecs{\\vvsinfsymbol}^\\top &= \\P^\\dagger\\tenmatgen{\\ten{H}^{(L)}_f}{L,1}\n\\end{align*}  \nby performing efficient computations in TT format for the products~\\cite{oseledets2011tensor} and pseudo-inverses~(see e.g. \\cite{klus2018tensor}).\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Real Data Experiment on Wind Speed Prediction}\\label{app:real}\nBesides the synthetic data experiments we showed in the paper, we have also conducted experiments on real data. The data that we use for the these experiments is from TUDelft\\footnote{http:\/\/weather.tudelft.nl\/csv\/}. Specifically, we use the data from Rijnhaven station as described in~\\cite{lin2016short}, which proposed a regression automata model and performed various experiments on the dataset we mentioned above. The data contains wind speed and related information at the Rijnhaven station from  2013-04-22 at 14:55:00 to 2018-10-20 at 11:40:00 and was collected every five minutes. To compare to the results in~\\cite{lin2016short}, we strictly followed the data preprocessing procedure described in the paper. We use the data from 2013-04-23 to 2015-10-12 as training data and the rest as our testing data.  The paper uses SAX as a preprocessing method to discretize the data. However, as there is no need to discretize data for our algorithm, we did not perform this procedure. For our method, we set the length $L = 3$ and we use the general algorithm described in Appendix~\\ref{app:lift}. We calculate hourly averages of the wind speed, and predict one\/three\/six hour(s) ahead, as in~\\cite{lin2016short}.  For our methods we use a linear 2-RNN with 10 states. Averages over 5 runs of this experiment for one-hour-ahead, three-hour-ahead, six-hour-ahead prediction error can be found in Table~\\ref{one_hour}, \\ref{three_hours} and \\ref{six_hours}. The results for RA, RNN and persistence are taken directly from~\\cite{lin2016short}. We see that while TIHT+SGD performs slightly worse than ARIMA and RA for one-hour-ahead prediction, it outperforms all other methods for three-hours and six-hours ahead predictions~(and the superiority w.r.t. other methods increases as the prediction horizon gets longer). \n\n\n\n\\begin{table}[h]\n\\caption{One-hour-ahead Speed Prediction Performance Comparisons}\n\\centering\n\\begin{tabular}{c|cccccc}\nMethod & TIHT     & TIHT+SGD & \\begin{tabular}[c]{@{}c@{}}Regression \\\\ Automata\\end{tabular} & ARIMA & RNN & Persistence\\\\ \\hline\nRMSE   & 0.573  & 0.519   & 0.500    & \\textbf{0.496 }& 0.606 & 0.508                                                    \\\\\nMAPE   & 21.35  & 18.79    & \\textbf{18.58}   & 18.74 & 24.48 & 18.61                                                    \\\\\nMAE    & 0.412  & 0.376    & 0.363    & \\textbf{0.361} & 0.471 & 0.367                                                    \n\\end{tabular}%\n\\label{one_hour}\n\\end{table}\n\\begin{table}[h]\n\\caption{Three-hour-ahead Speed Prediction Performance Comparisons}\n\\centering\n\\begin{tabular}{c|cccccc}\nMethod & TIHT     & TIHT+SGD & \\begin{tabular}[c]{@{}c@{}}Regression \\\\ Automata\\end{tabular} & ARIMA & RNN & Persistence\\\\ \\hline\nRMSE   & 0.868  & \\textbf{0.854}    & 0.872  & 0.882 & 1.002 & 0.893                                                       \\\\\nMAPE   & 33.98  & \\textbf{31.70}    & 32.52 & 33.165 & 37.24 & 33.29                                                      \\\\\nMAE    & 0.632  & \\textbf{0.624}    & 0.632 &0.642 & 0.764 & 0.649                                                       \n\\end{tabular}\n\\label{three_hours}\n\\end{table}\n\\begin{table}[h]\n\\caption{Six-hour-ahead Speed Prediction Performance Comparisons}\n\\centering\n\\begin{tabular}{c|cccccc}\nMethod & TIHT   & TIHT+SGD & \\begin{tabular}[c]{@{}c@{}}Regression \\\\ Automata\\end{tabular} & ARIMA & RNN & Persistence\\\\ \\hline\nRMSE   & 1.234  & \\textbf{1.145}    & 1.205  & 1.227 & 1.261 & 1.234                                                      \\\\\nMAPE   & 49.08  & \\textbf{44.88}    & 46.809  &48.02 & 47.03 & 48.11                                                      \\\\\nMAE    & 0.940  & \\textbf{0.865}    & 0.898  & 0.919 & 0.944 & 0.923            \n\\end{tabular}\n\\label{six_hours}\n\\end{table}\n\n\n\n\n\n\n\\section{Experiments}\\label{sec:xp}\n\n\n\n\n\n\n\\rev{\nIn this section, we perform experiments\\footnote{\\url{https:\/\/github.com\/litianyu1993\/learning_2RNN}} on two toy examples to compare how the choice of the recovery method~(\\texttt{LeastSquares}, \\texttt{NuclearNorm}, \\texttt{IHT} and \\texttt{TIHT}) affects the sample efficiency of Algorithm~\\ref{alg:2RNN-SL}. Additional experiments on real data can be found in Appendix~\\ref{app:real}.\nWe also include comparisons with RNNs with long short term memory~(LSTM) units~\\cite{hochreiter1997long} and report the performances\nobtained by  refining  the solution returned by our algorithm~(with the TIHT recovery method) using stochastic gradient descent~(\\texttt{TIHT+SGD}). \n}\n\n\n\n\n\n\n\nWe perform two experiments. In the first one, we randomly generate a linear 2-RNN with $5$ units computing a function $f:\\Rbb^3\\to\\Rbb^2$ by \ndrawing the entries of all parameters  $(\\vec{h}_0,\\Aten,\\vecs{\\vvsinfsymbol})$  independently  from the normal distribution\n$\\Ncal(0,0.2)$.\nThe training data consists of $3$ independently drawn sets \n$D_l = \\{((\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}),\\vec{y}^{(i)}) \\}_{i=1}^{N_l}\\subset(\\Rbb^d)^l\\times \\Rbb^p$ for $l\\in\\{L,2L,2L+1\\}$ with $L=2$,\nwhere  each $\\vec{x}^{(i)}_j\\sim\\Ncal(\\vec{0},\\mat{I})$ and where the outputs can be noisy, i.e.\\  \n$\\vec{y}^{(i)} = f(\\vec{x}^{(i)}_1,\\vec{x}_2^{(i)},\\cdots,\\vec{x}_l^{(i)}) + \\vecs{\\xi}^{(i)}$ where $\\vecs{\\xi}^{(i)}\\sim\\Ncal(0,\\sigma^2)$ for some noise variance $\\sigma^2$. \nIn the second experiment, the goal is to learn a simple arithmetic function computing the sum of the running differences between the two components of a sequence\nof $2$-dimensional vectors, i.e.\\  $f(\\vec{x}_1,\\cdots,\\vec{x}_k) = \\sum_{i=1}^k \\v^\\top\\vec{x}_i$ where $\\v^\\top = (-1\\ \\ 1)$. The $3$ training datasets are generated \nusing the same process as above and a constant entry equal to one is added to all the input vectors to encode a bias \nterm~(one can check that the resulting function can be computed by a linear 2-RNN with $2$ hidden units). \n\nWe run the experiments for different sizes of training data ranging from $N=20$ to $N=20,000$~(we set $N_L=N_{2L}=N_{2L+1}=N$) \nand we compare the different methods in terms of mean squared error~(MSE)\n on a test set of $1,000$ sequences of length $6$ generated in the same\nway as the training data~(note that the training data only contains sequences of length up to $5$). \n\n\\rev{We report the performances of (non-linear) RNNs with a single layer of LSTM with $20$  hidden units~(with $\\mathrm{tanh}$ activation \nfunctions\\footnote{We also tried training LSTMs with linear recurrent activation functions on the two tasks but they always performed worse than non-linear ones.}) and one fully-connected output layer, trained using\nthe Adam optimizer~\\cite{kingma2014adam} with learning rate 0.001. We also use Adam with learning rate 0.001 to refine the models returned by TIHT with stochastic gradient\ndescent SGD (we tried directly training a linear 2-RNN from random initializations using SGD as well but this approach always failed to return a good model). \nThe IHT\/TIHT methods sometimes returned aberrant models~(due to numerical instabilities), we used the following scheme to circumvent this issue: \nwhen the training MSE of the hypothesis was greater than the one of the zero function,\nthe zero function was returned instead (we applied this scheme to all other methods in the experiments).}\n\n\nThe results are reported in Figure~\\ref{fig:xp.random} and~\\ref{fig:xp.addition} where we see that all recovery methods of Algorithm~\\ref{alg:2RNN-SL} lead to consistent estimates of the\ntarget function given enough training data. This is the case even in the presence of noise~(in which case more samples are needed to achieve the same accuracy, as expected).\nWe can also see that \\texttt{IHT} and \\texttt{TIHT} are overall more sample efficient than the other methods~(especially with noisy data), showing\nthat taking the low rank structure of the Hankel tensors into account is profitable. Moreover, \\texttt{TIHT} tends to perform better than its matrix counterpart, \nconfirming our intuition that leveraging the tensor train \nstructure is beneficial. \n\\rev{While LSTMs obtain good performances on the addition task, they struggle to recover the random linear 2-RNN in the first task~(despite our efforts at hyper-parameter tuning and architecture search). \nIn the meantime, refining the TIHT models using SGD almost always leads to significant improvements~(especially under the noisy setting), matching or outperforming the\nperformances of RNNs on the two tasks.\nLastly, we show the effect of rank mis-specification in Figure~\\ref{fig:xp.misrank}: as one can expect, when the rank parameter $R$ \nis over-estimated Algorithm~\\ref{alg:2RNN-SL} still converges to the target function but it requires  more samples~(when the rank parameter was underestimated all  algorithms did not learn at all). \n}\n\n\\begin{figure}[t]\n\\begin{center}\n\\hspace*{0cm}\\includegraphics[width=0.5\\textwidth]{Mis_rank.jpeg}\n\\end{center}\n\\vspace*{-0.5cm}\n\\caption{ Comparison between different rank settings in terms of average MSE for the second experiment~(learning a simple arithmetic function) in the noisy\nsetting~($\\sigma^2=1$).}\n\\label{fig:xp.misrank}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sect_intro}\n\n \nCoronal mass ejections (CMEs) are associated with magnetic instabilities occurring in the solar corona, and they are\nexpelled during solar eruptive flare events.\nAs a consequence of these instabilities, large quantities of plasma and magnetic fields are expelled into the interplanetary space (IP), which can be observed by coronagraphs as CMEs. \nWhen their interplanetary manifestations (interplanetary CMEs or ICMEs) arrive at Earth, the observed intensity of energetic particles (\\textit{{\\it e.g.},}\\ Galactic Cosmic Rays, GCR) is modified.\nWhen they arrive at other planets (\\textit{{\\it e.g.},}\\ Mars) energetic particle fluxes can be also modified.\nICMEs\ncan also produce perturbations in the magnetosphere, triggering geomagnetic storms.\n \nA subset of ICMEs include a Magnetic Clouds (MC) which is characterised by a low proton temperature and an enhanced magnetic field intensity $|\\vec{B}|$ with a smooth rotation of its vector components, resembling that of a flux rope \\citep{burlaga81}. \nThe solar wind following an MC is expected to be perturbed, with characteristics similar to a wake.\n\n \nTypically, MCs travel faster than the local Alfv\\'en waves in the solar wind reference frame, producing a fast MHD shock ahead of them.\nThis shock produces an intermediate region of compressed plasma between the shock interface and the MC leading edge. \nThis region is characterised by high temperatures produced by the conversion of macroscopic to thermal energy at the shock, and therefore it generally presents high plasma $\\beta$ values.\nTypically, sheath regions also present large magnetic intensity and a high level of magnetic fluctuations.\n\n \nFluctuations around the shocks in the solar wind are\nobserved after the shock (downstream) as typically observed in fluids, but also upstream (before the arrival of the shock).  \nThis last case is mainly due to beam instabilities that are induced by shock-accelerated particles. \nThese instabilities can generate waves in the upstream region\n\\citep[\\textit{{\\it e.g.},}\\ ][]{blancocano_etal11,wang15,strumik15}, so that an increased level of fluctuations is expected before and after the shocks associated with ICMEs.\n\n  \nDuring their propagation away from the Sun, MCs interact with the plasma and magnetic field encountered in the interplanetary medium.\nThis interaction can induce reconnection between the sheath and the MC magnetic fields.  \nThis implies changes of the magnetic connectivity of the flux rope \\citep{mccomas88}, and a consequent peel off (erosion) of magnetic flux from the leading edge of the MC.\nThis also implies the formation of a 'back region' which involves field lines that were part of the MC before the erosion \\citep{dasso06,Ruffenach15}. \nThe back region is located in the MC wake and it typically has mixed plasma and magnetic field properties of ambient solar wind and MC. \n\n  \nSheaths in front of MCs significantly differ from planetary magneto-sheaths mainly because ICMEs not only propagate but also expand into the IP medium \\citep[\\textit{{\\it e.g.},}\\ ][]{Demoulin09,Gulisano10}.\nIn particular, the lateral deflection of the solar wind away from the nose of an MC is reduced due to the expansion, and the solar wind tends to pile up in front of an MC instead of flowing around it \\citep{Siscoe08}.\nThis effect is more important near the corona, where the expansion is stronger, than in the interplanetary medium \\citep{Das11}.\nMoreover, the drainage toward the sides can be enhanced due to magnetic reconnection between the magnetic field of the sheath and the flux rope.\n\n \nSummarising, a typical ICME is expected to be formed by the following sub-structures: \nslightly enhanced level of fluctuations (upstream waves turbulence), shock (driven by the MC), sheath (shocked, compressed, heated and turbulent material),  \nMC (flux rope), back (mixed flux rope and solar wind plasma when erosion has occurred), and wake (perturbed solar wind after the back region of the flux rope).\n\n \nTransient structures in the heliosphere affect the transport of energetic particles in the solar wind \\citep[\\textit{{\\it e.g.},}\\ ][and references therein]{masson12}. \nOn the one hand, acceleration at shocks driven by ICMEs is the main mechanism involved in the production of gradual energetic particle events in the inner heliosphere \\citep[\\textit{{\\it e.g.},}\\ ][]{vainio09}. \nOn the other hand, the decrease of the flux of energetic particles over a huge range of energies is associated with shocks and ICMEs. \nAt lower energy, this is observed in situ.  \nFor higher energies (\\textit{{\\it e.g.},}\\ galactic cosmic rays, GCRs) this is observed at ground level by neutron monitors \\citep[\\textit{{\\it e.g.},}\\ ][]{Simpson54}, by muon telescopes \\citep[\\textit{{\\it e.g.},}\\ ][]{arunbabu15}, or by water-Cherenkov detectors \\citep[]{auger11_scalers,Dasso12,Asorey16}.\n\n \nIn particular, the passage of ICMEs and their associated shocks have important effects on GCRs. \nIndeed, a Forbush decrease (FD) of galactic particles is typically observed during several days and in association with the passage of an ICME \\citep[\\textit{{\\it e.g.},}\\ see the review,][]{cane00}. \nAdditionally, a smaller amplitude and a higher-frequency variability in the GCR intensity compared with classical FDs can also be observed in situ\\ near Earth \\citep[\\textit{{\\it e.g.},}\\ ][]{mulligan09}. \nFurthermore, a decrease in the abundance of energetic particles associated with FD has also been in situ\\ observed by the Mars Science Laboratory's rover Curiosity, during its cruise phase from Earth to Mars \\citep{guo_etal15}.  \n\nThe full structure of ICMEs can perturb the transport of GCRs and has important effects both locally and globally on the density of GCRs.\nThese effects are associated respectively with \n  (1) strong changes of the local properties of the solar wind turbulence (mainly in the sheath) \n  which consequently affect the diffusion coefficients of these energetic particles and \n  (2) the presence of structures with smooth closed \\vec{B} field lines with typically high $|\\vec{B}|$ values inside MCs, which hardly allow diffusion transport across \\vec{B} \\citep[\\textit{{\\it e.g.},}\\ ][]{krittinatham09}.\n\n \nFrom a statistical study of interplanetary properties of ICMEs and muon data for cutoff rigidities \nbetween 14 and 24 GV (using the GRAPES-3 muon telescope), \\cite{arunbabu15} found that the enhancement of the interplanetary magnetic field inside the sheath regions is strongly associated with the observed FD profile.\nThey concluded that FDs are mainly caused by the cumulative diffusion of protons across $\\vec{B}$ in the sheath. \nHowever, it remains quantitatively unclear how FDs can be modelled, in particular how one can quantify the importance of $|\\vec{B}|$ and the level of turbulent fluctuations. \n\n \nIn this paper we characterise the mean properties of MCs at 1 AU and their relation to GCR transport.\nWe apply the superposed epoch analysis method on in situ\\ solar wind data and ground-based neutron monitor observations.\nThis method is powerful because it emphasises the common properties and removes peculiarities of some events. \nIt is similar to the superposed epoch developed by \\cite{lepping03b}, where they studied 19 MCs from 1995 to 1998 and created a profile of  magnetic field and plasma parameters of MCs, and even more to the method developed by \\cite{Yermolaev15} where they studied and compared the profiles of ejecta, MCs, their sheaths and corotating interaction regions.\nOur method is also similar to the one recently developed in \\cite{rodriguez16}, to analyse plasma, magnetic and composition properties, focussing on the physical mechanisms for the formation of structures inside and at the rear of MCs.\nIt is also close to the method used by \\cite{badruddin_etal16}, where the response of GCRs to corotating interaction regions and ICMEs was studied with a focus on their interfaces with the ambient solar wind.\n\nFinally, our study is related to the one of \\cite{belov_etal15} since they model the response of GCRs inside MCs as a function of the interplanetary conditions (plasma speed and magnetic field), and geomagnetic properties (Dst index).\nIn particular, they report the presence of\na local minimum of GCR inside MCs for strong-field events.\n\n\n\n \nIn \\sect{data_and_events} we present the ICMEs studied and the data used to analyse them.\nIn \\sect{superposed_epoch} the superposed epoch method, applied to our sample, provides the typical profiles of magnetic field and plasma parameters in ICMEs containing an MC.\nIn \\sect{splits_with_V} we further investigate how these profiles depend on the strength of the MC, in particular its mean velocity. \nFor that we split our sample in slow, mid and fast events.\nThe association of the event properties with the observed flux of GCRs is presented in \\sect{forbush}. \nThese results allow us to propose a novel quantitative model to describe the temporal evolution of GCR flux using parameters observed in the solar wind.\nFinally, in \\sect{conclusions}, we present our conclusions.\n\n\n\n\\section{Data and events selection}\n\\label{sect_data_and_events}\n\n\n\n\\begin{figure*}[ht]\n    \\sidecaption    \n    \\img{0.25}{example_01}  \n    \\img{0.25}{example_03}  \n    \\img{0.25}{example_02}  \n    \\caption{\n    Time profiles for different observables, from top to bottom: magnetic field $B$, bulk velocity $V$, proton density \\np, proton temperature \\Tp, plasma beta $\\beta$, normalised magnetic-fluctuation density $rmsBoB$ (\\eq{rmsBoB}), and absolute magnetic fluctuation $rmsB$ (\\eq{rmsB}).\n    We show three different events; one per column.\nEach of them belongs to each of the groups analysed in \\sect{splits_with_V}; from left to right, the events belong to the slow, intermediate, and fast groups.  \n    The date of the shock arrival is shown at the top of each column in UT.\n    The passage of the sheath is marked in orange, and the passage of the MC in blue.\n    The fastest event shows stronger compression at the MC front, with respect to the others.\n    The slowest event presents higher density values within the sheath. See \\sect{quantities_examples} for a description of the main important features shown in each panel.\n    }\n\\label{fig_examples}\n\\end{figure*}\n\nIn this section we present the data used for the analysis, the sample of studied events, and the observed physical quantities we explore.\n\n\\subsection{Events selection}\n\\label{sect_event_selection}\n\n \nWe use data of interplanetary magnetic field and plasma from the MAG and SWEPAM experiments onboard the ACE spacecraft \\citep{ACE98, mccomas98_ace}, using a 64 seconds time cadence, which are available at \\url{http:\/\/www.srl.caltech.edu\/ACE\/ASC\/level2\/}.\nThe intensity of GCRs is analysed using data from McMurdo neutron monitors\nwith one hour time cadence; these data are available at \\url{http:\/\/neutronm.bartol.udel.edu\/~pyle\/bri_table.html}.\n\n \nWe studied events taken from the list of \\cite{Richardson10}, including only ICMEs flagged as MCs from 1998 to 2006, \\url{http:\/\/www.srl.caltech.edu\/ACE\/ASC\/DATA\/level3\/icmetable2.htm}.\nSpecifically, we use only those events satisfying the following conditions: \na) ICMEs with flag \\texttt{2} in the list, which means that they are included in the WIND MC list compiled by R.P. Lepping, \\url{http:\/\/lepmfi.gsfc.nasa.gov\/mfi\/mag_cloud_S1.html}.\nb) no multiple MCs, to avoid complex structures involving interactions of different events.\nc) MCs with a sheath and with an associated shock.\n\n \nWe selected the shock events from the shock list of \\cite{wang10}.\nWe intersect both ICME and shock catalogs, only selecting ICMEs for which associated shocks are less than 3 hours before the ICME starting time.\nThe above procedure defines 44 events, and we summarise their main characteristics in \\tab{selec_events} (on line version).\n\n\n \nBecause the times listed in the table of \\cite{Richardson10} are referenced at Earth and our data are referenced at L1, we shift these listed times at L1, where the ACE data are observed.  \nWe use a global time shift as defined by the time difference of the shock observed at L1 and at Earth.\nWe made this shift for each ICME by identifying the shock discontinuity in the time series of the magnetic field and plasma parameters.\nThe time shifts are written in the column $5$, shown in \\tab{selec_events}. \n\n\n\\subsection{Analysed physical quantities}\n\\label{sect_quantities_examples}\n\nIn this section we describe some characteristics of three individual events as a context for further discussions of our results on average profiles.\nIn \\fig{examples} we show three MCs: one with a slow bulk velocity, one with a mid velocity value, and a fast one (shown on the left, central, and right column, respectively).\n\n \nThe upper panels show that the magnetic field strength has a comparable magnitude for the three selected MCs ($B_{\\rm max} \\approx 20$ nT). \nThe compression of the magnetic field (enhanced intensity) is present in the MC sheath, especially for the mid and fast MCs. \nThe faster MC presents also the most asymmetric B profile while the slow MC present a very symmetric B profile. \n\n \nThe second row of panels shows a gradient in the velocity profile.  \nThis typical velocity profile is associated with the MC expansion as it moves away from the Sun.\nThe slow and fast events present a very different change on the velocity during the MC passage (so that $\\Delta V =V_{end}-V_{start}$ is larger for the fast event compared to the slow one, a result consistent with the results of \\cite{Gulisano10} for MCs observed in the inner heliosphere).  \nFinally, we note that the event chosen with intermediate speed looks perturbed with a break in the velocity profile around the half time of the MC.\n\n \nIn the third and fourth rows, we see enhanced proton density \\np\\ and temperature \\Tp\\ inside the sheaths, as expected because these regions are formed of compressed and heated plasma behind the shock.\n\n \nIn the fifth row, we show the plasma $\\beta$. \nIts value is estimated with the same method as in the OMNI database, \\textit{i.e.}\\ by assuming that the temperature of alpha particles, $T_\\alpha$, is proportional to the proton temperature with: $T_\\alpha = 4 T_p$, and that the electron temperature is constant $T_e = 1.4 \\times 10^5 K$ (see \\url{http:\/\/omniweb.gsfc.nasa.gov\/ftpbrowser\/magnetopause\/Reference.html}).\nThese examples show that $\\beta$ can be greater than 10 inside the sheath, while inside the MC, $\\beta \\ll 1$ as expected \\citep[\\textit{{\\it e.g.},}\\ ][]{Dasso05}.\n\n \nWe also present the fluctuations of the unit vector of the magnetic field $\\vec{B}$, which we call the normalised magnetic fluctuations \n(i.e., rms of B over its magnitude B) and write it as $rmsBoB$ hereon.\nWe define it with:\n   \\begin{align}\n    \\centering\n    rmsBoB(t) &= rmsB(t) \/ B(t) \\label{eq_rmsBoB} \\\\\n    rmsB(t)   &= \\sqrt{\\sum_{i=1}^{3} <(B_i- \\langle B_i \\rangle)^2>} \n    \\label{eq_rmsB}\n   \\end{align}\nwhere $rmsB$ is computed in time windows of 16 seconds, using a high time cadence of 3 vectors per second, and the mean value of each component $\\langle B_i \\rangle$ is computed inside this time window (as provided by the ACE webpage).\n$rmsBoB$ is much lower inside MCs, than outside (sixth row of \\fig{examples}).\nIn the next section, we will see that this magnitude is a very robust quantity within MCs.\n\n\n \nThe magnetic fluctuations $rmsB$ are shown in the seventh (bottom) row. \nThey have maximum values within the sheath, while inside MCs it is roughly similar to the values observed before the shock arrival.\n\n \nBesides these expected features in these three cases, we obtain below typical profiles of ICME events, and quantify them.\nThis is the motivation to implement a superposed epoch analysis in the next section.\n\n\\begin{figure*}\n\\centering\n    \\img{0.9}{figs_all.global}\n    \\caption{\n    Superposed epoch for different quantities derived with the method described in \\sect{superposed_method}.   \n    The sheath is shown with an orange background colour (time range $0<t<1$) and the MC with a blue background colour ($1<t<4$).\n    The time $t$ is normalised with the sheath duration for $t<1$ and with the MC duration for $t>1$ (the MC duration is three times longer than the sheath duration as observed in average).\n    The black dots, connected with a line, are averaged values for each bin on the time axis, the grey band represents the error of the mean and the red line represents the median values in each time bin.\n    The average profiles were computed filtering those events that had less than 20\\% of data-gaps in the transient structure (MC or sheath).\n    The number of events after filtering is shown in each panel.\nThe panels represent: magnetic field $B$, bulk velocity $V$, proton density \\np, proton temperature \\Tp, plasma beta $\\beta$, normalised magnetic-fluctuation density $rmsBoB$ (\\eq{rmsBoB}), GCR intensity normalised to the pre-ICME level \\nGCR\\ and absolute magnetic fluctuation $rmsB$, \\eq{rmsB}.\n    A clear discontinuity of the magnetic fluctuation intensity, $rmsB$, is present at the shock position (front of the sheath).\n    The normalised magnetic fluctuations, $rmsBoB$, inside the MC are significantly lower than in the ambient solar wind (by a factor of $\\simeq$ 3).\n}\n    \\label{fig_avrg_profiles}\n\\end{figure*}\n\n\\section{Superposed epoch for the sheath and MC structures}\n\\label{sect_superposed_epoch}\n\n\\subsection{Method}\n\\label{sect_superposed_method}\n\nThe main aim of the superposed epoch is to obtain an average profile by taking a sample of individual profiles.\nIn order to obtain this average, each individual profile must have the same number of data points or bins in the time dimension.\nHowever, the primitive data have different durations for different events so we implement a re-binning such that within each MC we end up with 50 time bins. \nThen, the data within each bin are averaged to a single value per bin. \nIn the case when a primitive time series has a data gap in more than 20$\\%$ of the MC structure, we discard it from the average profile.\nWe also re-bin the data obtained after the MC, over the same time interval and with the same number of bins.\nNext, we average the data associated to the different events, by averaging bin by bin each quantity ($B$, $V$, $\\beta$, etc), which finally builds the average profiles. \n   \nWe repeat this procedure to obtain average profiles of the sheath structure.\nAgain, we resample all the time series to 50 bins inside each sheath.  \nSince the number of events having more than 20$\\%$ of gaps are not necessarily the same for the sheath analysis as for the MC analysis, we have indicated in each graph the total number of cases taken for both the sheath and the MC.\nThe same time normalisation as for the sheath is used to resample the data before the shock, over twice the time interval of each sheath. \n\nWe next build a combined profile which shows an average profile for: the background solar wind, the sheath, the MC and the MC wake. \nWe select the time origin at the shock.  \nFinally, we set the temporal lengths of the MC to sheath to a ratio 3:1 to better represent the typical relative durations of each structure. \n\nIn \\fig{avrg_profiles}, we show the average profiles (black lines) with the associated errors of the means (grey bands), and the median values (red lines).\nThe difference between the median and the mean profiles is a proxy of how non-symmetric the distributions of the quantities are.\nIndeed, in the central part of MCs, the observables are distributed in a log-normal manner \\citep{rodriguez16}.\nSimilar log-normal distributions also have been found for ICME observables in \\cite{guo_etal10} and \\cite{Mitsakou14}.\n\n\n\n\\begin{table*}\n\\centering\n\\caption{\n    Mean values of all quantities (column (1)) for all events (columns (2) and (3)), \nand for each group of events associated with MCs with low, mid and high velocities (columns (4) to (9)).\n    These mean values are calculated within sheaths and MCs.\n}\n\\begin{tabular}{c|cccccccccccccccrr}\n\\hline\n& & \\multicolumn{2}{c}{All} & &\\multicolumn{2}{c}{$V_{mc}^{low}$} & & \\multicolumn{2}{c}{$V_{mc}^{mid}$} & & \\multicolumn{2}{c}{$V_{mc}^{high}$} \\\\\n\\cline{3-4}\\cline{6-7}\\cline{9-10}\\cline{12-13}\nquantity                        &&sheath & MC    &&sheath & MC    &&sheath & MC    && sheath & MC    & \\\\\n (1) && (2) & (3) && (4) & (5) && (6) & (7) && (8) & (9)  \\\\\n\n\\hline\n$V$ [km\/s]                      && 561   & 525   && 441   & 402   && 520   & 493   &&  723   & 681 & \\\\\n$B$  [nT]                       && 15.4  & 14.3  && 11.4  & 14.0  && 13.4  & 11.9  &&  21.5  & 16.9  & \\\\\n$n_p$ [1$\/cm^3$]                && 14.3  & 6.5   && 18.5  & 9.5   && 14.1  & 5.3   &&  10.1  & 4.7   & \\\\\n\\Tp\\ [K] ($\\times 10^4$)        && 22.8  & 6.5   && 8.3   & 2.6   && 14    & 4.3   &&  46    & 12.4  & \\\\\n$\\beta$                         && 1.6   & 0.4   && 2.8   & 0.42  && 1.4   & 0.37  &&  0.7   & 0.25  & \\\\\n$rmsBoB$     ($\\times 10^{-2}$) && 6.8   & 2.7   && 6.3   & 2.3   && 6.3   & 3.0   &&  7.9   & 2.8   & \\\\\n$rmsB$ [nT]                     && 0.97  & 0.37  && 0.62  & 0.30  && 0.78  & 0.34  &&  1.53  & 0.47  & \\\\\n\\hline\n\\end{tabular}\n\\label{tab_avr_values}\n\\end{table*}\n\n\n\n\n\\subsection{Results for the average profiles}\n\\label{sect_profiles}\n\nFrom the mean profiles in \\fig{avrg_profiles}, we can clearly see sharp jumps at the arrival of the fast shock. \n$B$, $V$, $N_p$, and $T$ jump by factors of \n$\\sim$100$\\%$, $\\sim$30 $\\%$, $\\sim$150$\\%$, and $\\sim$300$\\%$, respectively.\nThen, all quantities behave differently whether in sheath or in the MC.\nWe describe shortly below the main results for each of the panels of \\fig{avrg_profiles} (except the bottom left one, which is described in \\sect{model_nGCR}).\nSome results in this subsection closely agree with those found in the sample of MCs studied in \\cite{Yermolaev15,rodriguez16}.\n\n \nThe upper left panel of \\fig{avrg_profiles} shows the piled-up magnetic field $B$ in the sheath. $B$ is enhanced by more than a factor 2 with respect to the pre-shock value.\nNext, the $B$ profile inside the MC is strongly asymmetric, with its maximum strongly shifted to the front of the MC.\nIndeed, due to the flux rope expansion a stronger $B$ is expected at the front compared to the rear since the spacecraft crosses the front earlier during the propagation of the MC than the rear \\citep[\\textit{{\\it e.g.},}\\ see Figures 3 and 4, and associated explanation in the main text of][]{Demoulin08}.  \nThis is the so-called aging effect.\nHowever, it was previously shown that expansion is typically not sufficient to explain such a $B$ asymmetry \\citep[see the upper panel of Figure 6 and associated main text of][]{Demoulin08}.\n\n \nThe speed profile (upper right panel of \\fig{avrg_profiles}) shows that across the shock there is a jump from 420-440 km\/s to 520-540 km\/s.\nThe first part of the sheath shows a fast compressing profile, while it later stabilises to be a roughly constant speed up to the boundary with the flux rope.\nInside the MC, the speed is a decreasing time function, in agreement with the expected typical expansion. \nThe linear decrease ends before the MC rear boundary. \nThis characteristic is present both in individual profiles and in other superposed epoch studies \\citep[\\textit{{\\it e.g.},}\\ ][]{lepping03b,rodriguez16}.\nA peak of $V$ is also present after the MC.\n\n\\begin{figure} \n    \\centering\n    \\img{0.6}{__paper__deviations_mix_Vmc.group}\n    \\caption{\n    Normalised values of average parameters corresponding to each subgroup (split according to the ordering parameter \\Vmc): \n    low \\Vmc\\ (blue triangle down), mid \\Vmc\\ (green circle), and high \\Vmc\\ (red triangle up).\n    Empty symbols are for sheaths and filled symbols for MCs.\n    For each quantity, we use the mean values of all events for the normalisation of each parameter (grey horizontal line).\n    From left to right the parameters are the means of: magnetic field $B$, bulk velocity $V$, proton density \\np, proton temperature \\Tp, plasma beta $\\beta$, normalised magnetic-fluctuation density $rmsBoB$ (\\eq{rmsBoB}), absolute magnetic fluctuation $rmsB$ (\\eq{rmsB}) and GCR intensity normalised to the pre-ICME level \\nGCR.\n    }\n    \\label{fig_Vmc_ordering}\n\\end{figure}\n\nIn the second left panel of \\fig{avrg_profiles} the high proton density, \\np, in the sheath contrasts with its lower value in the sheath surroundings.\nHowever, \\np\\ has no discontinuity at the interface between the sheath and the MC.\nOn the contrary, \\np\\ starts to decrease from the sheath-MC interface to inside the MC, up to around 15-20\\% of the MC size where its value becomes constant.  \nThis is not an effect of smearing the MC boundary in the averaging since this effect is also present in some individual MCs (\\textit{{\\it e.g.},}\\ \\fig{examples}).\nFinally, a clear density peak is present just after the rear boundary. \nIts physical origin was investigated in \\cite{rodriguez16}, where it was shown that several possible mechanisms, such as compression from a fast overtaking stream, or intrinsic mechanisms, such as the eruption conditions at the Sun, could account for this density peak.\n\n \nWhile in the sheath, the magnetic field, velocity and density are roughly constant (besides fluctuations), the proton temperature \\Tp\\ (right second panel of \\fig{avrg_profiles}) has a steep maximum near the shock.  \nThis is a consequence of the macroscopic to thermal energy conversion produced at the shock. \nLater on \\Tp\\ progressively decreases towards the MC.\nAs observed for \\np\\, the temperature profile is continuous around the sheath-MC interface without a clear break, up to a $\\sim$ 15-20 \\% of the mean MC size.\nLater on, \\Tp\\ stabilises to a roughly constant value up to almost the end of the MC.\n\n \nIn the density and temperature profiles (\\fig{avrg_profiles}), the transition between the sheath region and the \nMC is not as abrupt as expected, considering that the MC environment is much different from the sheath region. \nThis transition can be explained as follows.\n\\cite{gosling_etal95} discuss a mechanism where part of the field lines at the MC periphery are magnetically connected to the outer heliosphere after reconnection occurred.\nCase studies \\citep{dasso06,dasso07,Feng13} and a statistical analysis of a large sample of events \\citep{Ruffenach15} indicate that an MC can suffer magnetic erosion at its front during its travel in the heliosphere, forming a back region with mixed properties of both MC and ambient solar wind \\citep{dasso06,Ruffenach12}.\nThus, because MCs are 3D structures, it is possible that this erosion only happens at various specific locations along the flux rope.\nThen, the transition region between the sheath and the MC, explored by the spacecraft, can be magnetically connected to another region where erosion has already occurred. \nThis gives a possibility for solar wind material and heat to diffuse along the reconnected field lines, which leads to a smoother profile at the interface between those sub-structures.  \nIn this paper we refer to this mechanism as '3D reconnection'.\n\n\n\\def 0.20 {0.22}\n\\begin{figure*}[ht]\n    \\sidecaption\n    \\img{0.8}{figs_splitted_1}\n    \\caption{\n    Average profiles (black lines) distributed in three columns for the three subsets of events. Left: slow ($V<450$ km\/s), middle: medium ($450$ km\/s$<V<550$ km\/s), and right: fast MCs ($V>550$ km\/s).\n    The black line represents the average value, the grey band represents the error of the mean and the red line represents the median values in each time bin.\n    As in \\fig{avrg_profiles}, for $t<1$, time is normalised with the sheath average radial duration\nand for $t>1$, time is normalised with the average MC radial duration.\n    Orange delimits the sheath region ($0<t<1$), and blue the MC region ($1<t<4$).   \n    See the caption of \\fig{avrg_profiles} for more information.\n    From top to bottom the parameters are the means of: \n    magnetic field $B$, bulk velocity $V$, proton density \\np, proton temperature \\Tp, plasma beta $\\beta$ and normalised magnetic-fluctuation density $rmsBoB$, \\eq{rmsBoB}.\n    }\n    \\label{fig_split_i}\n\\end{figure*}\n\n\n \nThe plasma $\\beta$ and the normalised magnetic fluctuations $rmsBoB$ (left\/right third panels of \\fig{avrg_profiles}) are the quantities that most clearly mark the MC boundaries.\nBoth are roughly symmetric around the MC center with a flat minimum, with $rmsBoB$ being even more symmetric than $\\beta$.\n$rmsBoB$ shows a maximum at the shock, and then a smooth decrease\ntoward almost one-third of the MC size.\n\n \nThe level of absolute fluctuations $rmsB$ (right bottom panel of \\fig{avrg_profiles}) shows a sharp jump with a maximum at shock arrival, while it later continuously decreases to almost one-third of the MC.\nAfter the MC, it recovers towards typical solar wind values, but with slightly larger values. \n\n \nTo summarise this section, we describe the differences between mean values inside sheaths and MCs (columns 2 and 3 of \\tab{avr_values}). \nAll the quantities are higher in sheaths than in MCs, by: \n$7\\%$ for $V$, 8$\\%$ for $B$, $\\sim$ twice for $n_p$, $\\sim$ 3.5 times for $T_p$, \n$\\sim$ 4 times for $\\beta$, $\\sim$ 2.5 times for $rmsBoB$, and $\\sim$ 2.6 times for $rmsB$.\n\n\n\n\\section{Sheath and MC properties for slow and fast events}\n\\label{sect_splits_with_V}\n\n\\subsection{Criterion to define an ordering parameter}\n\\label{sect_criterium_split}\n\n \nThe mean velocity inside the ICME at 1 AU, its mean magnetic field intensity, and its time duration could be considered as different proxies for the strength of the event.\nMoreover, these values can be computed with a time window inside an ICME sub-structure such as the sheath or the MC; \nnamely \\Vsh, \\Bsh, \\dtsh, \\Vmc, \\Bmc, \\dtmc, where the suffixes ``sh'' and ``mc'' are associated to the sheath and the MC respectively.\n\n\n \nWhich is the most relevant parameter to order the events?\nTo answer this question, we first preliminarily consider one of the possible strength ordering parameter, \\Vmc . \nWe then split our sample of 44 events into 3 subgroups with roughly the same number of events: weak (slow), mid, and strong (fast).\nWe produce separate superposed epoch profiles for every physical quantity analysed (similar to those shown in \\fig{avrg_profiles}), \nand for each subgroup. \nFinally, we repeat this procedure for all the possible ordering parameters: \n\\Bmc, \\dtmc, \\Vsh, \\Bsh, and \\dtsh.\n\n \nThe criterion to select the best ordering parameter is as follows.\nFor every ordering parameter we construct a plot as the one shown in \n\\fig{Vmc_ordering} (where \\Vmc\\ was used in this case).\nThis figure shows normalised average values of all observables for each subgroup.\nThese values are derived from Table~\\ref{tab_avr_values}.\nThe average value of each subgroup is shown with: red triangle up \n(events with large velocities: \\Vmc $> 550$ km\/s), \ngreen circle (mid velocities events: $450$ km\/s < \\Vmc $<550$ km\/s), \nand blue triangle down (events with weak velocities: \\Vmc $<450$ km\/s).\nThese values are normalised with the global average using all events. \nThe criterion to select which strength parameter is the best is to observe: \ni) the largest separation between mean values of the 3 subgroups, and \nii) a coherent ordering between these values.\n\n \nAmong all the plots (not shown here) that we looked at, we realised that the best parameter to order the MCs is the mean MC speed \\Vmc\\ , which happened to be better for splitting our sample into three groups.\nAs is seen in \\fig{Vmc_ordering}, most of the parameters are well ordered (red, green and blue in succession for each parameter).  \n The same quality of ordering and spacing as \\fig{Vmc_ordering} was not found for any of the other explored ordering parameters, \\textit{{\\it e.g.},} \\Bmc.  \n\n \nThe parameters with largest spread are the proton density \\np\\ , the proton temperature \\Tp, the plasma $\\beta$, and the GCR percentage of depression \\nGCR\\  (see \\sect{forbush}).  \nOn the other hand, the parameter showing less split and no coherent order is $rmsBoB$.  \nThis implies that the normalised magnetic fluctuations are very similar in all groups within a $\\sim 20 \\%$ factor from the global mean.\nNext, we observe that when the splitting is large between the groups, the relative order and spread for the sheaths are typically the same as for the MCs (\\textit{{\\it e.g.},}\\ see $V$, \\np, and \\Tp\\ in \\fig{Vmc_ordering}).\nFinally, the fastest MCs (red triangle up) have: the largest $B$ intensity, the lowest proton density, the hottest temperature, the lowest plasma beta, the strongest magnetic fluctuations $rmsB$, and the deepest associated GCR depression.\n\n\n\\subsection{Superposed epoch for each MC subset}\n\\label{sect_results_split}\n\nWe analyze below the averaged profiles of the three subsets.\nIn the upper row of \\fig{split_i} we find large differences in the $B$ profiles: \nfor faster\/stronger events, $B$ is more peaked toward the start of the MC, while for slower\/weaker events, it is more peaked toward the MC center.\nAlso, $B$ jumps at the shock with $\\Delta B=7$, $8$, and $13$nT, for slow, mid, and fast events, respectively.\n\n \nIn the second row of \\fig{split_i}, the ambient solar wind before the front of the sheaths is slower for weaker\/slower events compared with stronger\/faster events, then they are not propagating in the same mean solar wind. \nNext, the speed profiles in the three groups have all a linear profile within the MCs, which indicates a common expansion process. \nFor sheaths, we also see expansion in slow events, but for fast events the sheaths have profiles showing compression.\nThis bimodal feature of the sheath, with an expansion and a compression, has not been reported before as far as we know, and it is shown pretty clearly in these superposed profiles with both processes having a comparable effect for MCs with mid velocities.\nThe fact that fast events are related to compressing sheaths is the result of the fast overtaking MC behind.\nFor the slow events, the magnetic field and the plasma had the time to reach a quasi-pressure balance.  \nThis implies an expansion similar to that of the flux rope \\citep[as modeled by ][]{Demoulin09}.\n\n \nIn the third row of \\fig{split_i}, slower events have higher proton density in all the sub-structures: sheath, MC, and its wake. \nBy contrast, the ambient solar wind at the front does not show systematic changes along the three groups.  \nWe interpret the higher density in the sheath of slow events as a selection effect due to the drag force, as follows.\nCMEs that are fast near the Sun and that later on encounter slow solar wind ahead, \nwhich is typically denser in the inner heliosphere \\citep[\\textit{{\\it e.g.},}\\ see][]{wolfe72}, accumulate more mass with a low velocity, resulting in a denser sheath and a slower MC by conservation of mechanic momentum.\nStudies in this direction have been made in \\cite{feng_etal15}, where they find that the SW pile-up of mass during the CME expulsion can be an important contribution to the mass increase determined by coronagraph observations.\nWe explore this observed feature in more details in \\sect{sigma_p_sheath}.\nNext, the low $n_p$ values observed in the wake of fast cases (right column of \\fig{split_i}), can be due to the sweeping of ambient solar wind plasma made by faster MCs \\citep[which are also typically larger, \\textit{{\\it e.g.},} , ][]{Janvier14c} combined with the difficulties to re-fill this space with new solar wind material just after the passage of the flux rope.\n\n \nIn the fourth row of \\fig{split_i} the proton temperature in the ambient solar wind is lower for slow events, which are also traveling in a slower solar wind. \nThese results are consistent because of the known direct correlation of speed and proton temperature in the quiet solar wind \\citep[\\textit{{\\it e.g.},}\\ ][]{Lopez86, Demoulin09d}.\nNext, the faster MCs present much hotter sheaths than slower ones; this is a consequence of the local heating near the shock.\nThe temperature in slow MCs is lower. \nThis is consistent with a relaxed flux rope which had time to adapt to the ambient solar wind pressure, to fully expand, and accordingly to cool.\n\n\\begin{figure}[ht!]\n    \\centering\n   \n    \\img{0.73}{Vmc__vs__sigma}\n    \\caption{\n    Left panel: Average proton surface densities of sheaths $\\sigma_{sh}$, \\eq{def_sigma}, as a function of the average MC speeds ($V_{mc}$) of all events. \n    The vertical dashed lines mark the thresholds between the three groups shown in \\figs{split_i}{split_ii}.\n    The quantity \\sigsh\\ is a proxy of the accumulated material ahead of MCs, within the sheaths.  \n    On average, slower MCs drag more material.\n    Right panel: histogram of \\sigsh\\ for the 44 events.\n    }\n\\label{fig_Vmc_vs_sigmas}\n\\end{figure}\n\n \nIn the fifth row of \\fig{split_i} the average plasma $\\beta$ values are higher for slower events, for all ICME sub-structures.\nEven in the ambient solar wind we note a tendency for slow events to travel in a less magnetised plasma (higher $\\beta$). \nWithin the sheaths, $\\beta$ is more constant and inside the MC the $\\beta$ profile is more symmetric for slow events.\nFor faster events $\\beta$ has a more asymmetric profile, according to the asymmetry of $B$ (first row of \\fig{split_i}).\n\n \nIn the last row of \\fig{split_i} we observe big differences between the $rmsBoB$ fluctuation profiles of slow and fast events.\nFor instance, the fluctuation intensity is much higher near the shock for fast events.\nAlso, these fluctuations start even before the shock arrival and with a clearer effect for faster events.\nThis can be related to foreshock waves excited by energetic particles near interplanetary shocks \\citep[\\textit{{\\it e.g.},}\\ see,][]{blancocano_etal11}.\nNext, the magnetic fluctuations have a steeper drop in the sheath-MC interface for the slow events. \nIn contrast, the fluctuations are roughly the same inside the MC for the three groups.\nFinally, the extension of low $rmsBoB$ values in the back suggests the presence of remnant MC-like structure due to magnetic reconnection at the front, as summarised in \\sect{profiles}.\n\n\n\\subsection{Surface density of protons in the sheath}\n\\label{sect_sigma_p_sheath}\n\nTo further explore the drag as the physical reason for finding some slow events, we estimate the accumulated material inside the sheath for every event by defining the ``surface density'' \\sigsh :\n \\begin{equation}\n \\label{eq_def_sigma}\n  \\sigma_{sh} = \\sum_{r~\\in\\ {\\rm sheath}} n_p \\Delta r  \n              = \\sum_{t~\\in\\ {\\rm sheath}} n_p V \\Delta t\n \\end{equation}\nwhere $\\Delta t$ is the time resolution of the data, and the sum approximates an integral along the radial direction in the ecliptic plane within the sheath boundaries.\n\\sigsh\\ characterises the total amount of material per unit surface (perpendicular to the radial from the Sun).\n\\sigsh\\ is associated with the material that has not escaped perpendicularly to the spacecraft crossing.\nThe right panel of \\fig{Vmc_vs_sigmas} shows an asymmetric distribution of \\sigsh, having more cases with \\sigsh\\ $\\lesssim 3 \\times 10^{14}$ cm$^{-2}$.\n\nIn the left side of \\fig{Vmc_vs_sigmas} we show a scatter plot of \\sigsh\\ versus the average MC speeds for all the studied events.\nFaster MCs tend to have less material ahead of them.\nThis result may seem surprising at first, since faster MCs are expected to overtake more solar wind and moreover the plasma has more difficulty to escape from the sides (faster and larger MCs).  \nHowever, this is consistent again with the drag scenario where the MCs are slowed down when they are ejected with a high speed in the corona and encounter a slow and dense solar wind.\nIn contrast, high speed-MCs encountering a fast solar wind ahead are less slowed down (lower mass pile-up with faster velocity). \nThese cases correspond to the lower right region of the scatter plot.\nFinally, the lower left part of the plot would correspond to MCs with a slow speed encountering a low solar wind density along their travel to 1 AU.\n\n\n\n\\section{Analysis of associated Forbush events}\n\\label{sect_forbush}\n\n\\subsection{Superposed epoch analysis}\n\\label{sect_Superposed_GCR}\n\n \nWe analyse below the cosmic rays observed with ground-based detectors.\nWe use data from McMurdo neutron monitors because they \nare close enough to one geomagnetic pole, and then they can measure particles with low energies. In particular, they can observe primary protons with energies as low as $\\sim$500 MeV \\cite[\\textit{{\\it e.g.},}\\ ][]{jordan11}.\nThe Larmor radius for these particles is $\\sim$5x10$^{-3}$ AU, when they are embedded in an interplanetary magnetic field of $\\sim$5 nT. \nFor normal conditions, particles observed by these detectors are mainly of galactic origin.\n\n \nAn additional treatment with respect to the previous quantities \n(\\sects{superposed_epoch}{splits_with_V}), is that we use the average GCR intensity before the MC sheath arrival as a reference level.\nIn essence, we take the data from 2 days prior the sheath arrival.\nIf a ground level enhancement (GLE) is present, we only take data from 6 hours prior to the sheath arrival (we verify for each event that during this time interval there is no GLE or intense noises).\nThen, we normalise the GCR flux of each event with this level taken before the MC sheath defining the percentage of variation \\nGCR.\nNext, we continue with the same treatment as in \\sect{superposed_method}.\n\n \nThe superposed epoch of all events is shown in the bottom left panel of \\fig{avrg_profiles}.  \nThere is a strong decrease of \\nGCR\\ in the sheath followed by a progressive recovery phase during and after the MC. \nThe importance of these effects depends on the MC speed as shown in the second row of \\fig{split_ii}. \nThe recovery time ($\\tau_{FD}$) and the amplitude ($A_{FD}$) of the GCR depressions are larger for the faster events; this result is respectively consistent with the following previous studies: \\cite{penna05} and \\cite{richardson11}.\nA minimum of \\nGCR\\ is present near the sheath rear for the three subsets. \nFor fast events, another minimum can be observed within the fluctuations, inside the MC window and near the MC center.\nThis second minimum is associated to the shielding of a strong magnetic field within the flux rope configuration (with closed field lines still connected to the Sun).\nThis result\nis consistent with \\cite{belov_etal15}, where they report that for strong-field ($>18$nT), there is a local minimum of \\nGCR\\ inside MCs.\n\n\n\\subsection{Model for the \\nGCR\\ typical profiles}\n\\label{sect_model_nGCR}\n\n\n\\def 0.20 {0.20}\n\\begin{figure*}[ht!]\n    \\centering\n    \\img{0.8}{figs_splitted_2}\n    \\caption{\n    Superposed epoch for the three subsets of events. Left: slow ($V<450$ km\/s), middle: medium ($450$ km\/s$<V<550$ km\/s), and right: fast MCs ($V>550$ km\/s).\n    {\\bf First row:} mean profiles of magnetic fluctuations $rmsB$, \\eq{rmsB}.\n    Typically, inside the MCs, the fluctuations are the same as in the ambient solar wind.\n    For slow and mid groups, $rmsB$ has higher values in the rear with respect to the ambient solar wind.\n    {\\bf Second row:} mean profiles of GCR intensity \\nGCR .\n    Before averaging events, the intensity was normalised by the pre-shock level.  \n    In the first and second rows the black line represents the average value, the grey band represents the error of the mean and the red line represents the median values in each time bin.\n    {\\bf Third row:} Fitted model (red lines) to the observed average profiles (see \\sect{model_nGCR}).\n     See the caption of \\fig{avrg_profiles} for more information.\n    }\n    \\label{fig_split_ii}\n\\end{figure*}\n\n\n\nA theoretical model to describe the profile of FD was developed by \\cite{wibberenz98}.\nHowever, they did not explicitly include small-scale interplanetary magnetic fluctuations, while it was shown that these can contribute to the variety of FD profiles \\citep[\\textit{{\\it e.g.},}\\ ][]{jordan11}.  \n\nIn order to quantitatively analyze the processes involved in the GCR shielding produced by ICMEs, we construct a mathematical model to reproduce the GCR profiles. \nIn particular, we use $B$ to account for the strong closed field lines of the MC, and $rmsB$ to account for the scattering turbulence.\n\n  \nFrom the superposed epoch profiles, we learned the following contributions (second row of \\fig{split_ii}):\\\\\ni) Inside the sheath region, \\nGCR\\ can be thought to be determined by an accumulation of interaction between the high energy particles and scattering centres. \nThe efficiency of the scattering is related to the level of turbulent fluctuations $rmsB$.\\\\\nii) The decrease of \\nGCR\\ is larger for a stronger magnetic field inside the MC.\\\\\niii) The recovery mechanism is effective after the passage of the scattering turbulence region.\\\\\niv) There is a jump in \\nGCR\\ at the interface between the sheath and the MC; this can also be seen in the individual profiles. \nThis feature can be related to the change of the cross diffusion coefficients associated to the plasma properties in these two different structures.\n\n  \nBy representing each of these mechanisms with a contributing term for the temporal evolution of the GCRs flux at Earth position, we propose the following model:\n\\begin{align}\n    \\centering\n    &\\frac{d n_{GCR}(t)}{d t} \n     &= \\text{ }& q_{\\xi} \\, \\xi(t)  & (i)  && \\nonumber \\\\\n    &&+ \\text{ }& q_{b} \\, b(t)      & (ii) && \\nonumber \\\\\n    &&+ \\text{ }& \\frac{-1}{\\tau_{FD}} n_{GCR}(t) \\, \\Theta(t-1) & (iii) &&\\nonumber\\\\\n    &&+ \\text{ }& \\Delta_o \\, \\delta (t-1)   & (iv) && \\forall t>0\n    \\label{eq_FD_model}\n\\end{align}\nwith,\n\\begin{align}\n    \\xi(t) &= \\textrm{max}(0, rmsB(t)-rmsB_o) \\nonumber \\\\\n    b(t)   &= \\textrm{max}(0, B(t)-B_o) \\,.\n\\end{align}\n$\\xi(t)$ is the $rmsB(t)$ value above its value in the quiet solar wind, $rmsB_o$, when $rmsB(t)$ is larger than $rmsB_o$. \n$b$ is the magnetic field above a reference value $B_o$ when $B>B_o$. \n$\\Theta$ is the Heaviside function indicating that the recovery mechanism starts at the beginning of the MC. \nThis simplification facilitates the mathematical fit and we checked that the recovery is negligible during the passage of the sheath.\n$\\delta$ is the Dirac delta function (more precisely distribution) which account for the observed jump of \\nGCR\\ at the sheath-MC interface. \nThere are 5 free parameters to fit: \n$q_{\\xi}$, $q_{b}$, $B_o$, $\\tau_{FD}$ and $\\Delta_o$.  \nThis model does not include explicitly the larger expansion of faster events nor a non local effect due to the sheath and MC spatial extension.  \nThe expansion rate is correlated to the field strength, so it is implicitly included, while we cannot include the effect of the global extension from local measurements.\n\n  \nIn order to perform the non-linear fit of the time integral of \\eq{FD_model} to the data, we employ the L-BFGS-B algorithm \\citep{zhu_etal97, byrd_etal95}.\nThis algorithm is suitable when working with a large number of fitting variables, and it uses the gradient information of a multiple variable-function to be minimised within given bounds.\nMainly, we use this method instead of other alternatives because we observe a better convergence for our problem.\n\n  \nThe third row of \\fig{split_ii} shows the fitted models to each of the average \\nGCR\\ profiles, where the detailed reproduction of the observations is notable.\nThe inclusion of the magnetic fluctuations allows the reproduction of the observed mean decrease of \\nGCR\\ within the sheath with only one adjusted parameter $q_{\\xi}$. \nMoreover, for the slow $V_{mc}$ group, we see a small decrease of \\nGCR\\ right after the MC passage, which according to our model, is due to the enhancement of $rmsB$ in the wake of the flux rope (see the left column of \\fig{split_ii}). \nThis behaviour is also present, but weaker, for the mid velocity group.\nThis result at the MC rear provides a further confirmation that magnetic fluctuations are locally affecting GCRs.\nIt is also in agreement with \\citet[][see their Fig. 7]{badruddin_etal16}, where they find a GCR decrease around the ICME trailing edge, and simultaneously find enhanced magnetic fluctuations.\nFinally, $|q_{\\xi}|$ is decreasing from the low to the fast velocity group (Table~\\ref{tab_fitted_vals}), which may be due to non-linearities in the diffusion process that are not taken into account in our model.\n\n  \nThe contribution of the magnetic field $B$ is important only for the group of fast \\Vmc , where \\nGCR\\ has a local minimum inside the MC. \nAs mentioned before, this can also be related to the larger expansion of the flux rope compared to the surrounding solar wind. \nThis leads to a decrease of the local density of the high energetic particles \\citep[\\textit{{\\it e.g.},}\\ ][]{Munakata06}.\n\nNext, the normalised time $\\tau_{FD}$, as well as the time $\\tau_{FD}$ in hours (taking into account the mean MC duration of each group) are both increasing by more than a factor 2 from the slow to the fast velocity group (Table~\\ref{tab_fitted_vals}).\nFinally, it is worth mentioning that without the inclusion of the jump parameter $\\Delta_o$, it is not possible to reproduce the recovery profile for the slow and fast $V_{mc}$ groups that clearly show this discontinuity.\n\n\n\n\\begin{table}\n\\centering\n\\caption{\n    Fitted values of the Forbush decrease model (see \\eq{FD_model}), for the three subset of events.\n    These are all free parameters in the model defined by \\eq{FD_model}.\n    \\tauFD\\ is normalised to the mean duration of the MCs for each group; see \\sect{model_nGCR} for informations.\n    The $B_o$ value in the first column is superfluous since $q_{b}=0$, in that case. \n}\n\\begin{tabular}{c|ccccccccccrr}\n\\hline\nfit parameter & \\multicolumn{1}{c}{$V_{mc}^{low}$} & \\multicolumn{1}{c}{$V_{mc}^{mid}$} & \\multicolumn{1}{c}{$V_{mc}^{high}$} \\\\\n\\hline\n$q_{\\xi}$  [nT$^{-1}$]    & -9.4   & -6.0   &  -5.5    & \\\\\n$q_{b}$  [nT$^{-1}$]      &~~0.0   & -0.9   &  -0.2    & \\\\\n$B_o$  [nT]               &~~--    & 11.9   &  14.5    & \\\\\n\\tauFD [1]                &~~2.4   &~~4.2   &~~5.8     & \\\\\n$\\Delta_o$  [\\%]          &~~0.9   &~~0.0   &~~1.0    & \\\\\n\\hline\n\\end{tabular}\n\\label{tab_fitted_vals}\n\\end{table}\n\n\n\n\\section{Summary and conclusions}\n\\label{sect_conclusions}\n\n  \nWe obtained superposed epoch profiles during the passage of MCs and their sheaths at 1~AU, for different physical quantities observed in situ. \nThis technique allows us to identify several phenomena that are common to most of the events.\nFirst, the magnetic field strength and the plasma velocity in the MC sheath are typically close to the values found at the beginning of the MC. \nSecond, the MC has a similar density than the background solar wind, while the sheath is a factor $\\approx 2.5$ denser. \n\n  \nWe next explored how the sheath and MC properties could be best differentiated according to the importance of the event.\nWe found that the mean MC velocity is the best parameter to order the sheath and MC properties. \nThis provides a measure of the strength of the events.\nThen, we separated our sample in three groups: slow\/intermediate\/fast events and compared their superposed epoch profiles.\n\n \nWe found that the slowest MCs have a larger proton density in their sheaths than faster events.\nWe next computed the density of protons per unit of surface along the sunward direction to better characterise the accumulated material that does not escape from the MC front to the sides. \nWe found that on average, slow events have more massive sheaths than fast events. \nWe attribute this result to a selection effect: a part of the slow MCs at 1 AU were initially fast events close to the Sun but they were slowed down as they encountered slow and dense plasma on their path to 1 AU.\nThen, such decelerated MCs have more massive sheaths at 1 AU.\n\nThe comparison of the three groups show that the slow MCs have properties compatible with a more relaxed configuration.\nFirst, they have an expanding sheath, with a very similar expansion rate as in the following MC. \nThen, the accumulated magnetic field in the sheath has the time to reach a quasi-equilibrium with the surrounding solar wind, which implies that the sheath expands with a similar rate as the driving flux rope.\nSecond, the slow MCs have a nearly symmetric magnetic profile around their center.\nFinally, they have a lower proton temperature (as they have the time to expand enough to be in quasi-equilibrium with the surrounding solar wind).\nAll these characteristics together indicate that the slower MCs are expected to be in a more relaxed force-free state than faster MCs. \n\n   \nThe proton density and temperature are significantly larger in sheaths than in MCs. \nStill, there is no sharp transition but a progressive sheath-MC transition which extends up to 20\\% of the MC duration. \nA very similar transition is also present for the amplitude of the magnetic fluctuations.  \nThese results are present for all three groups of events as well as in individual events, so they are not due to a smearing of sheath properties inside the MC due to inappropriate boundary definition. \nWe interpret these results as the consequence of magnetic reconnection at various locations along the flux rope between sheath and flux-rope magnetic fields.\nThen, while along the spacecraft trajectory the crossed magnetic field appears as belonging to the flux rope, some parts of the crossing can belong to a magnetic field region that has reconnected with the sheath magnetic field further away from the local spacecraft trajectory.\nThis allows plasma, heat and magnetic fluctuations to enter in this reconnected field implying, in average, sheath properties ``diffusing'' within the front of the MC.\n\n  \nWe next studied the typical effects that these sheaths and MCs have on the cosmic ray transport by studying the associated Forbush decrease profiles.\nFor all MCs a local minimum of the GCR flux is present inside the sheaths, while for fast MCs another minimum is also present within the MCs. \nIn the superposed epoch profiles of Forbush events, we found a discontinuity of the GCR flux at the sheath-MC interface, owing probably to the change of properties in the cross magnetic-field diffusion coefficients in these two structures.\nIndeed, GCRs encounter a structure with closed and intense magnetic field lines, where the high energy particles have difficulties to penetrate. \n\n\nWe finally derive a new semi-empirical model for the Earth observations of the Forbush profile that takes into account the former process \nas a simple superposition of \n(1) the enhancement of magnetic fluctuations over a threshold, \n(2) the intensity of the magnetic field over a threshold, \n(3) a jump at the time of the sheath-MC interface to consider the change of the magnetic connectivity, and \n(4) a recovery phase for times after the sheath-MC interface.\nThis model has five free parameters.  \nWe fit the model to each of the groups of events. \nBesides reproducing the first steep decrease of GCRs after the shock time and the recovery phase, we can reproduce a second decrease of GCRs in the wake of the slow MCs.\nThis second fall is due to the presence of scattering centres which we associate with the amplitude of the magnetic fluctuations. \nThis last result is in agreement with the recent study of \\cite{badruddin_etal16}.\n\nFurthermore, our model also reproduces a local \\nGCR\\ minima inside the MC for the sub-group of fast events.\nThis feature is consistent with \\cite{belov_etal15}, where they find that strong-field ($>18$ nT) are associated to local minima of \\nGCR\\ within MCs.\n\nFinally, we find that the recovery time \\tauFD\\ is increasing by a factor a bit larger than two from the slow to the fast events.\n\n\nThe results presented here improve the knowledge of MCs and their sheaths, their evolution in the solar wind, and also the relationship between MCs and their GCR shielding.\nIn particular, the semi-empirical model for the FD profile we presented will help to improve the understanding of energetic particle transport in the heliosphere, and can be used to put constrains on theoretical models that consider \n(1) global macro-scale magnetic configurations for MCs and (2) turbulent properties of the solar wind.\n\n\\onltab{\n\\begin{table*}\n\\centering\n\\caption{\n    List of the 44 events studied. Column: \n(1): event number,\n(2): time of shock arrival \\citep[as in][]{Richardson10},\n(3): time difference between MC leading edge and shock time (\\textit{i.e.}, the sheath duration),\n(4): MC duration, \n(5): difference between \\citeauthor{Richardson10} and the shifted shock time at L1 point, where the IP observations are compared,\n(6): amplitude of the associated GCR perturbation.\n}\n\\begin{tabular}{cccccc}\n\\hline\ncase & shock date        & sheath duration & MC duration & time shift & $A_{\\rm FD}$ \\\\\n     & [yyyy-mm-dd HH:MM]& [hours]         & [hours]     & [min]      & [\\% of the background] \\\\\n (1) &      (2)          & (3)    & (4) & (5)  &  (6)   \\\\\n\\hline\n   1 & 1998-03-04 11:56  &  ~~1.1 &  41 &  -60 & ~~-0.6 \\\\\n   2 & 1998-05-01 21:56  &   14.1 &  29 &  -30 & ~~-6.9 \\\\\n   3 & 1998-09-24 23:45  &   10.2 &  27 &  -30 &  -13.3 \\\\\n   4 & 1998-10-18 19:52  &  ~~8.1 &  10 &  -50 & ~~-2.1 \\\\\n   5 & 1998-11-08 04:51  &   20.1 &  24 &  -30 & ~~-6.5 \\\\\n   6 & 1999-02-18 02:46  &   11.2 &  22 &  -40 & ~~-5.6 \\\\\n   7 & 1999-04-16 11:25  &  ~~8.6 &  25 &  -45 & ~~-0.6 \\\\\n   8 & 1999-08-08 18:41  &   26.3 &  20 &  -60 & ~~-0.1 \\\\\n   9 & 2000-02-11 23:52  &   17.1 &   7 &  -40 & ~~-4.4 \\\\\n  10 & 2000-02-20 21:39  &   12.3 &  26 &  -60 & ~~-1.6 \\\\\n  11 & 2000-06-23 13:03  &   13.9 &  41 &  -40 & ~~-2.3 \\\\\n  12 & 2000-07-28 06:34  &   14.4 &  13 &  -50 & ~~-1.6 \\\\\n  13 & 2000-08-11 18:45  &   10.2 &  24 &  -40 & ~~-2.9 \\\\\n  14 & 2000-09-17 17:57  &  ~~8.1 &  16 &  -60 & ~~-5.2 \\\\\n  15 & 2000-10-03 00:54  &   16.1 &  21 &  -45 & ~~-1.9 \\\\\n  16 & 2000-10-28 09:54  &   11.1 &  25 &  -50 & ~~-6.5 \\\\\n  17 & 2000-11-06 09:48  &   12.2 &  20 &  -35 & ~~-6.4 \\\\\n  18 & 2001-04-04 14:55  &  ~~3.1 &  14 &  -40 & ~~-4.6 \\\\\n  19 & 2001-04-11 13:43  &   18.3 &  10 &  -30 & ~~-9.9 \\\\\n  20 & 2001-04-21 16:01  &  ~~7.0 &  26 &  -30 & ~~-1.3 \\\\\n  21 & 2001-04-28 05:01  &   21.0 &  11 &  -30 & ~~-7.7 \\\\\n  22 & 2001-05-27 14:59  &   21.0 &  22 &  -45 & ~~-3.5 \\\\\n  23 & 2001-10-31 13:48  &  ~~6.2 &  38 &  -60 &~~~~0.4 \\\\\n  24 & 2002-03-18 13:22  &   33.6 &  17 &  -60 & ~~-4.8 \\\\\n  25 & 2002-03-23 11:37  &   24.4 &  34 &  -45 & ~~-2.9 \\\\\n  26 & 2002-04-17 11:07  &   15.9 &  23 &  -45 & ~~-4.9 \\\\\n  27 & 2002-04-19 08:35  &   27.4 &  30 &  -30 & ~~-3.3 \\\\\n  28 & 2002-05-23 10:50  &   12.2 &  18 &  -40 & ~~-5.8 \\\\\n  29 & 2002-08-01 05:10  &  ~~6.8 &  11 &  -50 & ~~-1.3 \\\\\n  30 & 2002-08-01 23:09  &  ~~9.8 &  12 &  -50 & ~~-3.4 \\\\\n  31 & 2003-03-20 04:40  &  ~~7.3 &  10 &  -25 & ~~-2.5 \\\\\n  32 & 2003-08-17 14:21  &   20.6 &  18 &  -40 & ~~-1.9 \\\\\n  33 & 2003-11-20 08:03  &  ~~1.9 &  16 &  -35 & ~~-5.0 \\\\\n  34 & 2004-04-03 10:00  &   16.0 &  37 &  -60 & ~~-2.0 \\\\\n  35 & 2004-07-22 10:36  &  ~~4.4 &   6 &  -45 & ~~-4.6 \\\\\n  36 & 2004-07-26 22:49  &  ~~3.2 &  10 &  -25 &  -13.3 \\\\\n  37 & 2004-08-29 09:09  &  ~~9.8 &  27 &  -50 & ~~-0.6 \\\\\n  38 & 2004-11-07 18:27  &  ~~7.5 &  15 &  -30 & ~~-7.9 \\\\\n  39 & 2005-05-20 03:00  &  ~~4.0 &  22 &  -60 &~~~~0.3 \\\\\n  40 & 2005-06-12 07:45  &  ~~7.2 &  16 &  -60 & ~~-3.0 \\\\\n  41 & 2005-06-14 18:35  &   10.4 &  28 &  -45 & ~~-0.8 \\\\\n  42 & 2005-07-17 01:34  &   12.4 &  14 &  -60 & ~~-4.5 \\\\\n  43 & 2005-12-31 00:00  &   13.0 &  22 &  -40 & ~~-2.1 \\\\\n  44 & 2006-12-14 14:14  &  ~~7.8 &  22 &  -25 & ~~-8.0 \\\\\n\\hline\n\\end{tabular}\n\\label{tab_selec_events}\n\\end{table*}\n}\n\n\\begin{acknowledgements}\nThanks to the public service of NASA Space Physics Data Facility (SPDF) for the observations.\nWe thank Roger Clay and the anonymous referee for reading carefully the manuscript, and their valuable comments. \nThis work was partially supported by the Argentine grants UBACyT 20020120100220 (UBA), PICT-2013-1462 (FONCyT-ANPCyT), \nPIP-11220130100439CO (CONICET) and PIDDEF 2014-2017 nro. 8 (Ministerio de Defensa, Argentina), \nand by a one month invitation of SD by Paris Observatory.\nJJMM, SD, and LR acknowledge the cooperation project BE\/13\/07 (F.R.S.-FNRS\/MinCyT) between Belgium and Argentina.\nS.D. is member of the Carrera del Investigador Cien\\-t\\'\\i fi\\-co, CONICET. J.J. Mas\\'\\i as-Meza is a fellow of CONICET.\nL. R. acknowledges support from the Belgian Federal Science Policy Office through the ESA-PRODEX program. \nThis research has been partially funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (IAP P7\/08 CHARM). \n\\end{acknowledgements}\n\n   \\bibliographystyle{aa}\n   ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\section{Empirical accuracy of influence functions on constructed groups}\\label{sec:accuracy}\n\nHow well do influence functions estimate the effect of (removing) a group of training points?\nIf $n$ is large and we remove a subset $w$ uniformly at random, the new parameters $\\hat\\theta(\\bone - w)$ should remain close to $\\hat\\theta(\\bone)$ even when if fraction of removed points $\\alpha$ is non-negligible, so the influence error $|\\sI^{*}_f(w) - \\sI_f(w)|$ should be small.\nHowever, we are usually interested in removing coherent, non-random groups, e.g., all points from a data source or share some feature. In such settings, the parameters $\\hat\\theta(\\bone - w)$ and $\\hat\\theta(\\bone)$ might differ substantially, and the error $|\\sI^{*}_f(w) - \\sI_f(w)|$ could be large.\nPut another way, there could be a cluster of points such that removing one of those points would not change the model by much---so influence could be low---but removing all of them would.\n\nSurprisingly (to us), we found that even when removing large and coherent groups of points, the influence $\\sI_f(w)$ behaved consistently relative to the actual effect $\\sI^{*}_f(w)$ on test predictions, test losses, and self-loss, with two broad phenomena emerging:\n\\begin{enumerate}\n  \\item {\\bf Correlation}: $\\sI_f(w)$ and $\\sI^{*}_f(w)$ rank subsets of points $w$ similarly (e.g., high Spearman $\\rho$).\n  \\item {\\bf Underestimation}: $\\sI_f(w)$ and $\\sI^{*}_f(w)$ tend to have the same sign with $|\\sI_f(w)| < |\\sI^{*}_f(w)|$.\\footnote{\n  This holds with one exception: when measuring the change in test loss,\n  $f(\\theta) = \\ell({x_{\\mathrm{test}}}, {y_{\\mathrm{test}}}; \\theta)$,\n  underestimation only holds when actual effect $\\sI^{*}_f(w)$ is positive (\\reffig{acc}-Mid).\n  }\n\\end{enumerate}\nHere, we report results on 5 datasets chosen to span a range of applications, training set size $n$, and number of features $d$ (\\reftab{data-characteristics}).\\footnotemark{}\nIn an attempt to make the influence approximation as inaccurate as possible,\nwe constructed a variety of subsets, from small ($\\alpha = 0.25\\%$) to large ($\\alpha = 25\\%$), to be coherent and have considerable influence on the model.\nOn each dataset, we trained an $L_2$-regularized logistic regression model (or softmax for the multiclass tasks)\nand compared the influences and actual effects of these subsets.\n\n\\footnotetext{\nThe first 4 datasets involve hospital readmission prediction, spam classification, and object recognition, and were used in \\citet{koh2017understanding} to study the influence of individual points. The fifth dataset is a chemical-disease relationship (CDR) dataset \\citet{hancock2018babble}.\nIn \\refsec{applications}, we will also study the MultiNLI language inference dataset \\citep{williams2018broad}, which was omitted from the experiments here because its large size makes repeated retraining to compute the actual effect too expensive.\nSee \\refapp{ds-details} for dataset details.}\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{lrrrrrl}\nDataset          & Classes & $n$       & $d$     & $\\lambda\/n$ & Test acc. & Source \\\\\n\\hline\nDiabetes         &  $2$    & $20,000$  & $127$   & $2.2\\times 10^{-4}$    & $68.2\\%$  & \\citet{strack2014impact} \\\\\nEnron            &  $2$    & $4,137$   & $3,289$ & $1.0\\times 10^{-3}$    & $96.1\\%$  & \\citet{metsis2006spam} \\\\\nDogfish          &  $2$    & $1,800$   & $2,048$ & $2.2\\times 10^{-2}$    & $98.5\\%$  & \\citet{koh2017understanding} \\\\\nMNIST            &  $10$   & $55,000$  & $784$   & $1.0\\times 10^{-3}$    & $92.1\\%$  & \\citet{lecun1998gradient} \\\\\nCDR              &  $2$    & $24,177$  & $328$   & $1.0 \\times 10^{-4}$    & $67.4\\%$  & \\citet{hancock2018babble} \\\\\nMultiNLI         &  $3$    & $392,702$ & $600$     & $1.0 \\times 10^{-4}$    & $50.4\\%$  & \\citet{williams2018broad}\n\\end{tabular}\n\\caption{Dataset characteristics and the test accuracies that logistic regression achieves (with regularization $\\lambda$ selected by cross-validation). $n$ is the training set size and $d$ is the number of features.}\n\\label{tab:data-characteristics}\n\\end{table*}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-fig1.pdf}\n  \\vspace{3mm}\n  \\caption{\n  Influences vs. actual effects of coherent groups of points ranging from $0.25\\%$ to $25\\%$ in size. Each point corresponds to a group, and its color reflects how that group was constructed.\n  In Top and Mid, we show results for the test point with highest loss; other test points are similar (\\refapp{median}), though with more curvature for test loss (\\refapp{supp-curvature}).\n  The grey reference line has slope 1, and the red borders represent points that are not plotted because they are outside the x- or y-axis range.\n  We omit the top row for MNIST, as $\\theta^\\top {x_{\\mathrm{test}}}$ is not meaningful in the multi-class setting.\n  }\n  \\vspace{-12mm}\n  \\label{fig:acc}\n\\end{figure}\n\\paragraph{Group construction.}\nOur aim is to construct coherent groups that when removed will substantially change the model. To do so, we need to choose points that are similar in some way.\nSpecifically, for each dataset, we grouped points in 7 ways: 1) points that share feature values;\n2) points that cluster on their features or 3) on their gradients $\\nabla_\\theta\\ell(x, y, \\hat\\theta(\\bone))$);\n4) random points within the same class; 5) random points from any class.\nWe also grouped\n6) points with large positive and 7) negative influence on the test loss $\\ell({x_{\\mathrm{test}}}, {y_{\\mathrm{test}}}, \\hat\\theta(\\bone))$,\nsince intuitively, training points that all have high influence on a test point should act together to change the model substantially.\nOverall, for each dataset, we constructed\n1,700 subsets ranging in size from $0.25\\%$ to $25\\%$ of the training points. See \\refapp{expt-details} for more details.\n\n\n\\vspace{-1mm}\n\\paragraph{Results.} \\reffig{acc} shows that the influences and actual effects of all of these subsets on test prediction (Top), test loss (Mid), and self-loss (Bot) are highly correlated (Spearman $\\rho$ of $0.89$ to $0.99$ across all plots),\neven though the absolute and relative errors of the influence approximation can be quite large.\nMoreover, the influence of a group tends to underestimate its actual effect in all settings except for groups with negative influence on test loss (the left side of each plot in \\reffig{acc}-Mid).\nThese trends held across a wide range of regularizations $\\lambda$, though correlation increased with $\\lambda$ (\\refapp{reg}).\n\nIn \\refsec{applications}, we will use the CDR dataset \\citep{hancock2018babble} and the MultiNLI \\citep{williams2018broad} dataset to show that correlation and underestimation also apply to groups of data that arise naturally, and that influence functions can therefore be used to derive insights about real datasets and applications.\nBefore that, we first attempt to develop some theoretical insight into the results above.\n\n\\subsubsection*{Acknowledgments}\nWe are grateful to\nZhenghao Chen,\nBrad Efron,\nJean Feng,\nTatsunori Hashimoto,\nRobin Jia,\nStephen Mussmann,\nAditi Raghunathan,\nMarco T\\'ulio Ribeiro,\nNoah Simon,\nJacob Steinhardt,\nand Jian Zhang\nfor helpful discussions and comments.\nWe are further indebted to\nRyan Giordano,\nRuoxi Jia,\nand Will Stephenson for discussion about prior work,\nand\nSamuel Bowman,\nBraden Hancock,\nEmma Pierson,\nand Pranav Rajpurkar\nfor their assistance with applications and datasets.\nThis work was funded by an Open Philanthropy Project Award.\nPWK was supported by the Facebook Fellowship Program.\n\n\\section{Applications of influence functions on natural groups of data}\\label{sec:applications}\n\nThe analysis in \\refsec{theory} shows that\nthe cone constraint between predicted and actual group effects need not always hold.\nNonetheless, our experiments in \\refsec{accuracy} demonstrate that on real datasets, the correlation is much stronger than the theory predicts.\nWe now turn to using influence functions to predict group effects in two case studies where groups arise naturally.\n\n\n\n\\paragraph{Chemical-disease relation (CDR).}\nThe CDR dataset tackles the following task: given text about the relationship between a chemical and a disease, predict if the chemical causes the disease.\nIt was collected via data programming, where users provide labeling functions (LFs)---instead of labels---that take in an unlabeled point and either abstain or output a heuristic label \\citep{ratner2016data}. Specifically, \\citet{hancock2018babble} collected natural language explanations of provided classifications; parsed those explanations into LFs; and used those LFs to label a large pool of data (\\refapp{bl-details}).\n\nWe used influence functions to study two important properties of LFs: \\emph{coverage}, the fraction of unlabeled points for which an LF outputs a non-abstaining label; and \\emph{precision}, the proportion of correct labels output.\nWe associated each LF with the group of points that it labeled, and computed its influence; as expected, these correlated with actual effects on overall test loss (Spearman $\\rho=1$; \\reffig{babble-acc}).\nLFs with higher coverage had more influence (\\reffig{combined-apps}-Left; see also \\reffig{babble-size-corr}), but surprisingly, LFs with higher precision did not (\\reffig{combined-apps}-Mid).\nThe association with coverage stems at least partially from class balance:\neach LF outputs either all positive or all negative labels,\nso removing an LF with high coverage changes the class balance and consequently improves test performance on one class at the expense of the other (\\reffig{combined-apps}-Left).\nWhile these findings are not causal claims, they suggest that the coverage of an LF, rather than its precision, might have a stronger effect on its overall contribution to test performance.\n\n\n\n\\paragraph{MultiNLI.}\nThe MultiNLI dataset deals with natural language inference: determining if a pair of sentences agree, contradict, or are neutral. \\citet{williams2018broad} presented\ncrowdworkers with initial sentences from five genres and asked them to generate follow-on sentences that were neutral or in agreement\/contradiction (\\refapp{multinli-details}).\nWe studied the effect that each crowdworker had on the model's test set performance by computing the influence of the examples they created on overall test loss (Spearman $\\rho$ of $0.77$ to $0.86$ with actual effects across different genres; see \\reffig{mnli-acc}).\n\n\nStudying the influence of each crowdworker reveals that the number of examples a crowdworker created was not predictive of influence on test performance: e.g., the most prolific crowdworker contributed 35,000 examples but had negative influence, and we verified that removing all of those examples and retraining the model indeed made overall test performance worse (\\reffig{combined-apps}-Right).\nCuriously, this effect was genre-specific: crowdworkers who improved performance on some genres would lower performance on others (\\reffig{mnli-genre-categories}), even though the number of examples they contributed to a genre did not correlate with their influence on it (\\reffig{mnli-genre-contrib}).\nWe note that these results are obtained on a baseline logistic regression model built on top of a continuous bag-of-words representation.\nIdentifying precisely what makes a crowdworker's contributions useful, especially on higher-performing models, could help us improve dataset collection and credit attribution as well as better understand the biases due to annotator effects \\citep{geva2019annotator}.\n\n\n\\begin{figure}[t]\n  \\centering\n  \\vspace{-4mm}\n  \\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={right,top},capbesidewidth=58mm}}]{figure}[\\FBwidth]\n  {\\caption{\n    In CDR, the influence of a labeling function (LF) on test performance is predicted by its coverage (Left) but not its precision (Mid).\n    However, in MultiNLI, the number of examples contributed by a crowdworkers is not predictive of its influence (Right).\n    For CDR, LFs output either all $+$ or all $-$ labels; we plot the influence of each LF on the test points of the same class.\n  }\\label{fig:combined-apps}}\n  {\\includegraphics[width=0.55\\textwidth]{figs\/plots\/paper-combined-apps.pdf}}\n\\end{figure}\n\n\\section{Discussion}\\label{sec:discussion}\n\nIn this paper, we showed empirically that the influences of groups of points are highly correlated with, and consistently underestimate, their actual effects across a range of datasets, types of groups, and sizes.\nThese phenomena allows us to use influence functions to better understand the ``different stories that different parts of the data tell,'' in the words of \\citet{hampel1986robust}.\nWe showed that we can gain insight into the effects of a labeling function in data programming, or a crowdworker in a crowdsourced dataset, by computing the influence of their corresponding group effects.\n\nWhile these applications involved predefined groups, influence functions could potentially also discover coherent, semantically-relevant groups in the data.\nThey can also be used to approximate Shapley values,\nwhich are a different but related way of measuring the effect of data points;\nsee, e.g., \\citet{jia2019towards} and \\citet{ghorbani2019data}.\nSeparately, influence functions can also estimate the effects of \\emph{adding} training points. In this context, underestimation turns into overestimation, i.e., the influence of adding a group of training points tends to overestimate the actual effect of adding that group. This raises the possibility of using influence functions to evaluate the vulnerability of a given dataset and model to data poisoning attacks \\citep{steinhardt2017certified}.\n\nOur theoretical analysis showed that while correlation and underestimation hold in some restricted settings, they need not hold in general, realistic settings. This gap between theory and experiments opens up important directions for future work:\nWhy do we observe such striking correlation between predicted and actual effects on real data?\nTo what extent is this due to the specific model, datasets, or subsets used?\nDo these trends hold for non-convex models like neural networks?\nOur work suggests that there could be distributional assumptions that hold for real data and give rise to the broad phenomena of correlation and underestimation.\nOne promising lead is the surprising observation that the Newton approximation is much more accurate than influence at predicting group effects,\nwhich holds out the hope that we can understand group effects using just low-order terms (since the Newton approximation only uses the first and second derivatives of the loss) without needing to account for the whole loss function through higher order terms (as in \\citet{giordano2019higher}).\n\n\\section{Dataset details}\\label{sec:ds-details}\nWe used the same versions of the Diabetes, Enron, Dogfish, and MNIST datasets as \\citet{koh2017understanding}, since the examination of the accuracy of influence functions for large perturbations is a natural extension of their studies of small perturbations. Additionally, we applied influence to more natural settings in CDR and MultiNLI; here, we discuss their preprocessing pipelines.\n\n\\subsection{CDR}\\label{sec:bl-details}\n\n\\citet{hancock2018babble} established the BabbleLabble framework for data programming, following the following pipeline: They took labeled examples with natural language explanations, parsed the explanations into programmatic labeling functions (LFs) via a semantic parser, and filtered out obviously incorrect LFs. Then, they applied the remaining LFs to unlabeled data to create a sparse label matrix, from which they learned a label aggregator that outputs a noisily labeled training set. Finally, they ran $L_2$-regularized logistic regression on a set of basic linguistic features with the noisy labels.\n\nThey demonstrated their method on three datasets: Spouse, CDR, and Protein. The Protein dataset was not publicly available, and the vast majority of Spouse was labeled by a single LF, hence we chose to use CDR. This dataset's associated task involved identifying whether, according to a given sentence, a given chemical causes a given disease. For instance, the sentence\n``Young women on replacement \\textit{estrogens} for ovarian failure after cancer therapy may also have increased risk of \\textit{endometrial carcinoma} and should be examined periodically.''\nwould be labeled True, since it indicates that estrogens may cause endometrial carcinoma \\citep{hancock2018babble}.\nThe sentences and ground truth labels were sourced from the 2015 BioCreative chemical-disease relation dataset \\citep{wei2015overview}.\n\nIn our application, we began with their 28 LFs and the corresponding label matrix. For simplicity, we did not learn a label aggregator; instead, if an example $x$ was given labels $y_{i_1},y_{i_2},\\ldots,y_{i_k}$ by $k$ LFs $i_1,\\ldots,i_k$, then we created $k$ copies of $x$, each with weight $1\/k$. The subset of points corresponding to LF $i_1$ then included one instance of $x$ with weight $1\/k$. This weighting was taken into account in model training as well as in calculations of influence and actual effect. In addition, we used $L_1$-regularization for feature selection, reducing the number of features to 328 while still achieving similar F1 score to \\citep{hancock2018babble}; they reported an F1 of 42.3, while we achieved 42.0. After feature selection, we remove the $L_1$-regularization and train a $L_2$-regularized logistic regression model. We assume that the feature selection step is static and not affected by removing groups of data (though in general this assumption is not true); we therefore do not include feature selection in our influence calculations.\n\nWe note that in BabbleLabble, a given LF can never output positive on one example but negative on another. Hence, some LFs are positive (unable to output negative and only able to abstain or output positive), while the others are negative (unable to output positive and only able to abstain or output negative).\n\n\\subsection{MultiNLI}\\label{sec:multinli-details}\n\n\\citet{williams2018broad} created the MultiNLI dataset for the task of natural language inference: determining if a pair of sentences agree, contradict, or are neutral. To do so, they presented crowdworkers with initial sentences and asked them to generate follow-on sentences that were neutral or in agreement\/contradiction. For example, a crowdworker may be presented with ``Met my first girlfriend that way.'' and write the contradicting sentence ``I didn't meet my first girlfriend until later.'' \\citep{williams2018broad}. Thus, each of the 380 crowdworkers generated a subset of the dataset. We used these subsets in our application of influence.\n\nThe training set consisted of 392,702 examples from five genres. The development set consisted of 10,000 ``matched'' examples from the same five genres as the training set, as well as 10,000 ``mismatched'' examples from five new genres. The test set was put on Kaggle as an open competition, hence we do not have its labels and could not use it; therefore, we use the development set as the test set.\n\nThe continuous bag-of-words baseline in \\citet{williams2018broad} first converted the raw text of each sentence in the pair into a vector by treating the sentence as a continuous bag of words and simply averaging the 300-dimensional GloVe vector embeddings. This converted a pair of sentences into vectors $a,b$. They then concatenated $[a,b,a-b,a \\odot b]$ into a 1200D vector, where $a \\odot b$ denotes the element-wise product. Finally, they treated this as input to a neural network with three hidden layers and fine-tuned the entire model, including word embeddings (more details in \\citep{williams2018broad}).\n\nFor our application, we truncated their baseline and just used the concatenation of $a$ and $b$ as the representation for every example. By running logistic regression on this, we achieved test accuracy of $50.4\\%$ (vs. their baseline's $64.7\\%$; the performance difference comes from the additional dimensions in their vector embeddings and the finetuning through the neural network). Future work could explore influence in the setting of more complex and higher-performing models.\n\n\\section{Experimental details for comparing influence vs. actual effects on constructed groups}\\label{sec:expt-details}\n\\subsection{Model training}\nFor all experiments in \\refsec{accuracy}, we trained a logistic regression model (or softmax for multiclass) using\n\\texttt{sklearn.linear\\_model.LogisticRegression.fit},\n fitting the intercept but only applying $L_2$-regularization to the weights.\nTo choose the regularization strength $\\lambda$, we conducted 5-fold\ncross-validation across 10 possible values of $\\lambda\/n$ logarithmically\nspaced between $1.0\\times 10^{-4}$ and $1.0\\times 10^{-1}$, inclusive,\nselecting the regularization that yielded the highest cross-validation accuracy\n(except on the CDR dataset, where we selected regularization based on\ncross-validation F1 score to account for class imbalance as per\n\\citet{hancock2018babble}'s procedure).\n\n\\subsection{Group construction}\\label{sec:subsetgen}\nFor each dataset, we constructed groups of various sizes relative to the entire dataset by considering 100 sizes linearly spaced between $0.25\\%$ and $25\\%$ of the dataset. For each of these 100 sizes, we constructed one group with each of the following methods:\n\\begin{enumerate}\n  \\item Shared features: We selected a single feature uniformly at random and sorted the dataset along this selected feature. Next, we selected an training point uniformly at random. We then constructed a group of size $s$ that consisted of the $s$ unique training points that were closest to the chosen point, as measured by their values in the selected feature. We randomly sampled a feature and initial training point for each different group constructed in this way.\n\n  \\item Feature clustering: We clustered the dataset with respect to raw features via \\texttt{scipy.cluster.hierarchy.fclusterdata} with \\texttt{t} set to $1$, as well as with \\texttt{sklearn.cluster.KMeans.fit} with \\texttt{n\\_clusters} taking on values $4,8,16,32,64,128$. Since hierarchical clustering determines cluster sizes automatically with a principled heuristic and we try a range of values for \\texttt{n\\_clusters} in $k$-means, this recovers clusters with a large range of sizes. The clustering with $\\texttt{n\\_clusters}=4$ also guarantees (via the pigeonhole principle) that there is at least one cluster which contains at least $25\\%$ of the dataset. From all the clusters that are at least the size of the desired group, we chose one uniformly at random and chose the group uniformly at random and without replacement from the training points in this cluster.\n\n  \\item Gradient clustering: We followed the same procedure as ``Feature clustering,'' except that we clustered the dataset with respect to $\\nabla_\\theta \\ell(x,y;\\hat\\theta(\\bone))$, i.e. each training point was represented by the gradient of the loss on that point.\n\n  \\item Random within class: We considered all classes with at least as many training points as the size of the desired group. From these classes, we chose one uniformly at random. Then, we chose the group uniformly at random and without replacement from all training points in this class.\n\n\t\\item Random: We picked a group uniformly at random and without replacement from the entire dataset.\n\\end{enumerate}\n\nThe above methods gave us a total of 500 groups (100 groups per method) for each dataset, with the exception of the ``random within class'' method for MNIST. Since MNIST has 10 classes, each with only $10\\%$ of the data, we skipped over groups of size $>10\\%$ just for the ``random within class'' groups.\n\nIn addition, we selected 3 random test points and the 3 test points with highest loss; we intend these to represent the average case and the more extreme case that may be relevant to model developers who want to debug errors that their model outputs. For each of these 6 test points, we selected groups that had large positive influence on its test loss. More specifically, we proceeded in 3 stages:\n\\begin{enumerate}\n\t\\item We considered 33 group sizes linearly spaced between $0.25\\%$ and $2.5\\%$ of the dataset, and for any size $s$ out of these 33, we selected a group uniformly at random and without replacement from training points in the top $1.5\\times 2.5\\%$ of the dataset, ordered according to their influence on the test point of interest.\n\t\\item This was similar to the first stage, but with 33 sizes spaced between $0.25\\%$ and $10\\%$ and groups chosen from the top $1.5\\times 10\\%$ of the dataset.\n\t\\item Finally, we considered 34 sizes spaced between $0.25\\%$ and $25\\%$, with groups chosen from the top $1.5\\times 25\\%$ of the dataset.\n\\end{enumerate}\nLarger groups tend to have lower average influence than smaller groups, since by necessity, the group must contain points farther from the top.\nThis multi-stage approach ensured that we would select small groups with both a high average influence and also with a low average influence, so that we could compare them to larger groups and mitigate confounding the group size with its average influence.\n\nFinally, we repeated this last method of group construction for groups with large negative influence on test point loss.\n\nUsing these 6 test points, we generated 1,200 groups (100 subsets per group, with 6 test points, and drawing from the positive and negative tails). In total, we therefore generated 1,700 groups per dataset (except MNIST).\n\n\n\\subsection{Comparison of influence and actual effect}\\label{sec:compare}\n\nTo produce \\reffig{acc}, we selected groups as described in \\refapp{subsetgen}. We retrained the model once for each group, excluding the group in order to calculate its actual effect. To compute all groups' influences, we first calculated the influence of every individual training point using the procedure of \\citet{koh2017understanding}. Then, to compute the influence on test prediction or loss of some group, we simply added the relevant individual influences (in CDR, we weighted these individual influences according to that point's weight; see \\refapp{bl-details}). To compute the influence on self-loss of some group, we summed up the gradients of the loss of each training point to compute $g_\\bone(w)$, we calculated the inverse Hessian vector product $H_{\\lambda, \\bone}^{-1} g_\\bone(w)$ and took its dot product with $g_\\bone(w)$ (again, we modified this with appropriate weighting for individual points in CDR).\n\n\\section{Introduction}\\label{sec:intro}\n\nInfluence functions \\citep{jaeckel1972infinitesimal, hampel1974influence, cook1977detection} estimate the effect of removing an\nindividual training point on a model's predictions without the computationally-prohibitive cost of retraining the model.\nTracing a model's output back to its training data can be useful:\ninfluence functions have been recently applied to explain predictions \\citep{koh2017understanding},\nproduce confidence intervals \\citep{schulam2019can},\ninvestigate model bias \\citep{brunet2018understanding, wang2019repairing},\nimprove human trust \\citep{zhou2019effects},\nand even craft data poisoning attacks \\citep{koh2019stronger}.\n\nInfluence functions are based on first-order Taylor approximations that are accurate for estimating small perturbations to the model,\nwhich makes them suitable for predicting the effects of removing individual training points on the model.\nHowever, we often want to study the effects of removing \\emph{groups} of points, which represent large perturbations to the data.\nFor example, we might wish to analyze the effect of data collected from different experimental batches \\citep{leek2010tackling} or demographic groups \\citep{chen2018my};\napportion credit between crowdworkers, each of whom generated part of the data \\citep{arrieta2018should};\nor, in a multi-party learning setting, ensure that no individual user has too much influence on the joint model \\citep{hayes2018contamination}.\nAre influence functions still accurate when predicting the effects of (removing) these larger groups?\n\nIn this paper, we first show empirically that on real datasets and across a broad variety of groups of data, the predicted and actual effects are strikingly \\emph{correlated} (Spearman $\\rho$ of $0.8$ to $1.0$),\nsuch that the groups with the largest actual effect also tend to have the largest predicted effect. Moreover, the predicted effect tends to \\emph{underestimate} the actual effect, suggesting that it could be an approximate lower bound in practice.\nUsing influence functions to predict the actual effect of removing large, coherent groups of data can therefore still be useful,\neven though the violation of the small-perturbation assumption can result in high absolute and relative errors between the predicted and actual effects.\n\nWhat explains these phenomena of correlation and underestimation?\nPrior theoretical work focused on establishing the conditions under which this influence approximation is accurate, i.e., the error between the actual and predicted effects is small \\citep{giordano2019swiss, rad2018scalable}.\nHowever, in our setting of removing large, coherent groups of data,\nthis error can be quite large.\nAs a first step towards understanding the behavior of the influence approximation in this regime,\nwe characterize the relationship between the predicted and actual effects of a group via the one-step Newton approximation \\citep{pregibon1981logistic},\nwhich we find is a surprisingly accurate approximation in practice.\nWe show that correlation and underestimation arise under certain settings\n(e.g., removing multiple copies of a single training point),\nbut need not hold in general,\nwhich opens up the intriguing question of why we observe those phenomena across a wide range of empirical settings.\n\nFinally, we exploit the correlation of predicted and actual group effects in two example case studies: a chemical-disease relationship (CDR) task, where the groups correspond to different labeling functions \\citep{hancock2018babble}, and a natural language inference (NLI) task \\citep{williams2018broad}, where the groups come from different crowdworkers.\nOn the CDR task, we find that the influence of each labeling function correlates with its size (the number of examples it labels) but not its average accuracy, which suggests that practitioners should focus on the coverage of the labeling functions they construct. In contrast, on the NLI task, we find that the influence of each crowdworker is uncorrelated with the number of examples they contibute, which suggests that practitioners should focus on how to elicit high-quality examples from crowdworkers over increasing quantity.\n\n\n\n\\section{Background and problem setup}\\label{sec:setting}\nConsider learning a predictive model with parameters $\\theta \\in \\Theta$ that\nmaps from an input space $\\sX$ to an output space $\\sY$.\nWe are given $n$ training points $\\{(x_1, y_1), \\ldots, (x_n, y_n)\\}$\nand a loss function $\\ell(x, y, \\theta)$ that is twice-differentiable and convex in $\\theta$.\nTo train the model, we select the model parameters\n\\begin{align}\n  \\hat\\theta(\\bone) = {\\arg\\min}_{\\theta\\in\\Theta}  \\left[ \\sum_{i=1}^n \\ell(x_i, y_i; \\theta) \\right] + \\frac{\\lambda}{2} \\|\\theta\\|_2^2\n\\end{align}\nthat minimize the $L_2$-regularized empirical risk,\nwhere $\\lambda > 0$ controls regularization strength.\nThe all-ones vector $\\bone$ in $\\hat\\theta(\\bone)$ denotes that the initial training points all have uniform sample weights.\n\nOur goal is to measure the effects of different groups of training data\non the model: if we removed a subset of training points $W$,\nhow much would the model $\\hat\\theta$ change?\nConcretely, we define a vector $w \\in \\{0,1\\}^n$ of sample weights with $w_i = \\1((x_i, y_i) \\in W)$\nand consider the modified parameters\n\\begin{align}\n\\hat\\theta(\\bone - w) = {\\arg\\min}_{\\theta\\in\\Theta}  \\left[\\sum_{i=1}^n (1 - w_i) \\ell(x_i, y_i; \\theta) \\right] + \\frac{\\lambda}{2} \\|\\theta\\|_2^2\n\\end{align}\ncorresponding to retraining the model after excluding $W$.\nWe refer to $w$ as the subset (corresponding to $W$);\nthe number of removed points as $\\|w\\|_1$;\nand the fraction of removed points as $\\alpha = \\|w\\|_1 \/ n$.\n\nThe \\emph{actual effect} $\\sI^{*}_f:[0,1]^n \\to \\R$ of the subset $w$ is\n\\begin{align}\n  \\sI^{*}_f(w) = f(\\hat\\theta(\\bone - w)) - f(\\hat\\theta(\\bone)),\n\\end{align}\nwhere the evaluation function $f:\\Theta\\to\\R$ measures a quantity of interest. Specifically, we study:\n\\begin{itemize}\n  \\item The \\emph{change in test prediction}, with $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$. Linear models (for regression or binary classification) make predictions that are functions of $\\theta^\\top {x_{\\mathrm{test}}}$, so this measures the effect that removing a subset will have on the model's prediction for some test point ${x_{\\mathrm{test}}}$.\n\n  \\item The \\emph{change in test loss}, with $f(\\theta) = \\ell({x_{\\mathrm{test}}}, {y_{\\mathrm{test}}}; \\theta)$, which is similar to the test prediction.\n\n  \\item The \\emph{change in self-loss}, with $f(\\theta) = \\sum_{i=1}^n w_i \\ell(x_i, y_i; \\theta)$, measures the increase in loss on the removed points $w$. Its average over all subsets of size $\\|w\\|_1$ is the estimated extra loss that leave-$\\|w\\|_1$-out cross-validation (CV) measures over the training loss.\n\n\\end{itemize}\n\n\n\\subsection{Influence functions}\nThe issue with computing the actual effect $\\sI^{*}_f(w)$ is that retraining the model to compute $\\hat\\theta(\\bone - w)$ for each subset $w$ can be prohibitively expensive.\nInfluence functions provide a relatively efficient first-order approximation to $\\sI^{*}_f(w)$ that avoids retraining.\n\nConsider the function $q_w: [0, 1] \\to \\R$ with\n$q_w(t) = f\\bigl(\\hat\\theta(\\bone - tw)\\bigr)$,\nsuch that the actual effect $\\sI^{*}_f(w)$ can be written as $q_w(1) - q_w(0)$.\nWe define the \\emph{predicted effect} of the subset $w$ to be\nits \\emph{influence} $\\sI_f(w) = q_w'(0) \\approx q_w(1) - q_w(0)$;\nin this paper, we use the term predicted effect interchangeably with influence.\nIntuitively, influence measures the effect of removing an infinitesimal weight from each point in $w$ and then linearly extrapolates to removing all of $w$.\\footnote{In the statistics literature, influence typically refers to the effect of \\emph{adding} weight, so the sign is flipped.} By taking a Taylor approximation (see, e.g., \\citet{hampel1986robust} for details), the influence can be computed as\n\\begin{align}\n\\sI_f(w) \\eqdef q_w'(0)\\nonumber\n&= \\nabla_\\theta f\\bigl(\\hat\\theta(\\bone)\\bigr)^\\top \\left[ \\frac{d}{dt}\\hat\\theta(\\bone - tw)\\Bigr|_{\\substack{t=0}} \\right]\\nonumber\\\\\n&= \\nabla_\\theta f\\bigl(\\hat\\theta(\\bone)\\bigr)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w),\n\\end{align}\nwhere\n$g_\\bone(w) = \\sum_{i=1}^n w_i\\nabla_\\theta\\ell(x_i, y_i; \\hat\\theta(\\bone))$,\n$H_\\bone = \\sum_{i=1}^n \\nabla_\\theta^2\\ell(x_i, y_i; \\hat\\theta(\\bone))$,\nand $H_{\\lambda, \\bone} = H_\\bone + \\lambda I$.\nWhen measuring the change in test prediction or test loss, influence is additive: if $w = w_1 + w_2$, then $\\sI_f(w) = \\sI_f(w_1) + \\sI_f(w_2)$, i.e., the influence of a subset is the sum of influences of its constituent points,\nand we can efficiently compute the influence of any subset by pre-computing the influence of each individual point (e.g., by taking a single inverse Hessian-vector product, as in\n\\citet{koh2017understanding}).\nHowever, when measuring the change in self-loss, influence is not additive and requires a separate calculation for each subset removed.\n\n\\subsection{Relation to prior work}\\label{sec:related}\nInfluence functions---introduced in the seminal work of \\citet{hampel1974influence} and in \\citet{jaeckel1972infinitesimal}, where it was called the infinitesimal jackknife---have a rich history in robust statistics.\nThe use of influence functions in the ML community is more recent, though growing; in \\refsec{intro}, we provide references for several recent applications of influence functions in ML.\n\nRemoving a single training point, especially when the total number of points $n$ is large, represents a small perturbation to the training distribution,\nso we expect the first-order influence approximation to be accurate.\nIndeed, prior work on the accuracy of influence has focused on this regime: e.g.,\n\\citet{debruyne2008model, liu2014efficient, rad2018scalable, giordano2019swiss} give\nevidence that the influence on self-loss can approximate LOOCV, and \\citet{koh2017understanding} similarly examined the accuracy of estimating the change in test loss after removing single training points.\n\nHowever, removing a constant fraction $\\alpha$ of the training data represents a large perturbation to the training distribution.\nTo the best of our knowledge,\nthis setting has\nnot been empirically studied;\nperhaps the closest work is \\citet{khanna2019interpreting}'s use of Bayesian quadrature to estimate a maximally influential subset.\nInstead, older references have alluded to the phenomena of correlation and underestimation we observe:\n\\citet{pregibon1981logistic} note that influence tends to be conservative,\nwhile \\citet{hampel1986robust} say that ``bold extrapolations'' (i.e., large perturbations) are often still useful.\nOn the theoretical front, \\citet{giordano2019swiss} established\nfinite-sample error bounds that apply to groups, e.g., showing that the leave-$k$-out approximation is consistent as the fraction of removed points $\\alpha \\to 0$.\nOur focus is instead on the relationship of the actual effect $\\sI^{*}_f(w)$ and predicted effect (influence) $\\sI_f(w)$ in the regime where $\\alpha$ is constant and\nthe error $|\\sI^{*}_f(w) - \\sI_f(w)|$ is large.\n\n\\section{Proofs}\\label{sec:proofs}\n\\subsection{Notation}\\label{sec:notation}\nWe first review the notation given in \\refsec{setting} and introduce new definitions that will be useful in the sequel. We define the empirical risk as\n\\begin{align*}\n  L_s(\\theta) \\eqdef \\left[ \\sum_{i=1}^n s_i \\ell(x_i, y_i; \\theta)\\right] + \\frac{\\lambda}{2} \\|\\theta\\|_2^2,\n\\end{align*}\nsuch that the optimal parameters are $\\hat\\theta(s) \\eqdef {\\arg\\min}_{\\theta\\in\\Theta} L_s(\\theta)$.\n\nGiven sample weight vectors $r, s \\in \\R^n$, we define the derivatives\n\\begin{align*}\n  g_r(s) &\\eqdef \\sum_{i=1}^n s_i\\nabla_\\theta \\ell(x_i, y_i; \\hat\\theta(r))\\\\\n  H_r(s) &\\eqdef \\sum_{i=1}^n s_i \\nabla^2_\\theta \\ell(x_i, y_i; \\hat\\theta(r)).\n\\end{align*}\nIf the argument $s$ is omitted, it is assumed to be equal to $r$. For example,\n\\begin{align*}\n  H_\\bone \\eqdef \\sum_{i=1}^n \\nabla^2_\\theta \\ell(x_i, y_i; \\hat\\theta(\\bone)).\n\\end{align*}\nIf $H$ has a $\\lambda$ subscript, then we add $\\lambda I$. For example,\n\\begin{align*}\n  H_{\\lambda, \\bone} \\eqdef H_\\bone + \\lambda I.\n\\end{align*}\n\nFor a given dataset, we define the following constants:\n\\begin{align*}\n  C_\\ell &= \\max_{1 \\leq i \\leq n} \\twonorm{\\nabla_\\theta \\ell(x_i,y_i,\\hat\\theta(\\bone))},\\\\\n  \\sigma_{\\mathrm{min}} &= \\text{smallest singular value of}\\ H_\\bone,\\\\\n  \\sigma_{\\mathrm{max}} &= \\text{largest singular value of}\\ H_\\bone.\n\\end{align*}\n\nTo avoid confusion with the vector 2-norm, we will use the operator norm $\\opnorm{\\cdot}$ to denote the matrix 2-norm.\n\nIn the sequel, we study the order-3 tensor $\\nabla_\\theta^3 f(\\hat\\theta(\\bone))$. We define its product with a vector (which returns a matrix) as a contraction along the last dimension:\n\\begin{equation*}\n  \\left\\langle \\nabla_\\theta^3 f(\\hat\\theta(\\bone)), v \\right\\rangle_{ij} \\eqdef \\sum_k \\frac{\\partial^3 f(\\hat\\theta(\\bone))}{\\partial \\theta_i \\partial \\theta_j \\partial \\theta_k} v_k.\n\\end{equation*}\n\n\\subsection{Assumptions}\\label{sec:assumptions}\nWe make the following assumptions on the derivatives of the loss $\\ell(x, y, \\theta)$\nand the evaluation function $f(\\theta)$.\n\\begin{assumption}[Positive-definiteness and Lipschitz continuity of $H$]\\label{assump:l-const}\n  The loss $\\ell(x, y, \\theta)$ is convex and twice-differentiable in $\\theta$, with positive regularization $\\lambda > 0$.\n  Further, there exists $C_H\\in\\R$ such that\n  \\[\n    \\opnorm{\\nabla_\\theta^2 \\ell(x, y, \\theta_1) - \\nabla_\\theta^2\\ell(x, y, \\theta_2)} \\leq C_H \\twonorm{\\theta_1 - \\theta_2}\n  \\]\n  for all $(x, y) \\in \\sX \\times \\sY$ and $\\theta_1, \\theta_2 \\in \\Theta$. This is a bound on the third derivative of $\\ell$, if it exists.\n\\end{assumption}\n\\begin{assumption}[Bounded derivatives of $f$]\\label{assump:q-const}\n  $f(\\theta)$ is thrice-differentiable, with $C_f, C_{f, 3} \\in \\R$ such that\n  \\begin{align*}\n    C_{f} &= \\sup_{\\theta \\in \\Theta} \\twonorm{\\nabla_\\theta f(\\theta)},\n    &C_{f,3} &= \\sup_{v \\in \\Theta, \\twonorm{v} = 1} \\opnorm{\n    \\left\\langle \\nabla_\\theta^3 f(\\hat\\theta(\\bone)), v \\right\\rangle}.\n  \\end{align*}\n\\end{assumption}\nThese assumptions apply to all the results that follow below.\n\n\\subsection{Bounding the error of the one-step Newton approximation}\\label{sec:proof-newton}\n\n\n\\begin{repproposition}{prop:newton-actual}[Restated]\n  Let the Newton error be $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w) \\eqdef \\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w)$.\n  Then under \\refassumps{l-const}{q-const},\n  \\begin{align*}\n      \\abs{\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)} \\leq \\frac{n \\|w\\|_1^2 C_f C_H C_\\ell^2}{(\\sigma_{\\mathrm{min}} + \\lambda)^3}.\n  \\end{align*}\n  $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$ only involves third-order or higher derivatives of the loss, so it is 0 for quadratic losses.\n\\end{repproposition}\n\\begin{proof}\n  This proof is adapted to our setting from the standard analysis of the Newton method in convex optimization \\citep{boyd2004convex}.\n\n  First, note that $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w) = \\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w) = f(\\hat\\theta(\\bone - w)) - f(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w}))$.\n  We will bound the norm of the difference of the parameters $\\twonorm{\\hat\\theta(\\bone - w) - \\hat\\theta_{\\mathrm{Nt}}({\\bone - w})}$; the desired bound on $f$ then follows from the assumption that $f$ has gradients bounded by $C_f$ and is therefore Lipschitz.\n\n  Since $L_{\\bone - w}(\\theta)$ is strongly convex (with parameter $\\sigma_{\\mathrm{min}} + \\lambda$) and minimized by $\\hat\\theta(\\bone - w)$, we can bound the distance $\\twonorm{\\hat\\theta(\\bone - w) - \\hat\\theta_{\\mathrm{Nt}}({\\bone - w})}$ in terms of the norm of the gradient at $\\hat\\theta_{\\mathrm{Nt}}({\\bone - w})$:\n  \\begin{align*}\n    \\twonorm{\\hat\\theta(\\bone - w) - \\hat\\theta_{\\mathrm{Nt}}({\\bone - w})} &\\leq \\frac{2}{\\sigma_{\\mathrm{min}} + \\lambda}\\twonorm{\\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w})\\right)}.\n  \\end{align*}\n\n  Therefore, the problem reduces to bounding $\\twonorm{\\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w})\\right)}$.\n\n  We start by expressing the Newton step $\\Delta\\theta_{\\mathrm{Nt}}(w)$ in terms of the first and second derivatives of the empirical risk $L_{\\bone - w}(\\theta)$:\n  \\begin{align*}\n    g_\\bone(w) &= \\sum_{i=1}^n w_i \\nabla_\\theta\\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= -\\sum_{i=1}^n (1 - w_i) \\nabla_\\theta\\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= -\\nabla_\\theta L_{\\bone - w}(\\hat\\theta(\\bone)),\\\\\n    H_{\\lambda, \\bone}({\\bone - w}) &= \\sum_{i=1}^n (1 - w_i) \\nabla_\\theta^2 \\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= \\nabla_\\theta^2 L_{\\bone - w}(\\hat\\theta(\\bone)),\\\\\n    \\Delta\\theta_{\\mathrm{Nt}}(w) &= H_{\\lambda, \\bone}({\\bone - w})^{-1} g_\\bone(w)\\\\\n    &= -\\left[ \\nabla_\\theta^2 L_{\\bone - w}(\\hat\\theta(\\bone)) \\right]^{-1} \\nabla_\\theta L_{\\bone - w}(\\hat\\theta(\\bone)),\n  \\end{align*}\n  where the second equality for $g_\\bone(w)$ comes from the fact that at the optimum $\\hat\\theta(\\bone)$, the sum of the gradients $\\sum_{i=1}^n \\nabla_\\theta\\ell(x_i, y_i; \\hat\\theta(\\bone))$ is 0.\n\n  With these expressions, we bound the norm of the gradient $\\nabla_\\theta L_{\\bone - w}(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w}))$:\n  \\begin{align*}\n    &\\twonorm{\\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w})\\right)}\\\\\n    &= \\twonorm{\\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta(\\bone) + \\Delta\\theta_{\\mathrm{Nt}}(w)\\right)}\\\\\n    &= \\twonorm{\\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta(\\bone) + \\Delta\\theta_{\\mathrm{Nt}}(w)\\right)\n       - \\nabla_\\theta L_{\\bone - w}\\left(\\hat\\theta(\\bone)\\right)\n       - \\nabla_\\theta^2 L_{\\bone - w}\\left(\\hat\\theta(\\bone)\\right) \\Delta\\theta_{\\mathrm{Nt}}(w)}\\\\\n    &= \\twonorm{\\int_0^1 \\left(\\nabla_\\theta^2 L_{\\bone - w}\\left(\\hat\\theta(\\bone) + t\\Delta\\theta_{\\mathrm{Nt}}(w) \\right) - \\nabla_\\theta^2 L_{\\bone - w}\\left(\\hat\\theta(\\bone)\\right)\\right) \\Delta\\theta_{\\mathrm{Nt}}(w) \\ dt}\\\\\n    &\\leq \\frac{n C_H}{2} \\twonorm{\\Delta\\theta_{\\mathrm{Nt}}(w)}^2\\\\\n    &= \\frac{n C_H}{2} \\twonorm{\\left[ \\nabla_\\theta^2 L_{\\bone - w}(\\hat\\theta(\\bone)) \\right]^{-1} \\nabla_\\theta L_{\\bone - w}(\\hat\\theta(\\bone))}^2\\\\\n    &\\leq \\frac{n C_H}{2(\\sigma_{\\mathrm{min}} + \\lambda)^2} \\twonorm{\\nabla_\\theta L_{\\bone - w}(\\hat\\theta(\\bone))}^2\\\\\n    &\\leq \\frac{n \\onenorm{w}^2 C_H C_\\ell^2}{2(\\sigma_{\\mathrm{min}} + \\lambda)^2}.\n  \\end{align*}\n  Putting together the successive bounds gives the result.\n\\end{proof}\n\n\\subsection{Characterizing the difference between the Newton approximation and influence}\\label{sec:proof-decomp}\nBefore proving \\refprop{newton-inf}, we first prove a lemma about the spectrum of the error matrix $D(w)$.\n\\begin{lemma}\\label{lem:sv-bound-d}\nThe matrix $D(w) \\eqdef \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$ has singular values bounded between 0 and $\\frac{\\sigma_{\\mathrm{max}}}{\\lambda}$.\n\\end{lemma}\n\\begin{proof}\nWe first show that $H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}$ has singular values bounded between 0 and $\\frac{\\sigma_{\\mathrm{max}}}{\\sigma_{\\mathrm{max}} + \\lambda}$.\nThe lower bound of 0 comes from the fact that $H_{\\lambda, \\bone}^{-\\inv{2}} H_\\bone(w) H_{\\lambda, \\bone}^{-\\inv{2}}$ is symmetric and $H_\\bone(w) \\succeq 0$.\n\nTo show the upper bound, first note that $H_\\bone(w) \\preceq H_\\bone(w) + H_\\bone(\\bone - w) = H_\\bone$ (recalling that $w\\in\\{0,1\\}^n$),\nand let $U \\Sigma U^\\top$ be the singular value decomposition of $H_\\bone$. Since $H_{\\lambda, \\bone} = H_\\bone + \\lambda I$, we have\n\\begin{align*}\n  H_{\\lambda, \\bone}^{-\\inv{2}} H_\\bone(w) H_{\\lambda, \\bone}^{-\\inv{2}} &=\n  \\bigl(H_\\bone + \\lambda I\\bigr)^{-\\inv{2}} H_\\bone(w) \\bigl(H_\\bone + \\lambda I\\bigr)^{-\\inv{2}}\\\\\n  &\\preceq \\bigl(H_\\bone + \\lambda I\\bigr)^{-\\inv{2}} H_\\bone \\bigl(H_\\bone + \\lambda I\\bigr)^{-\\inv{2}}\\\\\n  &= \\bigl(U (\\Sigma + \\lambda I) U^\\top\\bigr)^{-\\inv{2}} U \\Sigma U^\\top \\bigl(U (\\Sigma + \\lambda I) U^\\top\\bigr)^{-\\inv{2}}\\\\\n  &= U (\\Sigma + \\lambda I)^{-\\inv{2}} \\Sigma (\\Sigma + \\lambda I)^{-\\inv{2}} U^\\top,\n\\end{align*}\nso its maximum singular value is upper bounded by $\\frac{\\sigma_{\\mathrm{max}}}{\\sigma_{\\mathrm{max}} + \\lambda}$.\n\nThe bound on the singular values of $H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}$ implies that the singular values of $I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}$ lie in $\\left[\\frac{\\lambda}{\\sigma_{\\mathrm{max}} + \\lambda}, 1\\right]$.\nIn turn, this implies that the singular values of $\\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1}$ lie in $\\left[1, \\frac{\\sigma_{\\mathrm{max}} + \\lambda}{\\lambda}\\right]$. Subtracting 1 from each end (for the identity matrix) gives the desired result.\n\\end{proof}\n\n\\begin{repproposition}{prop:newton-inf}[Restated]\n  Under \\refassumps{l-const}{q-const}, the Newton-influence error $\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w)$ is\n  \\begin{align*}\n    \\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w) &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\n    \\ + \\ \\underbrace{\\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w) + \\mathrm{Err}_{f,3}(w),}_\\text{\n      Error from curvature of $f(\\cdot)$\n    }\n  \\end{align*}\n  with $D(w) \\eqdef \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$\n  and $H_\\bone(w) \\eqdef \\sum_{i=1}^n w_i \\nabla_\\theta^2\\ell(x_i, y_i; \\hat\\theta(\\bone))$.\n  The \\emph{error matrix} $D(w)$ has eigenvalues between 0 and $\\frac{\\sigma_{\\mathrm{max}}}{\\lambda}$,\n  where $\\sigma_{\\mathrm{max}}$ is the largest eigenvalue of $H_\\bone$.\n  The residual term $\\mathrm{Err}_{f,3}(w)$ captures the error due to third-order derivatives of $f(\\cdot)$ and is bounded by $\\abs{\\mathrm{Err}_{f,3}(w)} \\leq {\\|w\\|_1^3 C_{f, 3}C_\\ell^3}\/{6(\\sigma_{\\mathrm{min}} + \\lambda)^3}$.\n\\end{repproposition}\n\\begin{proof}\n  From the second-order Taylor expansion of $f$ about $\\hat\\theta(\\bone)$, there exists $0 \\leq \\xi \\leq 1$ such that\n  \\begin{align*}\n    \\sI^\\mathrm{Nt}_f(w) &= f(\\hat\\theta_{\\mathrm{Nt}}({\\bone - w})) - f(\\hat\\theta(\\bone))\\\\\n    &= f(\\hat\\theta(\\bone) + \\Delta\\theta_{\\mathrm{Nt}}(w)) - f(\\hat\\theta(\\bone))\\\\\n    &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\Delta\\theta_{\\mathrm{Nt}}(w) + \\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w) +\\\\\n    &\\quad \\inv{6} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\left\\langle \\nabla_\\theta^3 f(\\hat\\theta(\\bone) + \\xi\\Delta\\theta_{\\mathrm{Nt}}(w)), \\Delta\\theta_{\\mathrm{Nt}}(w) \\right\\rangle \\Delta\\theta_{\\mathrm{Nt}}(w)\\\\\n    &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\Delta\\theta_{\\mathrm{Nt}}(w) + \\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w) + \\mathrm{Err}_{f,3}(w),\n    \\numberthis \\label{eqn:newton-inf-proof-1}\n  \\end{align*}\n  where we define $\\mathrm{Err}_{f,3}(w) \\eqdef \\inv{6} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\left\\langle \\nabla_\\theta^3 f(\\hat\\theta(\\bone) + \\xi\\Delta\\theta_{\\mathrm{Nt}}(w)), \\Delta\\theta_{\\mathrm{Nt}}(w) \\right\\rangle \\Delta\\theta_{\\mathrm{Nt}}(w)$ to be the error due to third-order and higher derivatives of $f$.\n\n  We can express the difference between the first-order Taylor term $\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\Delta\\theta_{\\mathrm{Nt}}(w)$ above and the first-order influence approximation $\\sI_f(w) = q'_w(0) = \\nabla_\\theta f\\bigl(\\hat\\theta(\\bone)\\bigr)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)$ as\n  \\begin{align*}\n    &\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\Delta\\theta_{\\mathrm{Nt}}(w) - \\sI_f(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\Delta\\theta_{\\mathrm{Nt}}(w) - \\nabla_\\theta f\\bigl(\\hat\\theta(\\bone)\\bigr)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\left(H_{\\lambda, \\bone}({\\bone - w})^{-1} - H_{\\lambda, \\bone}^{-1}\\right) g_\\bone(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top \\left( \\big(H_{\\lambda, \\bone} - H_\\bone(w)\\big)^{-1} - H_{\\lambda, \\bone}^{-1}\\right) g_\\bone(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} \\left(H_{\\lambda, \\bone}^{\\inv{2}}\\big(H_{\\lambda, \\bone} - H_\\bone(w)\\big)^{-1}H_{\\lambda, \\bone}^{\\inv{2}} - I\\right) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} \\left(\\big(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\big)^{-1} - I\\right) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\\\\\n    &=\\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\n    \\numberthis \\label{eqn:newton-inf-proof-2}.\n  \\end{align*}\n  Substituting \\refeqn{newton-inf-proof-2} into \\refeqn{newton-inf-proof-1},\n  we have that\n  \\begin{align*}\n    \\sI^\\mathrm{Nt}_f(w) - \\sI_f(w)\n    &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\\\\\n    &\\quad + \\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w)\n    + \\mathrm{Err}_{f,3}(w),\n  \\end{align*}\n  as desired.\n\n  We can bound $\\mathrm{Err}_{f,3}(w)$ as follows:\n  \\begin{align*}\n    \\abs{\\mathrm{Err}_{f,3}(w)} &\\leq\n    \\frac{C_{f, 3}}{6}\\twonorm{\\Delta\\theta_{\\mathrm{Nt}}(w)}^3\\\\\n    &\\leq \\frac{\\onenorm{w}^3 C_{f, 3}C_\\ell^3}{6(\\sigma_{\\mathrm{min}} + \\lambda)^3}.\n  \\end{align*}\n  Applying \\reflem{sv-bound-d} to bound the spectrum of $D(w)$ completes the proof.\n\\end{proof}\n\n\n\\subsection{The influence on self-loss}\\label{sec:proof-selfloss}\nWe first state two linear algebra facts that will be useful in the sequel.\n\\begin{lemma}\\label{lem:sv-bound}\n  Let $A \\succ 0, B \\succeq 0 \\in \\R^{d \\times d}$ be a pair of symmetric positive-definite and positive-semidefinite matrices, respectively. Let $\\sigma_{A, 1}$ be the largest eigenvalue of $A$, $\\sigma_{A, d}$ the smallest eigenvalue of $A$, and similarly let $\\sigma_{B, 1}$ and $\\sigma_{B, d}$ be the largest and smallest eigenvalues of $B$, respectively. Then\n  \\begin{align*}\n    \\frac{\\sigma_{B, d}}{\\sigma_{A, 1}} I\n    \\preceq A^{-\\inv{2}} B A^{-\\inv{2}}\n    \\preceq \\frac{\\sigma_{B, 1}}{\\sigma_{A, d}} I.\n  \\end{align*}\n\\end{lemma}\n\\begin{proof}\n  Note that $\\inv{\\sigma_{A, 1}}$ is the smallest eigenvalue of $A^{-1}$, while $\\inv{\\sigma_{A, d}}$ is its largest. The lemma follows from the fact that the smallest singular value of the product of two matrices is lower bounded by the product of the smallest singular values of each matrix, and similarly the largest singular value of the product is upper bounded by the product of the largest singular values of each matrix.\n\\end{proof}\n\nThe next fact is a consequence of the variational definition of eigenvalues.\n\\begin{lemma}\\label{lem:quadform-bound}\n  Given a symmetric matrix $A \\in \\R^{d \\times d}$ and a vector $v \\in \\R^d$, we have the following bounds on the quadratic form $v^\\top A v$:\n  \\[\n  \\sigma_d \\twonorm{v}^2 \\leq v^\\top A v \\leq \\sigma_1 \\twonorm{v}^2,\n  \\]\n  where $\\sigma_d$ is the smallest eigenvalue of $A$, and $\\sigma_1$ is the largest.\n\\end{lemma}\n\nWe are now ready to analyze the effect of removing a subset $w$ of $k$ training points on the total loss on those $k$ points.\n\\begin{repproposition}{prop:selfloss}[Restated]\n  Under \\refassumps{l-const}{q-const}, the influence on the self-loss $f(\\theta) = \\sum_{i=1}^n w_i \\ell(x_i, y_i; \\theta)$ obeys\n  \\begin{align*}\n    \\sI_f(w) + \\mathrm{Err}_{f,3}(w) &\\leq \\sI^\\mathrm{Nt}_f(w)\n    \\leq \\left(1 + \\frac{3\\sigma_{\\mathrm{max}}}{2\\lambda} + \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2}\\right) \\sI_f(w) + \\mathrm{Err}_{f,3}(w).\n  \\end{align*}\n\\end{repproposition}\n\\begin{proof}\n  Since $f(\\theta) = \\sum_{i=1}^n w_i \\ell(x_i, y_i; \\theta)$, we have that\n  \\begin{align*}\n    \\nabla_\\theta f(\\hat\\theta(\\bone)) &= \\sum_{i=1}^n w_i \\nabla_\\theta \\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= g_\\bone(w),\\\\\n    \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) &= \\sum_{i=1}^n w_i \\nabla_\\theta^2 \\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= H_\\bone(w).\n  \\end{align*}\n  Substituting these and $\\Delta\\theta_{\\mathrm{Nt}}(w) = H_{\\lambda, \\bone}({\\bone - w})^{-1} g_\\bone(w)$ into \\refprop{newton-inf}, we obtain\n  \\begin{align*}\n    &\\sI^\\mathrm{Nt}_f(w) - \\sI_f(w) - \\mathrm{Err}_{f,3}(w)\\\\\n    &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\n      + \\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w)\\\\\n    &= g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\n      + \\inv{2} g_\\bone(w)^\\top H_{\\lambda, \\bone}({\\bone - w})^{-1} H_\\bone(w) H_{\\lambda, \\bone}({\\bone - w})^{-1} g_\\bone(w)\\\\\n      &= g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-\\inv{2}} \\left[ D(w) + \\underbrace{\\inv{2}H_{\\lambda, \\bone}^{\\inv{2}}H_{\\lambda, \\bone}({\\bone - w})^{-1} H_\\bone(w) H_{\\lambda, \\bone}({\\bone - w})^{-1} H_{\\lambda, \\bone}^{\\inv{2}}}_{\\eqdef \\Lambda(w)}\\right] H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w),\n  \\end{align*}\n  where $D(w) \\eqdef \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$ has singular values bounded between 0 and $\\frac{\\sigma_{\\mathrm{max}}}{\\lambda}$.\n  From \\reflem{sv-bound}, $\\Lambda(w)$ has singular values bounded between $0$ and\n  $\\frac{\\sigma_{\\mathrm{max}}(\\sigma_{\\mathrm{max}} + \\lambda)}{2\\lambda^2}$.\n\n  Applying \\reflem{quadform-bound} and using $\\sI_f(w) = g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)$, we obtain\n  \\begin{align*}\n    0 &\\leq\n    g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-\\inv{2}} \\left[ D(w) + \\Lambda(w) \\right] H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\\\\\n    &\\leq \\left(\\frac{\\sigma_{\\mathrm{max}}}{\\lambda} + \\frac{\\sigma_{\\mathrm{max}}(\\sigma_{\\mathrm{max}} + \\lambda)}{2\\lambda^2}\\right) g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)\\\\\n    &= \\left(\\frac{3\\sigma_{\\mathrm{max}}}{2\\lambda} + \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2}\\right) \\sI_f(w),\n  \\end{align*}\n  which gives us\n  \\begin{align*}\n      \\sI_f(w) + \\mathrm{Err}_{f,3}(w) \\leq \\sI^\\mathrm{Nt}_f(w) \\leq \\left(1 + \\frac{3\\sigma_{\\mathrm{max}}}{2\\lambda} + \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2}\\right) \\sI_f(w) + \\mathrm{Err}_{f,3}(w).\n  \\end{align*}\n  Note that $\\mathrm{Err}_{f,2}(w) \\eqdef \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2} \\sI_f(w)$ can be bounded as\n  \\begin{align*}\n    \\abs{\\mathrm{Err}_{f,2}(w)} &= \\abs{\\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2} \\sI_f(w)}\\\\\n    &\\leq \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2} \\cdot \\abs{\\sI_f(w)}\\\\\n    &\\leq \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2} \\cdot \\abs{g_\\bone(w)^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)}\\\\\n    &\\leq \\frac{\\onenorm{w}^2 C_\\ell^2 \\sigma_{\\mathrm{max}}^2}{2 (\\sigma_{\\mathrm{min}} + \\lambda) \\lambda^2},\n  \\end{align*}\n  and $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w) = \\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w)$ grows as $O(1\/\\lambda^3)$ from \\refprop{newton-actual},\n  so we can also write\n  \\begin{align*}\n      \\sI_f(w) + O\\Bigl(\\inv{\\lambda^3}\\Bigr) \\leq \\sI^{*}_f(w) \\leq \\left(1 + \\frac{3\\sigma_{\\mathrm{max}}}{2\\lambda}\\right) \\sI_f(w) + O\\Bigl(\\inv{\\lambda^3}\\Bigr).\n  \\end{align*}\n\n\\end{proof}\n\n\n\\subsection{The influence on a test point}\n\\begin{repcorollary}{cor:xtest}[Restated]\n  Suppose $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$,\n  and define ${v_{\\mathrm{test}}} \\eqdef H_{\\lambda, \\bone}^{-\\inv{2}} {x_{\\mathrm{test}}}$ and $v_w \\eqdef H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)$.\n  Then under \\refassumps{l-const}{q-const}, $\\sI^\\mathrm{Nt}_f(w) = \\sI_f(w) + {v_{\\mathrm{test}}}^\\top D(w) v_w$,\n  where $D(w) = \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$ is the \\emph{error matrix} from \\refprop{newton-inf}.\n\\end{repcorollary}\n\\begin{proof}\n  Since $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$ is linear, we have that for any $\\theta \\in \\Theta$,\n  \\begin{align*}\n    \\nabla_\\theta f(\\theta) &= {x_{\\mathrm{test}}},\\\\\n    \\nabla_\\theta^2 f(\\theta) &= 0,\\\\\n    C_{f, 3} &= 0.\n  \\end{align*}\n  This in turn implies that $\\mathrm{Err}_{f,3}(w) = 0$. Substituting these expressions into  \\refprop{newton-inf} gives us the desired result.\n\\end{proof}\n\n\n\\begin{repproposition}{prop:single}[Restated]\n  Consider a binary classification setting with $y \\in \\{-1, +1\\}$ and a margin-based model with loss\n  $\\ell(x, y; \\theta) = \\phi(y\\theta^\\top x)$ for some $\\phi:\\R\\to\\R_+$.\n  Suppose $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$\n  and that the subset $w$ comprises $\\|w\\|_1$ identical copies of the training point $(x_w, y_w)$.\n  Then under \\refassumps{l-const}{q-const},\n  the Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ is related to the influence $\\sI_f(w)$ according to\n  \\begin{equation*}\n    \\sI^\\mathrm{Nt}_f(w) = \\frac{\\sI_f(w)}\n        {1 - \\onenorm{w} \\cdot \\phi''(y_w \\hat\\theta(\\bone)^\\top x_w) \\cdot x_w^\\top H_{\\lambda, \\bone}^{-1} x_w}.\n  \\end{equation*}\n  This implies the Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ is bounded between $\\sI_f(w)$ and\n  $\\bigl(1 + \\frac{\\sigma_{\\mathrm{max}}}{\\lambda}\\bigr) \\sI_f(w)$.\n\\end{repproposition}\n\\begin{proof}\n  From \\refcor{xtest},\n  \\begin{align*}\n    \\sI^\\mathrm{Nt}_f(w) &= \\sI_f(w) + {x_{\\mathrm{test}}}^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w).\n  \\end{align*}\n  With the additional assumptions on $w$ and $\\ell(x, y; \\theta)$, we have that\n  \\begin{align*}\n    \\nabla_\\theta \\ell(x, y; \\theta) &= y \\phi'(y\\theta^\\top x) x,\\\\\n    g_\\bone(w) &= \\sum_{i=1}^n w_i \\nabla_\\theta \\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= \\sum_{i=1}^n w_i y_i \\phi'(y_i\\hat\\theta(\\bone)^\\top x_i) x_i\\\\\n    &= \\sum_{i=1}^n w_i y_i \\phi_i' x_i\\\\\n    &= \\onenorm{w} y_w \\phi_k' x_w,\n  \\end{align*}\n  where in the last equality we use the assumption that we are removing $\\onenorm{w}$ copies of the point $(x_w, y_w)$. Similarly,\n  \\begin{align*}\n    \\nabla_\\theta^2 \\ell(x, y; \\theta) &= \\phi''(y\\theta^\\top x) xx^\\top,\\\\\n    H_\\bone(w) &= \\sum_{i=1}^n w_i \\nabla^2_\\theta \\ell(x_i, y_i; \\hat\\theta(\\bone))\\\\\n    &= \\sum_{i=1}^n w_i \\phi''(y_i\\hat\\theta(\\bone)^\\top x_i) x_i x_i^\\top\\\\\n    &= \\onenorm{w} \\phi''_k x_w x_w^T.\n  \\end{align*}\n  We thus have\n  \\begin{align*}\n    D(w) &= \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I\\\\\n    &= \\Bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}\n    \\onenorm{w} \\phi''_k x_w x_w^T\n    H_{\\lambda, \\bone}^{-\\inv{2}}\\Bigr)^{-1} - I\\\\\n    &= \\frac{\n      H_{\\lambda, \\bone}^{-\\inv{2}}\n      \\onenorm{w} \\phi''_k x_w x_w^T\n      H_{\\lambda, \\bone}^{-\\inv{2}}}\n    {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\\\\\n    &= \\frac{\n    \\onenorm{w} \\phi''_k H_{\\lambda, \\bone}^{-\\inv{2}}\n       x_w x_w^T\n      H_{\\lambda, \\bone}^{-\\inv{2}}}\n    {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w},\n  \\end{align*}\n  where the third equality comes from the Sherman-Morrison formula.\n  Substituting $D(w)$ into \\refcor{xtest}, we obtain\n  \\begin{align*}\n    \\sI^\\mathrm{Nt}_f(w) &= \\sI_f(w) + {x_{\\mathrm{test}}}^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\\\\\n    &= \\sI_f(w) +\n      \\frac{\n        \\onenorm{w} \\phi''_k {x_{\\mathrm{test}}}^\\top H_{\\lambda, \\bone}^{-1}\n        x_w x_w^T\n        H_{\\lambda, \\bone}^{-1} g_\\bone(w)\n        }\n        {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\\\\\n    &= \\sI_f(w) +\n      \\frac{\n        {x_{\\mathrm{test}}}^\\top H_{\\lambda, \\bone}^{-1} \\onenorm{w} y_w \\phi_k' x_w\n        \\cdot \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w\n        }\n        {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\\\\\n      &= \\sI_f(w) +\n        \\frac{\n          {x_{\\mathrm{test}}}^\\top H_{\\lambda, \\bone}^{-1} g_\\bone(w)\n          \\cdot \\onenorm{w} z_k^T H_{\\lambda, \\bone}^{-1} z_k\n          }\n          {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\\\\\n        &= \\sI_f(w) +\n          \\frac{\n            \\sI_f(w)\n            \\cdot \\onenorm{w} z_k^T H_{\\lambda, \\bone}^{-1} z_k\n            }\n            {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\\\\\n          &= \\frac{\\sI_f(w)}\n              {1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}.\n  \\end{align*}\n  To bound the denominator, we first use the trace trick to rearrange terms\n  \\begin{align*}\n    \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w &=\n    \\tr\\left(\\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w\\right)\\\\\n    &= \\tr\\left( H_{\\lambda, \\bone}^{-\\inv{2}} \\onenorm{w} \\phi''_k x_w x_w^T H_{\\lambda, \\bone}^{-\\inv{2}}\\right)\\\\\n    &= \\tr\\left(H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}} \\right).\n  \\end{align*}\n  Since $H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}$ has rank one under our assumptions, it only has at most one non-zero eigenvalue. We can therefore apply \\reflem{sv-bound-d} to conclude that\n  \\begin{align*}\n    \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w &= \\tr\\left(H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}} \\right)\\\\\n    &\\leq \\frac{\\sigma_{\\mathrm{max}}}{\\sigma_{\\mathrm{max}} + \\lambda},\n  \\end{align*}\n  which in turn implies that\n    $1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w\n    \\geq \\frac{\\lambda}{\\sigma_{\\mathrm{max}} + \\lambda}$,\n  so\n  \\begin{align*}\n    \\inv{1 - \\onenorm{w} \\phi''_k x_w^T H_{\\lambda, \\bone}^{-1} x_w}\n    \\leq \\frac{\\sigma_{\\mathrm{max}} + \\lambda}{\\lambda}=1 + \\frac{\\sigma_{\\mathrm{max}}}{\\lambda}.\n  \\end{align*}\n\n\\end{proof}\n\n\\subsubsection*{Reproducibility}\nThe code for replicating our experiments is available in the GitHub repository \\url{https:\/\/github.com\/kohpangwei\/group-influence-release}.\nAn executable version of this paper is also available on CodaLab at \\url{https:\/\/worksheets.codalab.org\/worksheets\/0xfed2ae0b9e5b44b7a1af8096365592a5}.\n\n\\chapter{#2}\\label{chp:#1}}\n\\newcommand\\Section[2]{\\section{#2}\\label{sec:#1}}\n\\newcommand\\Subsection[2]{\\subsection{#2}\\label{sec:#1}}\n\\newcommand\\Subsubsection[2]{\\subsubsection{#2}\\label{sec:#1}}\n\\ifthenelse{\\isundefined{\\definition}}{\\newtheorem{definition}{Definition}}{}\n\\ifthenelse{\\isundefined{\\assumption}}{\\newtheorem{assumption}{Assumption}}{}\n\\ifthenelse{\\isundefined{\\hypothesis}}{\\newtheorem{hypothesis}{Hypothesis}}{}\n\\ifthenelse{\\isundefined{\\proposition}}{\\newtheorem{proposition}{Proposition}}{}\n\\ifthenelse{\\isundefined{\\theorem}}{\\newtheorem{theorem}{Theorem}}{}\n\\ifthenelse{\\isundefined{\\lemma}}{\\newtheorem{lemma}{Lemma}}{}\n\\ifthenelse{\\isundefined{\\corollary}}{\\newtheorem{corollary}{Corollary}}{}\n\\ifthenelse{\\isundefined{\\alg}}{\\newtheorem{alg}{Algorithm}}{}\n\\ifthenelse{\\isundefined{\\example}}{\\newtheorem{example}{Example}}{}\n\\newcommand\\cv{\\ensuremath{\\to}}\n\\newcommand\\cvL{\\ensuremath{\\xrightarrow{\\mathcal{L}}}}\n\\newcommand\\cvd{\\ensuremath{\\xrightarrow{d}}}\n\\newcommand\\cvP{\\ensuremath{\\xrightarrow{P}}}\n\\newcommand\\cvas{\\ensuremath{\\xrightarrow{a.s.}}}\n\\newcommand\\eqdistrib{\\ensuremath{\\stackrel{d}{=}}}\n\\newcommand{\\E}{\\ensuremath{\\mathbb{E}}}\n\\newcommand\\KL[2]{\\ensuremath{\\text{KL}\\left( #1 \\| #2 \\right)}}\n\n\\section{Additional analysis on influence vs. actual effect on a test point}\\label{sec:supp-theory}\n\n\\subsection{Counterexamples}\\label{sec:theory-cex}\n\nFor \\reffig{cex}, we constructed two binary datasets in which the influence of a\ncertain class of subsets on the test prediction of a single test point exhibits\npathological behavior.\n\n\\paragraph{Rotation effect.}\nIn \\reffig{cex}-Left, our aim was to show that there can be a dataset with subsets such that the cone constraint discussed in \\refsec{theory-test} does not hold.\n\nThe rotation effect described in \\refcor{xtest} is due to the\nangular difference between the change in parameters predicted by the influence approximation,\n$\\Delta\\theta_{\\mathrm{inf}}(w) \\eqdef H_{\\lambda, \\bone}^{-1} g_\\bone(w) = H_{\\lambda, \\bone}^{-\\frac 1 2} v_w$,\nand the change in parameters predicted by the Newton approximation,\n$\\Delta\\theta_{\\mathrm{Nt}}(w) = H_{\\lambda, \\bone}^{-\\frac 1 2} \\bigl(D(w) + I\\bigr) v_w$.\nIf $\\Delta\\theta_{\\mathrm{inf}}(w)$ and $\\Delta\\theta_{\\mathrm{Nt}}(w)$ are linearly independent,\nthen for any pair of target values $a, b \\in \\R$,\nwe can find some ${x_{\\mathrm{test}}}$ such that $\\sI_f(w) = {x_{\\mathrm{test}}}^\\top \\Delta\\theta_{\\mathrm{inf}}(w) = a$ and $\\sI^\\mathrm{Nt}_f(w  = {x_{\\mathrm{test}}}^\\top \\Delta\\theta_{\\mathrm{Nt}}(w) = b$.\n\nTo exploit this, we constructed the \\emph{MoG} dataset as an equal mixture of two standard (identity covariance) Gaussian distributions in $\\R^{60}$, one for each class, and with means $(-1\/2, 0, \\ldots, 0)$ and $(1\/2, 0, \\ldots, 0)$, respectively. In particular:\n\\begin{enumerate}\n  \\item We sampled $60$ examples from each class for a total of $n = 120$ training points, and set the regularization strength $\\lambda = 0.001$.\n  \\item We then computed $\\Delta\\theta_{\\mathrm{inf}}(w)$ and $\\Delta\\theta_{\\mathrm{Nt}}(w)$ for each pair of training points and chose the $120$ pairs of training points with the largest angles between $\\Delta\\theta_{\\mathrm{inf}}(w)$ and $\\Delta\\theta_{\\mathrm{Nt}}(w)$.\n  \\item Finally, we solved a\n  least-squares optimization problem to find ${x_{\\mathrm{test}}}$ for which $\\sI_f(w)$ and\n  $\\sI^\\mathrm{Nt}_f(w)$ are approximately decorrelated.\n\\end{enumerate}\n\nNote that we adversarially chose which subsets to study in this counterexample, since our main goal was to show that there existed subsets for which the cone constraint did not hold.\nFor the next counterexample, we instead study all possible subsets in the restricted setting of removing copies of single points.\n\n\\paragraph{Scaling effect.} In \\reffig{cex}-Right, our aim was to construct a dataset such that even if we only removed subsets comprising copies of single distinct points, a low influence need not translate into a low actual effect.\n\nTo do so, we constructed the \\emph{Ortho} dataset that contains 2 repeated points of opposite classes on each of the 2 canonical axes of $\\R^2$ (for a total of 4 distinct points). By varying their\nrelative distances from the origin, we can control the influence of removing one of these points as well as the rate that the scaling factor $d(w)$ from \\refprop{single} grows as we remove more copies of the same point.\nFurthermore, because the axes are orthogonal, we can control $d(w)$ independently for each repeated\npoint. We fix the test point ${x_{\\mathrm{test}}} = (1, 1)$.\nMaximizing $d(w)$ for one axis\nand minimizing it for the other produces the two distinct lines in\n\\reffig{cex}-Right.\n\n\n\\subsection{Scaling effects when removing multiple points}\\label{sec:theory-D}\nIn the general setting of removing subsets of different points, the analogous\nfailure case to a varying scaling factor $d(w)$ (\\reffig{cex}-Right) is the\nvarying scaling effect that the error matrix $D(w)$ in \\refprop{newton-inf} can have on\ndifferent subsets $w$. The range of this effect is bounded by the spectral norm\nof $D(w)$. This norm is precisely equal to $d(w)$ in the single-point setting,\nand it is large when we remove a subset $w$ whose Hessian\n$H_\\bone(w)$ is almost as large as the full Hessian $H_{\\lambda, \\bone}$ in some direction.\nAs with $d(w)$, the spectral norm of $D(w)$ decreases with $\\lambda$ (\\refprop{newton-inf}), so as regularization increases, we expect that the influence of a group will track its actual effect more accurately.\n\n\\subsection{The relationship between influence and actual effect on the loss of a test point}\\label{sec:theory-loss}\nIn the margin-based setting, the loss $\\ell({x_{\\mathrm{test}}}, {y_{\\mathrm{test}}}; \\theta)$ is a monotone function of the linear prediction $\\theta^\\top {x_{\\mathrm{test}}}$. Thus, measuring $f(\\theta) = \\ell({x_{\\mathrm{test}}}, {y_{\\mathrm{test}}}; \\theta)$ will display the same rank correlation as measuring $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$ above,\nso the same results about correlation in the test prediction setting carry over.\n\nHowever, the second-order $f$-curvature term $\\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w)$ from \\refprop{newton-inf} is always non-negative, even if the influence is negative. Under the assumption that $\\mathrm{Err}_{f,3}(w)$ and $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$ are both small\nbecause they decay as $O(1\/\\lambda^3)$,\nthis implies that underestimation is only preserved when the influence is positive, as we observed empirically in \\reffig{acc}-Mid.\n\n\\section{Additional experiments}\\label{sec:supp-experiments}\n\nAs in \\reffig{acc}, in each of the plots below, the grey reference line has slope 1, and the red borders represent points that are not plotted because they are outside the x- or y-axis range.\n\n\\subsection{Representative test points}\\label{sec:median}\n\n\\reffig{median} is similar to \\reffig{acc} in the main text:\nit shows the influences vs. actual effects of groups on test points,\nbut with test points that are closer to the median (within the 40th to 60th percentile) of the test loss distribution.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-fig5.pdf}\n  \\caption{\n    Influences vs. actual effects of the same coherent groups in \\reffig{acc},\n    but on test points closer to the median (within the 40th to 60th\n    percentile) of the test loss distribution. We consider these to represent\n    average test points. On these, influence on the test prediction remains\n    well-correlated with the actual effect.\n  }\n  \\label{fig:median}\n\\end{figure}\n\n\n\\subsection{Regularization}\\label{sec:reg}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.9\\textwidth]{figs\/plots\/paper-fig3.pdf}\n  \\caption{\n    The effect of regularization for a representative test point.\n    Red frame lines indicate the existence of points exceeding those bounds. We did not include the test prediction for MNIST (small) because the margin is not well-defined for a multiclass model.\n  }\n  \\label{fig:regularization}\n\\end{figure}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.95\\textwidth]{figs\/plots\/paper-regs-spearman.pdf}\n  \\caption{\n    As regularization increases, correlation increases between the influences and actual effects on test prediction (Left), test loss (Middle), and self-loss (Right).\n  }\n  \\label{fig:regs-spearman}\n\\end{figure}\n\nIn \\refsec{theory}, our bounds show that influence ought to be closer to actual effect as regularization increases. Here, we support this claim empirically on Diabetes, Enron, Dogfish, and MNIST (small).\\footnotemark{} To do so, for each dataset, we selected a range of values for $\\lambda\/n$, and we selected subsets as described in \\refapp{subsetgen}. We then computed the influence and actual effect of each of these subsets on a representative test point's prediction, that point's loss, and on self-loss (\\reffig{regularization}).\n\n\\footnotetext{This experiment required us to retrain the model for every value of $\\lambda$ and for every subset. Thus, for computational purposes, we omitted CDR and MultiNLI, and we selected a random $10\\%$ subset of MNIST's training set to use in place of all of MNIST.}\n\nIn \\reffig{regs-spearman}, we observe the trend that correlation generally increases as $\\lambda$ does. Specifically, we computed the Spearman $\\rho$ between the influence and actual effect for each dataset, each value of $\\lambda$, and each evaluation function $f(\\cdot)$ of interest (i.e., test prediction, test loss, or self-loss).\n\n\n\\subsection{The effect of loss curvature on the accuracy of influence}\\label{sec:supp-curvature}\n\nOne takeaway from the results on test loss in \\reffig{acc}-Mid is that the curvature of $f(\\theta)$ can significantly increase\napproximation error; this is expected since the influence $\\sI_f(w)$\nlinearizes $f(\\cdot)$ around $\\hat\\theta(\\bone)$.\nWhen possible, choosing a $f(\\cdot)$ that has low curvature (e.g., the linear prediction) will result in higher accuracy.\nWe can mitigate this by using influence to approximate the parameters $\\hat\\theta(\\bone - w)$\nand then plug that estimate into $f(\\cdot)$ (\\reffig{pparam}), though this can be more\ncomputationally expensive.\n\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-fig6.pdf}\n  \\vspace{-4mm}\n  \\caption{\n    The top 3 rows show influence on test loss (with the same test points as in \\reffig{acc}),\n    while the bottom 3 show self-loss.\n    Within each set, the first row shows the influence vs. actual effect (as in \\reffig{acc});\n    the second shows the predicted effect obtained by estimating the change in parameters via influence and then\n    evaluating $f(\\cdot)$ directly on those parameters;\n    and the third shows the Newton approximation.\n    }\n    \\vspace{-4mm}\n  \\label{fig:pparam}\n\\end{figure}\n\n\nNote that \\reffig{pparam} shows that this technique does not help much for measuring self-loss.\nHowever, in the context of LOOCV,\nthe computational complexity of the Newton approximation for self-loss (described in \\refsec{theory})\nis similar to that of the influence approximation,\nso we encourage the use of the Newton approximation for LOOCV (as in \\citet{rad2018scalable});\n\\reffig{pparam} shows that this leads to more accurate approximations for self-loss.\n\n\\subsection{Additional analysis of influence functions applied to natural groups of data}\\label{sec:supp-applications}\nIn \\refsec{applications}, we considered the CDR and MultiNLI datasets, which contain the natural subsets of LFs and crowdworkers, respectively.\nTo draw inferences about these subsets, we took the $L_2$-regularized logistic regression model described in \\refapp{expt-details}, calculated the influence of the LF\/crowdworker subsets, and retrained the model once for each LF\/crowdworker.\n\n\\paragraph{CDR.}\nAs discussed in \\refapp{bl-details}, an LF is either positive or negative, where a positive LF can only give positive labels or abstain, and similarly for negative LFs. Because of this stark class separation, we indicate whether an LF is positive or negative, and we consider LF influence on the positive test examples separately from their influence on the negative test examples. To measure an LF's influence and actual effect on a set of test points, we simply add up its influence and actual effect on the set's individual test points.\n\nIn \\reffig{babble-acc}, we note that influence is a good approximation of an LF's actual effect, just as with other kinds of subsets as well as other datasets (\\reffig{acc}). Furthermore, we observe that positive LFs improve the overall performance of the positively labeled portion of the test set while hurting the negatively labeled portion of the test set, and vice versa for negative LFs. This dichotomous effect further motivates the analysis of influence on the positive test set separately from the negative test set, since the process of adding these two influences to study the influence on the entire test set would obscure the full story.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.65\\textwidth]{figs\/plots\/paper-suppfig-babble-acc-test.pdf}\n  \\caption{We observe both correlation and underestimation for LFs on the positive and negative test sets. We also see that positive LFs help the positive test set and hurt the negative test set; vice versa for negative LFs.\n  }\n  \\label{fig:babble-acc}\n\\end{figure}\n\nNext, we define an LF's \\textit{coverage} to be the proportion of the examples that it does not abstain on, which can be measured through the number of examples in its corresponding subset. In \\reffig{babble-size-corr}, we observe that the magnitude of influence correlates strongly with coverage.\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.65\\textwidth]{figs\/plots\/paper-suppfig-babble-sizes.pdf}\n  \\caption{The magnitude of LF influence correlates with coverage. This figure is an extension of \\reffig{combined-apps}-Left: there, we showed the influence of positive LFs on the positive test set and the influence of negative LFs on the negative test set. Here, we additionally show the influence of positive LFs on the negative test set and vice versa.\n  }\n  \\label{fig:babble-size-corr}\n\\end{figure}\n\nFinally, we define an LF's \\textit{precision} to be the number of examples it labels correctly divided by the number of examples it does not abstain on. Because the dataset had many more negative than positive examples, positive LFs had lower precision than negative LFs. Surprisingly, even when this effect was taken into account and we considered positive LFs separately from negative ones, precision did not correlate with influence (\\reffig{babble-acc-no-corr}).\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.65\\textwidth]{figs\/plots\/paper-suppfig-babble-accs.pdf}\n  \\caption{LF influence does not correlate with precision. Similar to \\reffig{babble-size-corr}, this figure is an extension of \\reffig{combined-apps}-Mid: there, we showed the influence of positive LFs on the positive test set and the influence of negative LFs on the negative test set. Here, we additionally show the influence of positive LFs on the negative test set and vice versa.\n  }\n  \\label{fig:babble-acc-no-corr}\n\\end{figure}\n\n\n\\paragraph{MultiNLI.}\nAs discussed in \\refapp{multinli-details}, the training set consisted of five genres, and the test set consisted of a matched portion with the same five genres, as well as a mismatched portion with five new genres. For succinctness, we refer to the influence\/actual effect of the set of examples generated by a single crowdworker as that crowdworker's influence\/actual effect.\n\nFirst, we note in \\reffig{mnli-acc} that influence is a good approximation of a crowdworker's actual effect for both matched and mismatched test sets, consistent with our findings in \\reffig{acc} for other subset types and datasets.\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-suppfig-mnli-acc-test.pdf}\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-suppfig-mnli-acc-nonfires.pdf}\n  \\caption{We observe strong correlation between crowdworkers' influence and actual effects.\n  }\n  \\label{fig:mnli-acc}\n\\end{figure}\n\nUnlike in CDR (\\reffig{babble-size-corr}), we do not find strong correlation between a crowdworker's influence and the number of examples they contributed; it is possible to contribute many examples but have relatively little influence (\\reffig{mnli-size-no-corr}).\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.42\\textwidth]{figs\/plots\/paper-suppfig-mnli-size.pdf}\n  \\caption{Size does not correlate strongly with influence. The hidden point is the crowdworker that contributed 35,000 examples. This is the figure presented in \\reffig{combined-apps}-Right.\n  }\n  \\label{fig:mnli-size-no-corr}\n\\end{figure}\n\nThe most prolific crowdworker contributed 35,000 examples and had large negative influence on the test set. A closer analysis revealed that they had positive influence on the fiction genre but lowered performance on many other genres, despite contributing roughly equally to each genre. This genre-specific trend tended to hold more broadly among the workers: there appear to be two categories of genres (fiction, facetoface, nineeleven vs. travel, government, verbatim, letters, oup) such that each worker tended to have positive influence on all genres in one category and negative influence on all genres in the other (\\reffig{mnli-genre-categories}). Moreover, the number of examples a worker contributed to a given genre was not a good indicator for their influence on that genre (\\reffig{mnli-genre-contrib}).\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=0.55\\textwidth]{figs\/plots\/paper-suppfig-mnli-genre-effect.pdf}\n  \\caption{Workers tended to have positive influence on fiction, facetoface, and nineeleven and negative influence on travel, government, verbatim, letters, and oup (or vice versa). In this plot, we allowed for full color saturation when the magnitude of the total influence on the test set (matched) exceeded 0.8.\n  }\n  \\label{fig:mnli-genre-categories}\n\\end{figure}\n\n\\begin{figure}[h!]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-suppfig-mnli-genre-contrib.pdf}\n  \\caption{Influence on a genre does not correlate with number of contributions to that genre.\n  }\n  \\label{fig:mnli-genre-contrib}\n\\end{figure}\n\n\\section{Theoretical analysis}\\label{sec:theory}\n\nThe experimental results above show that there is consistent underestimation and high correlation between the predicted effects, based on influence functions, and the actual effects of groups across a variety of datasets, despite the influence approximation incurring large absolute and relative error.\nAs we discussed in \\refsec{related}, this is outside the regime of existing theory.\n\nAs an initial step towards understanding the high-error regime,\nwe establish conditions under which the actual effect $\\sI^{*}_f(w)$ lies approximately between $\\sI_f(w)$ and $C_\\mathrm{max} \\sI_f(w)$ for some $C_\\mathrm{max} > 0$.\nThis cone constraint---so called because it implies that all points on the graph of influence vs. actual effect lie within a cone---implies underestimation and,\nif $C_\\mathrm{max}$ is small, some degree of correlation.\nWe first show that this constraint holds in restricted settings---when measuring self-loss, or when removing multiple copies of the same point---and that $C_\\mathrm{max}$ varies inversely with the regularization term $\\lambda$, which is expected since stronger regularization reduces the change in the model.\nHowever, the cone constraint is stronger than necessary because it bounds the degree of underestimation,\nand we construct counterexamples to show that it need not hold in more general settings.\n\nOur analysis centers on the \\emph{one-step Newton approximation}, which estimates the change in parameters\n\\begin{align}\n  \\hat\\theta(\\bone - w) - \\hat\\theta(\\bone) &\\approx \\Delta\\theta_{\\mathrm{Nt}}(w) \\nonumber\n  \\eqdef \\bigl(H_{\\lambda, \\bone}({\\bone - w})\\bigr)^{-1} g_\\bone(w),\n\\end{align}\nwhere $H_{\\lambda, \\bone}({\\bone - w}) = (\\sum_{i=1}^n (1 - w_i) \\nabla_\\theta^2\\ell(x_i, y_i; \\hat\\theta(\\bone))) + \\lambda I$ is the\nregularized empirical Hessian at $\\hat\\theta(\\bone)$ but reweighted after removing the subset $w$.\nThis change in parameters gives the Newton approximation of the effect\n$\\sI^\\mathrm{Nt}_f(w) = f\\bigl(\\hat\\theta(\\bone) + \\Delta\\theta_{\\mathrm{Nt}}(w)\\bigr) - f(\\hat\\theta(\\bone)))$ and the corresponding Newton error $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w) = \\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w)$, which measures its gap from the actual effect. Specifically, we decompose the error between the actual effect $\\sI^{*}_f(w)$ and influence $\\sI_f(w)$ as\n\\begin{align}\n  \\sI^{*}_f(w) - \\sI_f(w) \\ =\\  \\underbrace{\\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w)}_{\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)} \\ +\\\n  \\underbrace{\\sI^\\mathrm{Nt}_f(w) - \\sI_f(w).}_{\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w)}\\label{eqn:decomp}\n\\end{align}\nIn \\refsec{theory-newton}, we first show that the Newton-actual error $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$\ndecays at a rate of $O\\bigl(1 \/ (\\sigma_{\\mathrm{min}} + \\lambda)^3\\bigr)$, where $\\lambda$ is regularization strength and $\\sigma_{\\mathrm{min}}$ is the smallest eigenvalue of the empirical Hessian $H_\\bone$. Empirically, this error is small on our datasets, so we focus on characterizing the Newton-influence error $\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w)$ in \\refsec{theory-decompose}.\nWe use this characterization to study the behavior of influence relative to the actual effect on self-loss (\\refsec{theory-selfloss}) and test prediction (\\refsec{theory-test}). For margin-based models, the test loss is a monotone function of the test prediction, so the analysis is similar (\\refapp{theory-loss}).\n\n\n\\subsection{Bounding the error of the one-step Newton approximation}\\label{sec:theory-newton}\n\nThe Newton approximation is computationally expensive\nbecause it computes $(H_{\\lambda, \\bone}({\\bone - w}))^{-1}$ for each $w$\n(instead of the fixed $H_{\\lambda, \\bone}^{-1}$ in the influence calculation).\nHowever, it provides more accurate estimates (e.g., \\citet{pregibon1981logistic}, \\citet{rad2018scalable}),\nand we show that its error can be bounded as follows (all proofs in \\refapp{proofs}):\n\\begin{proposition}\\label{prop:newton-actual}\n  Let the Newton error be $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w) \\eqdef \\sI^{*}_f(w) - \\sI^\\mathrm{Nt}_f(w)$.\n  Assume that the evaluation function $f(\\theta)$ is $C_f$-Lipschitz\n  and that the Hessian $\\nabla_\\theta^2 \\ell(x, y, \\theta)$ is $C_H$-Lipschitz.\n  Then\n  \\begin{align*}\n      \\abs{\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)} \\leq \\frac{n \\|w\\|_1^2 C_f C_H C_\\ell^2}{(\\sigma_{\\mathrm{min}} + \\lambda)^3},\n  \\end{align*}\n  where we define\n  $C_\\ell \\eqdef \\max_{1 \\leq i \\leq n} \\|\\nabla_\\theta \\ell(x_i,y_i,\\hat\\theta(\\bone))\\|_2$ to be the largest norm of a training point's gradient at $\\hat\\theta(\\bone)$,\n  and $\\sigma_{\\mathrm{min}}$ to be the smallest eigenvalue of $H_\\bone$.\n  $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$ only involves third-order or higher derivatives of the loss, so it is 0 for quadratic losses.\n\\end{proposition}\n\\refprop{newton-actual} tells us that the Newton approximation is accurate\nwhen $\\lambda$ is large or the third derivative of $\\ell(x, y; \\cdot)$ (controlled by $C_H$) is small. Empirically,\nthe Newton error $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$ is strikingly small in most of our settings (\\reffig{newton}), even though the overall error of the influence approximation $\\sI^{*}_f(w) - \\sI_f(w)$ is still large.\nIn the remainder of this section, we therefore focus on characterizing the Newton-influence error $\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w)$, under the assumption that the Newton approximation is similar to the actual effect (within a factor of $O(1\/\\lambda^3)$).\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\textwidth]{figs\/plots\/paper-fig2.pdf}\n  \\caption{\n    The Newton approximation accurately captures the actual effect for our datasets (though there is more error on the Diabetes dataset),\n    with the same test point as in \\reffig{acc}-Top.\n    We omit MNIST and MultiNLI for computational reasons. See \\reffig{pparam} for plots of test loss and self-loss.\n  }\n  \\label{fig:newton}\n\\end{figure}\n\n\\subsection{Characterizing the difference between the Newton approximation and influence}\\label{sec:theory-decompose}\nWe next characterize the Newton-influence error $\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w) = \\sI^\\mathrm{Nt}_f(w) - \\sI_f(w)$:\n\\begin{proposition}\\label{prop:newton-inf}\n  Under the assumptions of \\refprop{newton-actual} and the additional assumption that the third derivative of $f(\\theta)$ exists and is bounded in norm by $C_{f, 3}$, the Newton-influence error $\\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w)$ is\n  \\begin{align*}\n    \\mathrm{Err}_\\mathrm{Nt\\text{-}inf}(w) &= \\nabla_\\theta f(\\hat\\theta(\\bone))^\\top H_{\\lambda, \\bone}^{-\\inv{2}} D(w) H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)\n    \\ + \\ \\underbrace{\\inv{2} \\Delta\\theta_{\\mathrm{Nt}}(w)^\\top \\nabla_\\theta^2 f(\\hat\\theta(\\bone)) \\Delta\\theta_{\\mathrm{Nt}}(w) + \\mathrm{Err}_{f,3}(w),}_\\text{\n      Error from curvature of $f(\\cdot)$\n    }\n  \\end{align*}\n  with $D(w) \\eqdef \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$\n  and $H_\\bone(w) \\eqdef \\sum_{i=1}^n w_i \\nabla_\\theta^2\\ell(x_i, y_i; \\hat\\theta(\\bone))$.\n  The \\emph{error matrix} $D(w)$ has eigenvalues between 0 and $\\frac{\\sigma_{\\mathrm{max}}}{\\lambda}$,\n  where $\\sigma_{\\mathrm{max}}$ is the largest eigenvalue of $H_\\bone$.\n  The residual term $\\mathrm{Err}_{f,3}(w)$ captures the error due to third-order derivatives of $f(\\cdot)$ and is bounded by $\\abs{\\mathrm{Err}_{f,3}(w)} \\leq {\\|w\\|_1^3 C_{f, 3}C_\\ell^3}\/{6(\\sigma_{\\mathrm{min}} + \\lambda)^3}$.\n\\end{proposition}\nWe can interpret \\refprop{newton-inf} as a formalization of \\citet{hampel1986robust}'s observation that influence approximations are accurate when the model is robust and the curvature of the loss is low.\nIn general, the error decreases as $\\lambda$ increases and $f(\\cdot)$ becomes less curved;\nin \\reffig{regularization}, we show that increasing $\\lambda$ reduces error and increases correlation in our experiments.\n\n\\subsection{The relationship between influence and actual effect on self-loss}\\label{sec:theory-selfloss}\nLet us now apply \\refprop{newton-inf} to analyze the behavior of influence under different choices of evaluation function $f(\\cdot)$. We start with the self-loss $f(\\theta) = \\sum_{i=1}^n w_i \\ell(x_i, y_i; \\theta)$, as its influences and actual effects are always non-negative, and it is the cleanest to characterize:\n\\begin{proposition}\\label{prop:selfloss}\n  Under the assumptions of \\refprop{newton-inf}, the influence on the self-loss obeys\n  \\begin{align*}\n    \\sI_f(w) + \\mathrm{Err}_{f,3}(w) &\\leq \\sI^\\mathrm{Nt}_f(w)\n    \\leq \\left(1 + \\frac{3\\sigma_{\\mathrm{max}}}{2\\lambda} + \\frac{\\sigma_{\\mathrm{max}}^2}{2\\lambda^2}\\right) \\sI_f(w) + \\mathrm{Err}_{f,3}(w).\n  \\end{align*}\n\\end{proposition}\nThe constraint in \\refprop{selfloss} implies that up to $O(1\/\\lambda^3)$ terms, influence underestimates the Newton approximation and therefore the actual effect.\nThis explains the previously-unexplained downward bias observed when using influence to approximate LOOCV \\citep{debruyne2008model, giordano2019swiss}.\nEquivalently, all points on the graph of influences vs. actual effects\nlie within the cone bounded by the lines with slope $1$ and slope $\\frac{\\lambda}{\\lambda + 3\\sigma_{\\mathrm{max}}\/2}$ lines, up to $O(1\/\\lambda^3)$ terms.\nAs $\\lambda$ grows, these lines will converge, and the error terms $\\mathrm{Err}_{f,3}(w)$ and $\\mathrm{Err}_\\mathrm{Nt\\text{-}act}(w)$ will decay at a rate of $O(1\/\\lambda^3)$,\nforcing the influences and actual effects to be equal.\n\nHowever, $\\lambda \/ \\sigma_{\\mathrm{max}}$ is quite small in our experiments in \\refsec{accuracy},\nso the actual correlation of influence is better than predicted by this theory:\nin \\reffig{acc}-Bot, the sizes of the theoretically-permissible cones can be quite large, but the points in the graphs nevertheless trace a tight curve through the cone.\n\n\n\\subsection{The relationship between influence and actual effect on a test point}\\label{sec:theory-test}\n\nWe now turn to measuring the test prediction $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$. Here, we show that correlation and underestimation need not hold, and that we cannot obtain a cone constraint similar to \\refprop{selfloss} except in a restricted setting.\nDefine ${v_{\\mathrm{test}}} = H_{\\lambda, \\bone}^{-\\inv{2}} {x_{\\mathrm{test}}}$ and $v_w = H_{\\lambda, \\bone}^{-\\inv{2}} g_\\bone(w)$.\n\\refprop{newton-inf} gives:\n\\begin{corollary}\\label{cor:xtest}\n  Suppose $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$.\n  Then $\\sI^\\mathrm{Nt}_f(w) = \\sI_f(w) + {v_{\\mathrm{test}}}^\\top D(w) v_w$,\n  where $D(w) = \\bigl(I - H_{\\lambda, \\bone}^{-\\inv{2}}H_\\bone(w)H_{\\lambda, \\bone}^{-\\inv{2}}\\bigr)^{-1} - I$ is the \\emph{error matrix} from \\refprop{newton-inf}.\n\\end{corollary}\nUnfortunately, \\refcor{xtest} implies that no cone constraint applies: in general, we can find ${x_{\\mathrm{test}}}$ such that the influence $\\sI_f(w) = {v_{\\mathrm{test}}}^\\top v_w = 0$ but the Newton approximation $\\sI^\\mathrm{Nt}_f(w) = {v_{\\mathrm{test}}}^\\top D(w) v_w$ is large.\nAs a counterexample, \\reffig{cex}-Left shows that on synthetic data, $\\sI_f(w)$ and $\\sI^\\mathrm{Nt}_f(w)$ can even have opposite signs on some subsets $w$.\n\n\\begin{figure}[t]\n  \\centering\n  \\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={right,top},capbesidewidth=6.5cm}}]{figure}[\\FBwidth]\n  {\\caption{\n    Influence $\\sI_f(w)$ vs. Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ on the test prediction on two  counterexamples detailed in \\refapp{theory-cex}.\n    Left: We adversarially choose a set of $w$'s such that $\\sI_f(w)$ and $\\sI^\\mathrm{Nt}_f(w)$ can have different signs and need not correlate.\n    Right: When we only remove copies of single points, underestimation holds. However, we can control the scaling factor $d(w)$ between $\\sI_f(w)$ and $\\sI^\\mathrm{Nt}_f(w)$ on different groups, so correlation need not hold.\n  }\\label{fig:cex}}\n  {\\includegraphics[width=0.5\\textwidth]{figs\/plots\/paper-fig4.pdf}}\n\\end{figure}\n\n\nWe can recover a cone constraint similar to \\refprop{selfloss} if we restrict our attention to\nthe special case where we use a margin-based model and remove (possibly multiple copies) of a single point:\n\\begin{proposition}\\label{prop:single}\n  Consider a binary classification setting with $y \\in \\{-1, +1\\}$ and a margin-based model with loss\n    $\\ell(x, y; \\theta) = \\phi(y\\theta^\\top x)$ for some $\\phi:\\R\\to\\R_+$.\n    Suppose $f(\\theta) = \\theta^\\top {x_{\\mathrm{test}}}$\n    and that the subset $w$ comprises $\\|w\\|_1$ identical copies of the training point $(x_w, y_w)$.\n    Then under the assumptions of \\refprop{newton-actual},\n    the Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ is related to the influence $\\sI_f(w)$ according to\n  \\begin{equation*}\n    \\sI^\\mathrm{Nt}_f(w) = \\frac{\\sI_f(w)}\n        {1 - \\onenorm{w} \\cdot \\phi''(y_w \\hat\\theta(\\bone)^\\top x_w) \\cdot x_w^\\top H_{\\lambda, \\bone}^{-1} x_w}.\n  \\end{equation*}\n  This implies the Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ is bounded between $\\sI_f(w)$ and\n  $\\bigl(1 + \\frac{\\sigma_{\\mathrm{max}}}{\\lambda}\\bigr) \\sI_f(w)$.\n\\end{proposition}\nSimilar to \\refprop{selfloss}, \\refprop{single} shows that up to $O(1\/\\lambda^3)$ terms, the influence underestimates the actual effect when removing copies of a single point.\nMoreover, all points on the graph of influences vs. actual effects lie within the cone bounded by the lines with slope $1$ and slope $\\lambda \/ (\\lambda + \\sigma_{\\mathrm{max}})$,\nup to $O(1\/\\lambda^3)$ terms.\nAs $\\lambda \/ \\sigma_{\\mathrm{max}}$ grows, the cone shrinks, and correlation increases.\n\nHowever, if $\\lambda \/ \\sigma_{\\mathrm{max}}$ is small (as in our experiments in \\refsec{accuracy}),\nthe cone is wide,\nand the scaling factor $d(w) = 1 \/ (1 - \\onenorm{w} \\cdot \\phi_k'' x_k^\\top H_{\\lambda, \\bone}^{-1} x_k)$ in \\refprop{single} can be quite large for some subsets $w$ but not for others.\nIn particular, $d(w)$ is large when there are few remaining points in the direction of the removed points.\nIn \\reffig{cex}-Right, we exploit this fact to show that the influence $\\sI_f(w)$ and Newton approximation $\\sI^\\mathrm{Nt}_f(w)$ can exhibit low correlation (e.g., low $\\sI_f(w)$ need not mean low $\\sI^\\mathrm{Nt}_f(w)$),\neven in the simplified setting of removing copies of single points.\nWe comment on the analogue of $d(w)$ in the general multiple-point setting in \\refapp{theory-D}, and on the influence on test loss (instead of test prediction) in \\refapp{theory-loss}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}