{"text":"\\section{\\@startsection {section}{1}{\\z@}%\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nThe kernel trick in machine learning allows for non-linear analysis of data using many techniques such as PCA and SVM which were originally designed for linear analysis.  The ``trick'' is that these procedures only access data through inner products between data points,\nhere we consider this non-linear inner product defined by a kernel $K(\\cdot,\\cdot)$.  \nNow given $n$ data points, one can compute the $n \\times n$ gram matrix $G$ of all pairwise inner products; that is so $G_{i,j} = K(x_i, x_j)$ for all $x_i, x_j$ in input data set $X$.  \nThen the analysis can proceed using just the gram matrix $G$.  \n\nHowever, for large data sets, constructing this $n \\times n$ matrix is a computational bottleneck, so methods have been devised for lifting $n$ data points $P \\subset \\ensuremath{\\mathbb{R}}^d$ to a high-dimensional space $\\ensuremath{\\mathbb{R}}^m$ (but where $m \\ll n$) so that the Euclidean dot-product in this space approximates the non-linear inner product.  \n\nFor reproducing kernels $K$, there exists a lifting $\\phi : \\ensuremath{\\mathbb{R}}^d \\to \\Eu{H}_K$, where $\\Eu{H}_K$ is the reproducing kernel Hilbert space.  It is in general infinite dimensional, but for every finite subset of points $\\Phi(X) = \\{\\phi(x) \\mid x \\in X\\}$ spans an $n$-dimensional Euclidean space.  That is $K(x,y) = \\langle \\phi(x), \\phi(y) \\rangle$.  \nMoreover, we can define the norm of a point in $\\Eu{H}_K$ as $\\|\\phi(x)\\|_{\\Eu{H}_K} = \\sqrt{\\langle \\phi(x), \\phi(x) \\rangle}$ using the inner product, and then due to linearity, a distance (the \\emph{kernel distance}) between two points is defined:\t\t\n\\begin{align*}\nD_K(x,y) \n=& \n\\sqrt{\\|\\phi(x)\\|_{\\Eu{H}_K}^2 + \\|\\phi(y)\\|_{\\Eu{H}_K}^2 - 2 \\langle \\phi(x), \\phi(y) \\rangle}\\\\\n=&\n\\sqrt{K(x,x) + K(y,y) - 2K(x,y)}.\n\\end{align*}\nFor reproducing kernels (actually a slightly smaller set called characteristic kernels) this is a metric~\\cite{SGFSL10,muller97}.  \n\nThus we may desire an approximate lifting $\\hat \\phi : \\ensuremath{\\mathbb{R}}^d \\to \\ensuremath{\\mathbb{R}}^m$ such that with probability at least $1-\\delta$ and all $x,y \\in X$\n\\[\n(1-\\varepsilon) \\leq \\frac{D_K(x,y)}{\\|\\hat \\phi(x) - \\hat \\phi(y)\\|} \\leq (1+\\varepsilon).  \n\\]\nIt turns out, one can always construct such a lifting with $m = O((1\/\\varepsilon^2) \\log (n\/\\delta))$ by the famous Johnson-Lindenstrauss (JL) Lemma~\\cite{JL84}.  However, unlike the JL Lemma, there is not always known an implicit construction.  In general, we must first construct the $n \\times n$ gram matrix, revealing an $n$-dimensional subspace (through an $O(n^3)$ time eigendecomposition) and then apply $m = O((1\/\\varepsilon^2) \\log (n\/\\delta))$ random projections.  \n\nSo in recent years there have been many types of kernels considered for these implicit embeddings with various sorts of error analysis, such as for Gaussian kernels~\\cite{rahimi2007random,lopez2014randomized,SS15,SS15-UAI}\ngroup invariant kernels~\\cite{li2010random}, \nmin\/intersection kernels~\\cite{maji2009max}, \ndot-product kernels~\\cite{kar2012random}, \ninformation spaces~\\cite{AKMVV16},\nand polynomial kernels~\\cite{hamid2013compact,ANW14}.  \n\nIn this document we reanalyze one of the most widely used and first variants, the Random Fourier Features, introduced by \\cite{rahimi2007random}.  It applies to symmetric shift-invariant kernels which include Laplace, Cauchy, and most notably Gaussian.  We will primarily focus on  Gaussian kernels, defined $K(x,y) = e^{-\\frac{\\|x - y\\|^2}{2 \\sigma^2}}$, and use this definition for $K$ henceforth unless otherwise specified.  It is characteristic, hence $D_K$ is a metric.  \n\n\\subsection{Existing Properties of Gaussian Kernel Embeddings}\n\n\\cite{rahimi2007random} defined two approximate embedding functions: \n$\\tilde \\phi : \\ensuremath{\\mathbb{R}}^d \\to \\ensuremath{\\mathbb{R}}^m$ and $\\hat \\phi : \\ensuremath{\\mathbb{R}}^d \\to \\ensuremath{\\mathbb{R}}^m$.  Only the former appears in the final version of paper, but the latter is also commonly used throughout the literature~\\cite{SS15-UAI}.  \nBoth features use random variables $\\omega_i \\in \\ensuremath{\\mathbb{R}}^d$ drawn uniformly at random from the Fourier transform of the kernel function; in the case of the Gaussian kernel, the Fourier transform is again a Gaussian specifically $\\omega_i \\sim \\mathcal{N}_d(0,\\sigma^{-2})$.  \n\nIn the former case, they define $m$ functions of the form\n$\\tilde f_i(x) = \\cos(\\omega_i^T x + \\gamma_i)$, where $\\gamma_i \\sim \\textsf{Unif}(0,2\\pi]$, uniformly at random from the interval $(0,2\\pi]$, is a random shift.   \nApplying each $\\tilde f_i$ on a datapoint $x$ gives the $i$th coordinate of $\\tilde \\phi(x)$ in $\\ensuremath{\\mathbb{R}}^{m}$ as $\\tilde \\phi(x)_i = \\tilde f_i(x) \/ \\sqrt{m}$. \n\nIn the latter case, they define $t = m\/2$ functions of the form\n\n\\[\n\\hat f_i(x) = \\left[\\begin{array}{c}\\cos(\\langle \\omega_i, x \\rangle) \\\\ \\sin(\\langle \\omega_i, x \\rangle)\\end{array}\\right]\n\\]\n\nas a single $2 \\times 1$ dimensional vector, and one feature pair.  Then applying $\\hat f_i$ on a data point $x$ yields the $(2i)$th and $(2i+1)$th coordinate of $\\hat \\phi(x)$ in $\\ensuremath{\\mathbb{R}}^m$ as $[\\hat \\phi(x)_{2i}; \\hat \\phi(x)_{2i+1}] = \\hat f_i(x) \/ \\sqrt{t}$.  \n\n\nThey showed that $\\s{\\textbf{E}}[\\tilde \\phi(x)^T \\tilde \\phi(y)] = K(x,y)$ for any $x,y \\in \\ensuremath{\\mathbb{R}}^d$, and that this implied\n\\[\n\\Pr[| \\langle \\tilde \\phi(x), \\tilde \\phi(y) \\rangle - K(x,y) | \\geq \\varepsilon ] \\leq \\delta\n\\]\n\\begin{itemize} \\itemsep -2pt\\parsep=-1pt\\partopsep -2pt\n\t\\item with $m = O((1\/\\varepsilon^2) \\log(1\/\\delta))$ for each $x,y \\in \\ensuremath{\\mathbb{R}}^d$, \n\n\n\n\t%\n\t\\item with $m = O((1\/\\varepsilon^2) \\log(n\/\\delta))$, for all $x,y \\in X$, for $X \\subset \\ensuremath{\\mathbb{R}}^d$ of size $n$, \n\n\n\n\t%\n\t\\;\\;\\; or\n\t\\item with $m = O((d\/\\varepsilon^2) \\log(\\mathcal{M}\/\\delta))$, for all $x,y \\in X$, for $X \\subset \\ensuremath{\\mathbb{R}}^d$ so $\\mathcal{M} = \\displaystyle{\\max_{x,y \\in X}} \\|x-y\\|\/\\sigma$.  \n\n\n\n\\end{itemize}\nRecently \\cite{SS15} tightened the above asymptotic bounds to show actual constants.  \nIt is folklore (apparently removed from final version of \\cite{rahimi2007random}; reproved in Section \\ref{sec:basic}) that also  $\\s{\\textbf{E}}[\\hat \\phi(x)^T \\hat \\phi(y)] = K(x,y)$, and thus all of the above PAC bounds hold for $\\hat \\phi$ as well.  \n\n\nAlso recently, \\cite{SS15-UAI} compared $\\tilde \\phi$ and $\\hat \\phi$ (they used symbol $\\breve \\phi$ in place of our symbol $\\hat \\phi$), and demonstrated that $\\hat \\phi$ performs better (for the same $m$) and has provably lower variance in approximating $K(x,y)$ with $\\hat \\phi(x)^T \\hat \\phi(y)$ as opposed to with $\\tilde \\phi(x)^T \\tilde \\phi(y)$.  \nHowever, these results do \\emph{not} obtain a bound on $\\|\\hat \\phi(x) - \\hat \\phi(y)\\| \/ D_K(x,y)$ since for very small distances, the additive error bounds on $K(x,y)$ are not sufficient to say much about $D_K(x,y)$.  \n\n\n\n\n\n\n\n\n\\subsection{Our Results}\n\nIn this paper we show that $\\hat \\phi$ probabilistically induces a kernel \n$\\hat K(x,y) = \\langle \\hat \\phi(x), \\hat \\phi(y)\\rangle$ and a distance\n\\[\nD_{\\hat K}(x,y) = \\sqrt{\\|\\hat \\phi(x)\\|^2 + \\|\\hat \\phi(y)\\|^2 - 2 \\hat K(x,y)} = \\|\\hat \\phi(x) - \\hat \\phi(y)\\|,\n\\]\nwhich has strong relative error bounds with respect to $D_K(x,y)$, namely for a parameter $\\varepsilon \\in (0,1)$\n\\begin{equation} \\label{eq:rel-err}\n(1-\\varepsilon) \\leq \\frac{D_K(x,y)}{D_{\\hat K}(x,y)} \\leq (1+\\varepsilon).\n\\end{equation}\n\nIn Section \\ref{sec:basic} we show (\\ref{eq:rel-err}) holds for each $x,y$ such that $\\|x-y\\|\/\\sigma \\geq 1$, with probability at least $1-\\delta$, with $m = O((1\/\\varepsilon^2) \\log(1\/\\delta))$.  We also review known or folklore properties about $\\hat \\phi$ and $D_K$.  \n\nWe first prove bounds that depend on the size $n$ of a data set $X \\subset \\ensuremath{\\mathbb{R}}^d$.  \nWe show that $m = O((1\/\\varepsilon^2) \\log n)$ features are necessary (Section \\ref{sec:local-l2}) and sufficient (Section \\ref{sec:small-data}) to achieve (\\ref{eq:rel-err}) with high probability (e.g., at least $1-1\/n$), when $d$ and $X$ are otherwise unrestricted.  \n\nIn Section \\ref{sec:lowD} we prove bounds for $X \\subset \\ensuremath{\\mathbb{R}}^d$ where $d$ is small, but the size $n = |X|$ is unrestricted.  Let $\\mathcal{M} = \\max_{x,y \\in X} \\|x-y\\|\/\\sigma$.  \nWe show that $m = O((d\/\\varepsilon^2) \\log(\\frac{d}{\\varepsilon} \\frac{\\mathcal{M}}{\\delta}))$ is sufficient to show (\\ref{eq:rel-err}) with probability $1-\\delta$.   Then in Section \\ref{sec:LowDimLB} we show that $m = \\Omega(\\frac{d}{\\varepsilon^2 \\log(1\/\\varepsilon)} \\log (\\frac{\\mathcal{M}}{\\log(1\/\\varepsilon)}))$ is necessary for any feature map.  \n\n\nWe also empirically confirm the relative error through some simulations in Section \\ref{sec:exp}.  \nThis includes showing that kernel PCA using these approximate features obtains relative error.  \n\n\\paragraph{Further Implications in Machine Learning.}\nThere has been extensive recent effort to find oblivious subspace embeddings (OSE) of data sets into Euclidian spaces that preserve relative error. Such strong guarantees are, for instance, required to prove results about regression on the resulting set since we may not know the units on different coordinates; additive error bounds do not make sense in directions which are linear combinations of several coordinates.\n\nThe obliviousness of the features (they can be defined without seeing the data, and in some cases are independent of the data size) are essential for many large-scale settings such as streaming or distributed computing where we are not able to observe all of the data at once.\n\nOur results to do not describe unrestricted OSEs, as are possible with polynomial kernels \\cite{ANW14}. Rather our lower bounds show that any OSE must have dimension depend on $n$ or $\\mathcal{M}$.\n\nHowever, we show that random feature mappings allow for a finer notion of approximating the geometry of RKHS than previously known.\nIn particular, the approximation error of inner products is proportional to the approximation error of distances. This is because both $\\phi$ and $\\ensuremath{\\hat{\\phi}}$ map every input point to a unit vector; thus $D_K(x,y)^2 = 2 - 2K(x,y)$ and $D_{\\hat K}(x,y)^2 = 2 - 2 \\hat{K}(x,y)$, for any distinct $x,y \\in \\ensuremath{\\mathbb{R}}^d$. Therefore $|K(x,y) - \\hat{K}(x,y)|$ is the same as $\\frac{1}{2} | D_{K}(x,y)^2 - D_{\\hat{K}}(x,y)^2 |$.  Hence approximation error of the Gram matrix is bounded in terms of the sum of errors in pairwise distances\n\\[ \n\\| G - \\hat{G} \\|_1 \\leq \\frac{1}{2}\\sum_{x,y \\in X} |D_{K}(x,y)^2 - D_{\\hat{K}}(x,y)^2|. \n\\]\nMoreover, with our low-dimensional bounds in Section \\ref{sec:lowD}, we can see that if an object $U \\subset \\ensuremath{\\mathbb{R}}^d$ (such as a non-linear decision boundary) and training data $S \\subset \\ensuremath{\\mathbb{R}}^d$ both lie within a ball with finite radius $\\mathcal{M}$, then for any point $x \\in S$, the minimum kernel distance between $U$ and $x$ is approximately preserved in the random feature space as $\\min_{y \\in U} \\|\\phi(x) - \\phi(y)\\|$. This suggests better performance guarantees for kernelized learning problems regarding the minimization of $\\ell_2$ distances, such as in kernel SVM (hinge-loss) and in kernel PCA (recovery error); see Section \\ref{sec:exp}.  \n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Basic Bounds}\n\\label{sec:basic}\n\n\nWe first review some properties of the Gaussian kernel and the theory of Random Fourier Features. \n\nAn earlier version of Rahimi-Recht~\\cite{rahimi2007random} seemed to prove the following lemma.  We repeat it for completeness.  \n\\begin{lemma}  \\label{lemma:ExpectationInnerProduct}\nGiven a random $d$-dimensional Gaussian $\\omega_i \\sim \\mathcal{N}_d(0, \\sigma^{-2})$, and any two $x,y \\in \\ensuremath{\\mathbb{R}}^d$ then \n\\[\n\\s{\\textbf{E}}_{\\omega_i}[\\langle \\hat f_i(x), \\hat f_i(y) \\rangle] = K(x,y).\n\\] \n\\end{lemma}\n\\begin{proof}\n\tFirst we expand\n\t\\begin{align*}\n\t\\langle \\hat f_i(x), \\hat f_i(y) \\rangle = &\n\t\\cos(\\langle \\omega_i, x \\rangle) \\cos(\\langle \\omega_i, y \\rangle) +  \\sin(\\langle \\omega_i, x \\rangle) \\sin(\\langle \\omega_i, y \\rangle)\\\\\n\t= &\\cos(\\langle \\omega_i, x-y \\rangle).\n\t\\end{align*}\n\tBy the stability and scaling of the Gaussian distribution, we can write $\\langle \\omega_i, x-y \\rangle$ as $ \\omega_{i,x,y} \\frac{\\| x - y \\|}{\\sigma}$ where $\\omega_{i,x,y} \\sim \\mathcal{N}(0,1)$. Now note \n\n\t\\[\n\t\\s{\\textbf{E}}_{\\omega_i}[\\langle \\hat f_i(x), \\hat f_i(y) \\rangle]\\\\ = \\int_{z \\in \\mathbb{R}} \\! \\frac{1}{\\sqrt{2\\pi} } \\exp(-z^2\/2 )) \\cos(z \\frac{\\| x - y \\|}{\\sigma}) \\, \\mathrm{d}z.\n\n\t\\]\n\n\tWe next show that the right hand side evaluates to $K(x,y)$.  \n\tUsing $\\Delta = \\frac{x-y}{\\sigma}$, let $g(\\|\\Delta\\|) := \\int_{z \\in \\mathbb{R}} \\exp(-z^2\/(2 )) \\cos(z \\|\\Delta\\|) dz$. Now with integration by parts,\n\t\\begin{align*}\n\t\\frac{\\mathrm{d}}{\\mathrm{d}\\|\\Delta\\|} g(\\|\\Delta\\|)  =  &\n\t- \\int_{ \\in \\mathbb{R}} \\! e^{-\\frac{z^2}{2}} \\cdot z \\sin(z \\|\\Delta\\|) \\, \\mathrm{d}z \n\t\\\\ = & \n\t\\int_{z \\in \\mathbb{R}} \\! \\frac{\\partial}{\\partial z} \\left( e^{-\\frac{z^2}{2}} \\sin(z  \\|\\Delta\\|) \\right) -  \\|\\Delta\\| e^{-\\frac{z^2}{2}} \\cos(z \\|\\Delta\\|) \\, \\mathrm{d}z \\\\\n\t= &-  \\|\\Delta\\| \\int_{z \\in \\mathbb{R}} \\! e^{-\\frac{z^2}{2}} \\cos(z  \\|\\Delta\\|) \\, \\mathrm{d}z = -  \\|\\Delta\\| g(\\|\\Delta\\|).  \n\t\\end{align*}\n\t\n\tSo $g(\\|\\Delta\\|) = \\kappa e^{-\\frac{\\|\\Delta\\|^2}{2}}$ for some constant $\\kappa$. But $\\kappa = g(0)=\\int_{z \\in \\mathbb{R}} e^{-\\frac{z^2}{2}} \\mathrm{d}z = \\sqrt{2\\pi}$, therefore:\n\t\\begin{align*}\n\t\\s{\\textbf{E}}_{\\omega_i}[\\langle \\psi_i(x), \\psi_i(y) \\rangle]= & \\sqrt{2\\pi} \\cdot \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{\\|\\Delta\\|^2}{2}} = e^{-\\frac{\\|x-y\\|^2}{2\\sigma^2}} = K(x,y). \\qedhere\n\t\\end{align*}\n\\end{proof}\n\n\n\nWe can then state the next simple corollary.  \n\n\\begin{corollary}\nThe higher dimensional random feature maps $\\ensuremath{\\hat{\\phi}}$ also satisfy for any $x,y \\in \\ensuremath{\\mathbb{R}}^d$\n\\[\n\\s{\\textbf{E}}_{\\omega_1, \\ldots, \\omega_d}[ \\langle \\ensuremath{\\hat{\\phi}}(x), \\ensuremath{\\hat{\\phi}}(y) \\rangle ] = K(x,y).\n\\]\n\\end{corollary}\n\nFor any $x,y \\in \\ensuremath{\\mathbb{R}}^d$ and any $\\omega_i \\sim \\mathcal{N}(0,\\varsigma)$ we have that \n$\n\\langle \\psi_i(x), \\psi_i(y) \\rangle \\in [-2,2].  \n$\nThus we can apply a standard Chernoff-Hoeffding bound to show the next corollary.\n\n\\begin{corollary}\nGiven a random feature map $\\ensuremath{\\hat{\\phi}}$ defined by $t = O((1\/\\varepsilon^2) \\log(1\/\\delta))$ iid random $d$-dimensional Gaussian $\\omega_i \\sim \\mathcal{N}_d(0,\\sigma^{-2})$, for $\\varepsilon \\in (0,1\/2)$ and $\\delta \\in (0,1)$, then for any $x,y \\in \\ensuremath{\\mathbb{R}}^d$\n\\[\n\\Pr\\left[ \\left| K(x,y) - \\langle \\ensuremath{\\hat{\\phi}}(x), \\ensuremath{\\hat{\\phi}}(y) \\rangle \\right| \\leq \\varepsilon\\right] \\geq 1-\\delta.  \n\\]\n\\end{corollary}\n\n\n\n\n\nSince $\\hat K(x,x,) = 1$, then $D_{\\hat K}(x,y)^2 = 2 - 2 \\hat K(x,y)$, and additive error bounds between $D_K(x,y)^2$ and $D_{\\hat K}(x,y)^2$ follow directly.  But we can also state some relative error bounds when $\\|x-y\\|$ is large enough.  \n\\begin{lemma}\n\\label{lem:bigD-rel}\nFor each $x,y \\in \\ensuremath{\\mathbb{R}}^d$ such that $\\|x - y\\| \\geq \\sigma$ and $m = O((1\/\\varepsilon^2) \\log(1\/\\delta))$ with $\\varepsilon \\in (0, 1\/10)$ and $\\delta \\in (0,1)$.  \nThen with probability at least $1-\\delta$, we have \n$\n\\frac{D_K(x,y)}{D_{\\hat K}(x,y)} \\in [1-\\varepsilon,1+\\varepsilon].\n$\n\\end{lemma}\n\\begin{proof}\nBy choosing $m = O((1\/\\varepsilon^2) \\log (1\/\\delta))$ so that $|K(x,y) - \\hat K(x,y)| \\leq \\varepsilon\/4$ (via Lemma \\ref{lemma:ExpectationInnerProduct}), we have that $|D^2_K(x,y) - D^2_{\\hat K}(x,y) | \\leq \\varepsilon\/2$.  \nWe also note that when $\\|x-y\\| \\geq \\sigma$ then $K(x,y) \\leq \\exp(- \\sigma^2\/(2\\sigma^2)) = \\frac{1}{\\sqrt{e}} \\leq 0.61$.  \nHence $D^2_K(x,y) \\geq 2(1-0.61) = 0.78 \\geq 0.5$, and we have that \n\\[\n| D^2_K(x,y) - D^2_{\\hat K}(x,y) | \\leq \\varepsilon\/2 \\leq \\varepsilon D^2_K(x,y).  \n\\]\nThen \n$| 1 - \\frac{D^2_{\\hat K}(x,y)}{D^2_K(x,y)} | \\leq \\varepsilon$, \nimplying\n$1-\\varepsilon \\leq \\frac{D^2_{\\hat K}(x,y)}{D^2_K(x,y)} \\leq 1+\\varepsilon$.  \nTaking the square root of all parts completes the proof via $\\sqrt{1+\\varepsilon} < (1+\\varepsilon)$ and $\\sqrt{1-\\varepsilon} > (1-\\varepsilon)$.  \n\\end{proof}\n\n\nSo to understand the relative error in $D_K(x,y)$, what remains is the case when $\\|x-y\\|$ is small. As we will see, when $\\|x-y\\|$ is small, then $D_K(x,y)$ behaves like $\\|x-y\\|$ and we can borrow insights from $\\ell_2$ embeddings.  \nThen combining the two cases (when $\\|x-y\\|$ is large and when $\\|x-y\\|$ is small) we can achieve ``for all bounds\" either via simple union bounds, or through careful ``continuous net'' arguments when $X$ is in a bounded range.  Similarly, we will show near-matching lower bounds via appealing to near-$\\ell_2$ properties or via net arguments.  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Approximations and Relation to $\\ell_2$ on Small Distances}\n\\label{sec:local-l2}\n\nFor the remainder of the paper, it will be convenient to let $\\Delta = (x-y)\/\\sigma$ be the scaled vector between some pair of points $x,y \\in X$.   Define $D_K(\\Delta) = D_K(x,y) = \\sqrt{2 - 2 e^{\\frac{1}{2} \\|\\Delta\\|^2}}$.\nLemma \\ref{lem:bigD-rel} already established that when $\\|\\Delta\\| \\geq 1$, then additive error bounds imply relative error bounds.  In this section we consider the alternate case of when $\\|\\Delta\\| \\leq 1$, and show that in this setting that $D_{\\hat K}$ is indeed close to $D_K$, and to varying degrees also approximates $\\ell_2$.  \n\n\nAs a warm up, and as was similarly observed in \\cite{PWZ15}, a simple Taylor expansion when $\\|\\Delta\\| \\leq 1$, implies that\n\\[\n\\|\\Delta\\|^2 - \\frac{1}{4} \\|\\Delta\\|^4\n\\leq\nD_K(\\Delta)^2 = 2 - 2\\exp(\\|\\Delta\\|^2\/2) \\leq \\|\\Delta\\|^2,\n\\]\nand by $\\frac{1}{4} \\|\\Delta\\|^4 \\leq \\frac{1}{4} \\|\\Delta\\|^2$ and a square root \n\\begin{equation} \\label{eq:near-linear}\n0.86 \\|\\Delta\\| \\leq D_K(\\Delta) \\leq \\|\\Delta\\|.\n\\end{equation}\nMoreover, when $\\|\\Delta\\| \\leq 2 \\sqrt{\\varepsilon}$ then \n\\begin{equation} \\label{eq:eps-linear}\n(1-\\varepsilon) \\|\\Delta\\|^2 \\leq D_K(\\Delta)^2 \\leq \\|\\Delta\\|^2.\n\\end{equation}\nIn what follows we derive more nuanced and powerful bounds relating $D_K(x,y)$ and $D_{\\hat K}(x,y)$; which transitively relates $D_{\\hat K}(x,y)$ to $\\|x-y\\|$.  \n\n\n\n\n\\paragraph{Useful expansions.}\nWe first observe that by $\\cos(a)\\cos(b) + \\sin(a)\\sin(b) = \\cos(a-b)$ that\n\\[\n\\langle \\hat f_i(x), \\hat f_i(y) \\rangle \n= \n\\cos(\\langle \\omega_i, x \\rangle) \\cos(\\langle \\omega_i, y \\rangle) + \\sin(\\langle \\omega_i, x \\rangle) \\sin(\\langle \\omega_i, y \\rangle) \n= \n\\cos(\\langle \\omega_i, (x-y) \\rangle).  \n\\]\nHence by $\\langle \\hat f_i(x), \\hat f_i(x) \\rangle = \\cos(\\langle \\omega_i, 0) = 1$ we have $D_{\\hat K}(x,y)^2 = 2 - 2 \\frac{1}{t} \\sum_i^t \\cos (\\langle \\omega_i, ( x - y ) \\rangle)$.  \n\nBy the rotational stability of the Gaussian distribution we can replace $\\langle \\omega_i, ( x - y ) \\rangle$ with $\\omega_{i,x,y} \\frac{\\| x - y\\|}{\\sigma}$ where $\\omega_{i,x,y} \\sim \\mathcal{N}(0,1)$.  It will be more convenient to write $\\omega_{i,x,y}$ as $\\omega_{i,\\Delta}$, so $\\langle \\omega_i, ( x - y ) \\rangle = \\omega_{i, \\Delta} \\|\\Delta\\|$. \nThus\n$\\langle \\hat f_i(x), \\hat f_i(y) \\rangle = \\cos(\\omega_{i,\\Delta} \\|\\Delta\\|)$.  \nMoreover, we can define\n$\nD_{\\hat K}(\\Delta) = D_{\\hat K}(x,y) = \\sqrt{2 - 2 \\frac{1}{t} \\sum_{i=1}^t \\cos(\\omega_{i,\\Delta} \\|\\Delta\\|)}\n$.\n\nNow considering \n\\[\n\\frac{D_{\\hat K}(\\Delta)^2}{D_K(\\Delta)^2} = \\frac{1 - \\frac{1}{t} \\sum_{i=1}^t \\cos(\\omega_{i,\\Delta} \\|\\Delta\\|)}{1- e^{\\frac{1}{2} \\|\\Delta\\|^2}}, \n\\]\nthe following Taylor expansion, for $\\omega_{i,\\Delta} \\|\\Delta\\| \\leq 1$, will be extremely useful:\n\\[\n \\frac{\\frac{1}{t} \\sum_{i=1}^t \\frac{1}{2} \\omega^2_{i,\\Delta} \\|\\Delta\\|^2 }{\\frac{1}{2}{\\|\\Delta\\|^2} - \\cdot \\frac{1}{4} \\|\\Delta\\|^4}\n\\geq\n\\frac{D_{\\hat{K}}(\\Delta)^2}{D_{K}(\\Delta)^2}\n\\geq\n\\frac{\\frac{1}{t} \\sum_{i=1}^t \\left( \\frac{1}{2}\\omega^2_{i,\\Delta} \\|\\Delta\\|^2 -  \\frac{1}{24} (\\omega^4_{i,\\Delta} \\|\\Delta\\|^4 )\\right)}{\\frac{1}{2}{\\|\\Delta\\|^2}}.\n\\]\nSimplifying gives\n\\begin{equation}     \\label{equation:RelativeErrorObjective}\n\\frac{1}{1 - \\frac{1}{2} \\|\\Delta\\|^2} \\left(\\frac{1}{t} \\sum_{i=1}^t\\omega^2_{i,\\Delta}\\right)\n    \\geq \\frac{D_{\\hat{K}}(\\Delta)^2}{D_{K}(\\Delta)^2} \\geq \n    \\left(\\frac{1}{t} \\sum_{i=1}^t \\omega^2_{i,\\Delta}\\right) - \\frac{\\| \\Delta \\|^2}{12 } \\cdot \\frac{1}{t} \\sum_{i=1}^t \\omega^4_{i,\\Delta}.\n\\end{equation}\n\n\\paragraph{Very small distances.}\nNext we can understand what happens in the limit as we shrink the region containing $X$.  We do so by using a scaling parameter $\\lambda$, and observe what happens to the ratio $D_{\\hat K}(\\lambda \\Delta)^2 \/ D_K(\\lambda \\Delta)$ in the limit as $\\lambda \\to 0$.  \n\n\\begin{lemma}  \\label{lem:lim-err}\nFor scalar scaling parameter $\\lambda$, \n \\[\n \\lim_{\\lambda \\rightarrow 0} \\frac{D_{\\hat K}(\\lambda \\Delta)^2}{D_K(\\lambda \\Delta)^2} = \\frac{1}{t} \\sum_{i=1}^t \\omega^2_{i,\\Delta}.\n \\]\n\\end{lemma}\n\\begin{proof}\nObserve that $\\omega_{i,\\Delta} = \\omega_{i,\\lambda \\Delta}$, for any $\\lambda > 0$.  \nThus in equation (\\ref{equation:RelativeErrorObjective}), $\\lim_{\\lambda \\to 0} 1\/(1-\\frac{1}{2}\\|\\lambda \\Delta\\|^2)$ goes to $1$ so the left hand-side approaches $\\frac{1}{t} \\sum_{i=1}^t \\omega_{i,\\Delta}^2$.  \nSimilarly, $\\lim_{\\lambda \\to 0} \\|\\lambda \\Delta\\|^2 \/12$ goes to $0$, and the right-hand side also approaches $\\frac{1}{t} \\sum_{i=1}^t \\omega_{i,\\Delta}^2$.  \n\\end{proof}\n\nIf we fix $\\Delta$ then $\\omega_{i,\\Delta}, 1 \\leq i \\leq t$ are i.i.d Gaussian variables with mean $0$ and standard deviation $1$. Thus $\\sum_{i=1}^t\\omega^2_{i,\\Delta}$ is a $\\chi^2$-variable with $t$ degrees of freedom. \nThis observation paired with Lemma \\ref{lem:lim-err} will be useful for a lower bound.  \n\nBut to prove an upper bound, we do not need to go all the way to the limit.  Common concentration results for $\\chi^2$ variables give us the following.\n\n\n\\begin{lemma}   \\label{lem:OmegaChi}\nFor $\\varepsilon\\in (0,1)$ and $\\delta \\in (0,1\/2)$, if $t \\geq 8 \\frac{1}{\\varepsilon^2} \\ln (2\/\\delta)$ then \\[\n\\Pr \\left[\\frac{1}{t}\\sum_{i=1}^t\\omega^2_{i,\\Delta} \\notin [1 - \\varepsilon, 1 + \\varepsilon] \\right] \\leq \\delta.  \n\\]\n\\end{lemma}\n\\begin{proof}\nHere we use Lemma 1 from \\cite{10.2307\/2674095}; if $X$ is a $\\chi^2$ random variable with $t$ degrees of freedom\n\\[ \n\\Pr[t - 2 \\sqrt{tx} \\leq X \\leq t + 2 \\sqrt{tx} + 2x] \\geq 1 - 2e^{-x}.  \n\\] \nHere we can set $x := \\frac{1}{8} t \\varepsilon^2$ then \n$t - 2 \\sqrt{tx} = t - \\varepsilon t\/\\sqrt{2}$, and \n$t + 2 \\sqrt{tx} + 2x = t + \\varepsilon t\/\\sqrt{2} + \\frac{1}{4} \\varepsilon^2 t < t + \\varepsilon t$.\n \nAlso, \n$2e^{-x} = 2 e^{-\\frac{1}{8} t \\varepsilon^2} = 2 e^{- \\ln (2\/\\delta)} = \\delta\/2 \\leq \\delta$ for $\\delta \\leq 1\/2$.  \n\nTherefore, $\\frac{1}{t}\\sum_{i=1}^t\\omega^2_{i,\\Delta} \\notin [1 - \\varepsilon, 1 + \\varepsilon]$ with probability at most $\\delta$.    \n\\end{proof}\n\nThis result bounds to $[1-\\varepsilon, 1+\\varepsilon]$ for some $t = O((1\/\\varepsilon^2) \\log(1\/\\delta)$, the main terms of equation (\\ref{equation:RelativeErrorObjective}).  \nHowever, the other parts ($\\|\\Delta\\|^2\/2$ and the term containing $\\omega^4_{i,\\Delta}$) require a further restriction on $\\|\\Delta\\|$ to be  handled directly, as we show in the following.\n\n\\begin{lemma}   \\label{lem:SpikePreservation}\nFor $\\varepsilon\\in (0,1)$ and $\\delta \\in (0,1\/2)$, if $\\|\\Delta\\| \\leq \\frac{\\sqrt{\\varepsilon}}{\\log (1\/\\delta)}$, and $t = \\Omega(\\frac{1}{\\varepsilon^2} \\log (1\/\\delta))$, then with probability at least $1 - O(\\delta)$, \n for all $\\lambda \\in [0,1]$ we have \n$\n\\frac{D_{\\hat{K}}(\\lambda \\cdot \\Delta)^2}{D_{K}(\\lambda \\cdot \\Delta)^2} \\in [1 - \\varepsilon, 1 + \\varepsilon].\n$\n\\end{lemma}\n\n\\begin{proof}\nIf $\\omega$ is a standard Gaussian variable, then $|\\omega| \\leq  C \\cdot q \\sqrt{\\log (1\/\\delta)}$ with probability at least $1 - \\delta^q$ for any $q > 0$, for some constant $C$. \nThis means, if \n$\\|\\Delta\\| \\leq \\frac{\\sqrt{\\varepsilon}}{\\log (1\/\\delta)}$\nthen \n$\\omega_{i,\\Delta} \\|\\Delta\\| \\leq \\sqrt{\\frac{\\varepsilon}{\\log (1\/\\delta)}}$ \nwith probability at least $1 - O(\\delta)$.  Also then $\\omega_{i,\\Delta} \\|\\Delta\\| \\leq 1$, which satisfies the conditions for ($\\ref{equation:RelativeErrorObjective}$).\n\nThis also bounds the term\n$\\omega^4_{i,\\Delta} \\leq \\frac{\\varepsilon^2}{\\log^2 (1\/\\delta)}$\nused in equation  (\\ref{equation:RelativeErrorObjective}).  \nIn particular, along with $\\|\\Delta\\| \\leq \\frac{\\sqrt{\\varepsilon}}{\\log (1\/\\delta)}$, this implies\n\\[\n\\frac{\\| \\Delta \\|^2}{12} \\cdot \\frac{1}{t} \\sum_{i=1}^t \\omega^4_{i,\\Delta} \n\\leq \n\\frac{1}{12} \\|\\Delta\\| \\cdot \\frac{\\varepsilon^2}{\\log^2 (1\/\\delta)} \n=\n\\frac{\\varepsilon}{12} \\frac{\\varepsilon}{\\log^2 (1\/\\delta)} \\frac{1}{\\|\\Delta\\|^2}\n\\leq\n\\frac{\\varepsilon}{12}.\n\\]\nThen along with Lemma~\\ref{lem:OmegaChi} and RHS of (\\ref{equation:RelativeErrorObjective}), we have $\\frac{D_{\\hat{K}}(\\Delta)^2}{D_K(\\Delta)^2} \\geq 1 - O(\\varepsilon)$. \n \nSimilarly, Lemma~\\ref{lem:OmegaChi} and $\\frac{1}{2}\\|\\Delta\\|^2 \\leq \\frac{\\varepsilon}{2 \\log^2 (1\/\\delta)}$ imply the LHS of (\\ref{equation:RelativeErrorObjective}) is bounded above by $1 + O(\\varepsilon)$. Thus, after adjusting constants in $t$, we have $\\frac{D_{\\hat{K}}(\\Delta)^2}{D_{K}(\\Delta)^2} \\in [1 - \\varepsilon, 1 + \\varepsilon]$.\n \nFor $\\frac{D_{\\hat{K}}(\\lambda \\Delta)^2}{D_{K}(\\lambda \\Delta)^2} \\in [1 - \\varepsilon, 1 + \\varepsilon]$, note that the above analysis still holds if we scale $\\|\\Delta\\|$ to be smaller, i.e. as long as $\\lambda \\in [0,1]$.  In particular, $\\omega_{i, \\Delta}$ is unchanged by the scaling $\\lambda$.  \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Lower Bounds, Based on Very Small $\\|x-y\\|$}\n\nLemma \\ref{lem:lim-err} implies that when $\\|x-y\\|$ is small, $D_{\\hat K}(x,y)$ behaves like a Johnson-Lindenstrauss (JL) random projection of $\\|x-y\\|$, and we can invoke known JL lower bounds.  \n\nIn particular, Lemma \\ref{lem:lim-err} implies if the input data set $X \\subset \\ensuremath{\\mathbb{R}}^d$ is in a sufficiently small neighborhood of zero, the relative error is preserved only when $\\sum_{i=1}^t \\omega^2_{i,x,y}\\|\\lambda(x-y)\\|^2 \\in [(1-\\epsilon)\\|\\lambda(x-y)\\|^2,(1+\\epsilon)\\|\\lambda(x-y)\\|^2]$ for all $x,y \\in X$, and for all arbitrary $\\lambda \\in \\ensuremath{\\mathbb{R}}$. Which implies for arbitrary $x,y \\in X$, and $\\lambda \\in \\ensuremath{\\mathbb{R}}$ that\n\\[\n\\sqrt{\\sum^t_{i=1} \\left|\\omega_{i}\\cdot \\lambda(x-y)\\right|^2 } \n= \n\\sqrt{\\sum^t_{i=1} \\omega^2_{i,x,y}\\lambda\\|x-y\\|^2 } \n\\in \n\\left[ \\left(1-\\varepsilon\\right)\\lambda\\left\\| x -y \\right\\|,\\left(1+\\varepsilon\\right)\\lambda\\left\\| x -y \\right\\| \\right].  \n\\]\n\nThe far left hand side is in fact the norm $\\| g(x)-  g(y)\\|$ where $g(x)$ is the vector with coordinates $(\\omega_1 \\cdot \\lambda x, ..., \\omega_t \\cdot \\lambda x)$. Thus these are the exact conditions for relative error bounds on embedding $\\ell_2$ via the Johnson-Lindenstrauss transforms, which gives the following.\n\n\\begin{lemma} \\label{lem:RFF-JL}\n If for any $n,d > 0, X \\subset \\ensuremath{\\mathbb{R}}^d$ s.t. $|X| = n$, using $t(n,\\varepsilon)$ random Fourier features, $\\frac{D_{\\hat{K}}(x,y)}{D_{K}(x,y)} \\in [1 - \\varepsilon, 1 + \\varepsilon]$ with probability $1 - \\delta$, then there exists a random linear embedding with $t(n,\\varepsilon)$ projected dimensions preserving the $\\ell_2$-norm for all pairs $x,y \\in S$ up to relative error with probability at least $1 - \\delta$.\n\\end{lemma}\n\n\\begin{theorem}\nThere exists a set of $n$ points $X \\subset \\ensuremath{\\mathbb{R}}^d$ so that $t = \\Omega(\\frac{1}{\\varepsilon^2} \\log n)$ dimensions are necessary so for any $x,y \\in X$ that $\\frac{D_{\\hat{K}}(x,y)}{D_{K}(x,y)} \\in [1 - \\varepsilon, 1 + \\varepsilon]$.\n\\label{thm:eps-2logn-LB}\n\\end{theorem}\n\\begin{proof}\nA lower bound of $\\Omega(\\frac{1}{\\varepsilon^2} \\log n)$ projected dimensions for linear embeddings in $\\ell_2$ is given by e.g. \\cite{DBLP:journals\/corr\/LarsenN14}.\n\\end{proof}\n\nSection \\ref{sec:LowDimLB} shows another lower bound for point sets with unbounded $n$, but that are contained in a ball of bounded radius $\\mathcal{M}$ and bounded dimension $d$.  \n\n\\vspace{.1in}\\noindent\n\\emph{Remark:}\nA new result of Larsen and Nelson~\\cite{LN16} provides a $t = \\Omega(\\frac{1}{\\varepsilon^2} \\log n)$ lower bound for even \\emph{non-linear} embeddings of a size $n$ point set in $\\ensuremath{\\mathbb{R}}^d$  into $\\ensuremath{\\mathbb{R}}^t$ that preserve distances within $(1 \\pm \\varepsilon)$.  It holds for any $\\varepsilon \\in (1\/\\min\\{n,d\\}^{0.4999},1)$.  Since, there exists an isometric embedding of any set of $n$ points in any RKHS into $\\ensuremath{\\mathbb{R}}^n$, then this $t = \\Omega(\\frac{1}{\\varepsilon^2} \\log n)$ lower bound suggests that it applies to $\\hat \\phi$ and $\\tilde \\phi$ or any other technique, for almost any $\\varepsilon$.  However, it is not clear that \\emph{any} point set (including the ones used in the strong lower bound proof~\\cite{LN16}), can result from an isomorphic (or approximate) embedding of RKHS into $\\ensuremath{\\mathbb{R}}^n$.  Hence, this new result does not immediately imply our lower bound.  \n\nMoreover, the above proof of Theorem \\ref{thm:eps-2logn-LB} retains two points of potential interest.  First it holds for a (very slightly) larger range of $\\varepsilon \\in (0,1)$.  Second, Lemma \\ref{lem:RFF-JL} highlights that at very small ranges, $\\hat \\phi$ is indistinguishable from the standard JL embedding.  \n\n\n\n\n\n\n\n\n\n\n\n\\section{Relative Error Bounds For Small Distances and Small Data Sets}\n\\label{sec:small-data}\n\n\n\nThe Taylor expansion in equation (\\ref{equation:RelativeErrorObjective}) and additive errors via Lemma \\ref{lemma:ExpectationInnerProduct} are only sufficient to provide us bounds for $\\|\\Delta\\| \\leq \\sqrt{\\varepsilon}\/\\log n$ or for $\\|\\Delta\\| \\geq 1$.  In this section we need to use a more powerful technique or moment generating functions to fill in this gap.  \n\n\n\nIn particular, $1 - \\cos \\left( \\omega_{i,\\Delta} \\|\\Delta\\| \\right)$ is a sub-Gaussian random variable so it is expected to have a good concentration, but this fact is not enough for relative error bounds. We will use a more precise bound of the moment generating function of $1 - \\cos \\left( \\omega_{i,\\Delta} \\|\\Delta\\| \\right)$. Recall that the moment generating function $M(s)$ of a random variable $X$ is given by $\\s{\\textbf{E}}[e^{sX}]$.\n\n\\begin{lemma} \\label{lemma:BoundMGF}\nFor $\\omega \\sim \\mathcal{N}(0,1)$ and $0 \\leq \\|\\Delta\\| \\leq 1$,  let $M(s)$ be the moment generating function of $1 - \\cos \\left( \\omega \\|\\Delta\\| \\right) - \\left( 1 - e^{-\\frac{1}{2}\\|\\Delta\\|^2} \\right) = e^{-\\frac{1}{2}\\|\\Delta\\|^2} - \\cos \\left( \\omega \\|\\Delta\\| \\right)$. Then \nfor all $s \\in [0,\\frac{1}{2 \\|\\Delta\\|^2})$, \n\\[\n\\ln M(s) \\leq \\frac{1}{4} s^2 \\|\\Delta\\|^4. \n\\]\n\\end{lemma}\n\\begin{proof}\n First we note two Taylor approximations which hold for all $x \\in \\ensuremath{\\mathbb{R}}$:  \n\\[ \n\\cos x    \\geq 1 - \\frac{1}{2} x^2 \n\\;\\;\\;\\;\\; \\text{ and }  \\;\\;\\;\\;\\;       \ne^{-x}  \\leq 1 -  x + \\frac{1}{2} x^2.  \n\\]\n Now\n \\begin{align*}\nM(s)  \n & = \n\\s{\\textbf{E}} \\left[\\exp \\left(s (e^{-\\frac{1}{2}\\|\\Delta\\|^2} - \\cos(\\omega \\|\\Delta\\|)) \\right) \\right] \n\\\\  & \\leq  \n\\s{\\textbf{E}} \\left[\\exp \\left( s(1 - \\frac{1}{2} \\|\\Delta\\|^2 + \\frac{1}{8} \\|\\Delta\\|^4) - s (1 -  \\frac{1}{2} \\omega^2 \\|\\Delta\\|^2) \\right) \\right]\n\\\\ & =  \n\\s{\\textbf{E}} \\left[\\exp \\left(- \\frac{s}{2} \\|\\Delta\\|^2 + \\frac{s}{8} \\|\\Delta\\|^4 + \\frac{s}{2} \\omega^2 \\|\\Delta\\|^2 \\right) \\right]\n\\\\ & =\n\\exp( \\frac{s}{8} \\cdot \\|\\Delta\\|^4 - \\frac{s}{2}\\|\\Delta\\|^2) \\cdot \\s{\\textbf{E}} \\left[ e^{- s \\frac{1}{2} \\omega^2 \\|\\Delta\\|^2} \\right].  \n  \\end{align*}\nBut \n \\begin{align*}\n\\s{\\textbf{E}}\\left[ e^{- s \\frac{1}{2}\\omega^2\\|\\Delta\\|^2} \\right]\n    & =  \\int_{-\\infty}^{\\infty} \\! \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{u^2}{2}} \\cdot e^{- s \\frac{1}{2}u^2 \\|\\Delta\\|^2} \\mathrm{d}u \\\\\n    & =  \\int_{-\\infty}^{\\infty} \\! \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{1}{2} u^2 ( 1 + s \\|\\Delta\\|^2)} \\mathrm{d}u \\\\\n    & =  \\frac{1}{\\sqrt{1 + s \\|\\Delta\\|^2}} \\int_{-\\infty}^{\\infty} \\! \\sqrt{1 + s \\|\\Delta\\|^2} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{1}{2} u^2 (1 + s \\|\\Delta\\|^2)} \\mathrm{d}u \\\\\n    & = \\frac{1}{\\sqrt{1 + s \\|\\Delta\\|^2}}.  \n  \\end{align*}\nNoting that $\\ln (1 + x) \\geq x - \\frac{x^2}{2}$ for $x \\in [0,\\frac{1}{2})$, then whenever $s \\in [0, \\frac{1}{2\\|\\Delta\\|^2})$:\n\\begin{align*}\n\\ln M(s) \n& \\leq \n\\ln \\left( \\frac{\\exp(\\frac{s}{8} \\|\\Delta\\|^4 - \\frac{s}{2}\\|\\Delta\\|^2)}{\\sqrt{1 + s \\|\\Delta\\|^2}} \\right) \n\\\\ & = \n\\frac{s}{8}  \\|\\Delta\\|^4 - \\frac{s}{2}\\|\\Delta\\|^2 - \\frac{1}{2} \\ln (1 + s \\|\\Delta\\|^2) \n\\\\ &\\leq \n\\frac{s}{8} \\|\\Delta\\|^4 - \\frac{s}{2}\\|\\Delta\\|^2 - \\frac{1}{2} (s \\|\\Delta\\|^2 - \\frac{1}{2} s^2 \\|\\Delta\\|^4) \n\\\\ & =\n\\frac{s^2}{4} \\|\\Delta\\|^4 - \\frac{s}{8}\\|\\Delta\\|^4  - s \\|\\Delta\\|^2\n\\\\ &\\leq\n\\frac{s^2}{4} \\|\\Delta\\|^4. \\qedhere\n\\end{align*}\n\\end{proof}\n\n\nWe next combine this result with an existing bound on sub-exponential random variables~\\cite{opac-b1132854}(Lemma 4.1 in Chapter 1).  \nLet $X$ be a random variable, and let $M(s)$ be the moment generating function of $X - \\s{\\textbf{E}}[X]$. Let $\\bar{X}_t := \\frac{1}{t}\\sum_{i=1}^t X_i $ where $X_1,...,X_t$ are i.i.d. samples of $X$.\nIf $\\ln M(s) \\leq \\frac{s^2 p}{2}$ for all $s \\in [0,\\frac{1}{q})$, then\n\\begin{equation}\n P( | \\bar{X}_t - \\s{\\textbf{E}}[X] | \\geq \\varepsilon \\s{\\textbf{E}}[X]) \\leq\n 2 \\exp \\left(-\\min \\left(t \\frac{\\varepsilon^2 \\s{\\textbf{E}}[X]^2}{2 p}, t \\frac{\\varepsilon \\s{\\textbf{E}}[X]}{2 q}\\right) \\right).  \n \\label{eq:MGFHoeffding}\n\\end{equation}\n\n\n\n\\begin{lemma}\nIf $\\|x - y\\| \\leq \\sigma$, $m = \\Omega(\\varepsilon^{-2} \\log \\frac{1}{\\delta})$, then $\\frac{D_{\\hat{K}}(x,y)}{D_K(x,y)} \\in [1 - \\varepsilon, 1 + \\varepsilon]$ with probability at least $1 - \\delta$.\n\\end{lemma}\n\\begin{proof}\nRecall that $\\langle \\hat f_i(x), \\hat f_i(y) \\rangle = \\cos(\\langle \\omega_i, (x-y) \\rangle)$ and $(1\/2) D_{\\hat K}(x,y)^2 = \\frac{1}{t} \\sum_{i=1}^t (1- \\cos(\\langle \\omega_i (x-y) \\rangle))$.  \nThen define random variable $X_i = (1- \\cos(\\langle \\omega_i (x-y) \\rangle))$, \nand $X = \\frac{1}{t} \\sum_{i=1}^t X_i$.  \nSince $\\s{\\textbf{E}}[X] = \\s{\\textbf{E}}[D_{\\hat K}(x,y)^2] = D_K(x,y)^2$, then $\\s{\\textbf{E}}[X_i] = 1-\\exp(-\\frac{1}{2}\\|\\Delta\\|^2)$.  \n\nFor $M(s)$ the moment generating function of $X_i - \\s{\\textbf{E}}[X_i]$, by Lemma \\ref{lemma:BoundMGF} we have $\\ln (M(s)) \\leq \\frac{1}{2} s^2 p$ for $p = \\frac{1}{2} \\|\\Delta\\|^4$ for $s \\in [0, \\frac{1}{q}]$ with $q = 2 \\|\\Delta\\|^2$.  \nAlso recall by equation (\\ref{eq:near-linear}) we have for any $x,y \\in \\ensuremath{\\mathbb{R}}^d$ with $\\|x-y\\| \\leq \\sigma$, that  $0.86 \\leq \\frac{D_K(x,y)}{\\|\\Delta\\|} \\leq 1$.  \n\nPlugging these values into equation (\\ref{eq:MGFHoeffding}) with $t = \\frac{6}{\\varepsilon^2}  \\frac{\\|\\Delta\\|^4}{D_K(x,y)^4} \\ln(2\/\\delta) = O(\\frac{1}{\\varepsilon^2} \\log \\frac{1}{\\delta})$, we obtain that \n\\begin{align*}\n\\Pr[ | D_{\\hat K}(x,y) - D_{K}(x,y) | \\geq \\varepsilon D_{K}(x,y)]\n& =\n\\Pr[ | X - \\s{\\textbf{E}}[X] | \\geq \\varepsilon \\s{\\textbf{E}}[X]]\n\\\\ \\leq &\n2 \\exp\\left( - \\min \\left( t \\frac{\\varepsilon^2 \\s{\\textbf{E}}[X]^2}{2p}, t \\frac{\\varepsilon \\s{\\textbf{E}}[X]}{2 q} \\right) \\right)\n\\\\ = &\n2 \\exp \\left(- \\min \\left( t \\frac{\\varepsilon^2 \\s{\\textbf{E}}[X]^2}{\\|\\Delta\\|^4}, t \\frac{\\varepsilon \\s{\\textbf{E}}[X]}{4 \\|\\Delta\\|^2} \\right) \\right).  \n\\\\ = &\n2 \\exp \\left(- \\min \\left( 6, \\frac{3}{2\\varepsilon} \\frac{\\|\\Delta\\|^2}{D_K(x,y)^2} \\right) \\ln \\frac{2}{\\delta} \\right)\n\\\\ \\leq &\n2 \\exp \\left(- \\min \\left( 6, \\frac{1}{\\varepsilon}\\right) \\ln \\frac{2}{\\delta} \\right) \n\\leq \\delta.  \\qedhere \n\\end{align*}\n\\end{proof}\n\nTogether with Lemma \\ref{lem:bigD-rel} (for $\\|x-y\\| \\geq \\sigma$), we apply a union bound over all $n \\choose 2$ pairs vectors from a set of $n$ vectors.\n\n\\begin{theorem}\n\\label{cor:eps-2logn-UB}\nFor any set $X \\subset \\ensuremath{\\mathbb{R}}^d$ of size $n$, then $m = \\Omega(\\frac{1}{\\varepsilon^2} \\log n)$ dimensions are sufficient so $\\frac{D_{\\hat K}(x,y)}{D_K(x,y)} \\in [1-\\varepsilon,1+\\varepsilon]$ with high probability (e.g., at least $1-1\/n$).  \n\\end{theorem}\n\n\n\n\n\n\\section{Relative Error Bounds for Low Dimensions and Diameter}\n\\label{sec:lowD}\n\n\nA common approach in subspace embeddings replaces $n$ with the size of a sufficiently fine net.  Given a smoothness condition, once the error is bound on the net points, the guarantee is extended to the `gaps' in between. To bound the size of the gaps, we derive the Lipschitz constant of $D_{\\hat{K}}(\\cdot)^2$, with respect to the vector $\\Delta$ (not individual points in $\\ensuremath{\\mathbb{R}}^d$).  \n\nAs opposed to previous work regarding the norms of the Euclidean space, the Gaussian kernel distance is non-linear; nonetheless we observe that it is near linear close to $0$, and make use of a special construct that allows us to take advantage of this insight.\n\n\\begin{lemma}\n For any $\\Delta \\in \\mathbb{R}^d$, $|\\nabla D_{\\hat{K}}(\\Delta)^2| \\leq \\sum^t_{i=1} \\sqrt{d} \\|\\omega^2_{i,\\Delta}\\|\\| \\Delta \\|$. \n\\end{lemma}\n\n\\begin{proof}\n We denote by $\\omega^{(j)}_{i,\\Delta}$ the $j$-th coordinate of $\\omega_{i,\\Delta}$.\n \\begin{align*}\n  \\left|\\nabla D_{\\hat{K}}(\\Delta)^2\\right|\n  &= \n  \\left| \\sum^t_{i=1}\\sum^d_{j=1} \\omega^{(j)}_{i,\\Delta} \\sin (\\omega_{i,\\Delta} \\cdot \\Delta)\\right| \n \n  \\leq  \n  \\sum^t_{i=1}\\sum^d_{j=1} |\\omega^{(j)}_{i,\\Delta}| \\left| \\sin (\\omega_{i,\\Delta} \\cdot \\Delta)\\right|\n  \\\\&\\leq  \n  \\sum^t_{i=1} \\|\\omega_{i,\\Delta}\\|_1 |\\omega_{i,\\Delta} \\cdot \\Delta|\n \n  \\leq  \n  \\sum^t_{i=1} \\| \\omega_{i,\\Delta} \\|_1 \\| \\omega_{i,\\Delta} \\| \\| \\Delta \\|\n  \\leq  \n  \\sum^t_{i=1} \\sqrt{d} \\| \\omega_{i,\\Delta} \\|^2 \\| \\Delta \\|\n \\end{align*} \n\\end{proof}\n\n\n\\begin{corollary}\nFor any $c \\geq 0$, over the region $\\| \\Delta \\| \\leq c$, the Lipschitz constant of $D_{\\hat{K}}(\\Delta)^2$ is bounded above by $O(c \\cdot t \\sqrt{d} \\log n)$ with probability at least $1 - O(\\frac{1}{n})$.\n\\label{corollary:lipschitz}\n\\end{corollary}\n\\begin{proof}\nWe can bound $\\| \\omega_{i,\\Delta} \\|^2 \\leq O(\\log n)$ with probability at least $1 - \\frac{1}{n^q}$ for any large fixed $q > 0$. \nSo the gradient is bounded by \n$\\sqrt{d} \\|\\Delta\\| \\sum_{i=1}^t \\|\\omega_{i,\\Delta}\\|^2\n\\leq\n\\sqrt{d} \\|\\Delta\\| t \\cdot O(\\log n)\n\\leq O(c \\cdot t \\sqrt{d} \\log n)$, \nwhich also bounds the Lipschitz constant.\n\\end{proof}\n\nIn case $c = \\frac{\\sqrt{\\varepsilon}}{\\log n}$, the Lipschitz constant is $O(t \\sqrt{\\varepsilon d})$.  \nWe then have enough ingredients for the following main result. Intuitively, what separates typical net arguments from ours is the scaling $\\lambda$ in Lemma~\\ref{lem:SpikePreservation}; our `net' contains, which we call a $\\lambda$-urchin, a set of line segments extending from the origin, in addition to discrete points.  \n\n\\begin{lemma}\nIf $t = \\Omega(\\frac{1}{\\varepsilon^2}d \\log \\left( \\frac{d}{\\varepsilon}\\frac{1}{\\delta} \\right))$, then with probability at least $1 - \\delta$, for all $\\Delta$ such that $\\|\\Delta\\| \\leq \\frac{\\sqrt{\\varepsilon}}{\\log n}$, then $\\frac{D_{\\hat{K}}(\\Delta)^2}{D_K(\\Delta)^2} \\in [1 - \\varepsilon, 1 + \\varepsilon]$.\n\\end{lemma}\n\\begin{proof}\nThe proof will first consider distances $\\Delta$ such that $\\{ \\Delta : \\|\\Delta\\| = \\frac{\\sqrt{\\varepsilon}}{\\log n} \\}$, \nand then generalize to smaller distances using Lemma \\ref{lem:SpikePreservation} and a construction we call a $\\lambda$-urchin.  \n\n\\noindent\\textbf{Fixed distance case: }  \nConsider two points $\\Delta_1, \\Delta_2$ from the surface $\\{ \\Delta : \\|\\Delta\\| = \\frac{\\sqrt{\\varepsilon}}{\\log n} \\}$.\nIf $\\|\\Delta_1 - \\Delta_2\\| \\leq \\frac{\\varepsilon^{1.5}}{t \\sqrt{d} \\log^2 n}$ then Corollary~\\ref{corollary:lipschitz} implies\n\\begin{align*}\n \\left| D_{\\hat{K}}(\\Delta_1)^2 - D_{\\hat{K}}(\\Delta_2)^2 \\right|\n& \\leq  \nO(t \\sqrt{\\varepsilon d}) \\cdot \\|\\Delta_1 - \\Delta_2\\|\n\\leq \nO(t\\sqrt{\\varepsilon d}) \\cdot \\frac{\\varepsilon^{1.5}}{t \\sqrt{d} \\log^2 n}\n\\\\ & =\nO\\left(\\varepsilon \\cdot \\left(\\frac{\\sqrt{\\varepsilon}}{\\log n}\\right)^2 \\right)\n=\nO(\\varepsilon) \\cdot D_K(\\Delta_1)^2.  \n\\end{align*}\n\n\nNow let $\\Gamma_\\gamma$ be a $\\gamma$-net  over \n$\\{ \\Delta : \\|\\Delta\\| = \\frac{\\sqrt{\\varepsilon}}{\\log n} \\}$ \nwhere \n$\\gamma \\leq \\frac{\\varepsilon^{1.5}}{t\\sqrt{d}\\log^2 n}$. \nFor any \n$\\Delta_1 \\in \\{\\Delta : \\|\\Delta\\| = \\frac{\\sqrt{\\varepsilon}}{\\log n} \\}$, \nthere exists $\\Delta_2 \\in \\Gamma_\\gamma$ such that \n$\\|\\Delta_1 - \\Delta_2\\| \\leq \\gamma$. \nThen the above implies\n\\begin{equation}\n(1 - O(\\varepsilon)) D_{\\hat{K}}(\\Delta_2)^2 \n\\leq \nD_{\\hat{K}}(\\Delta_1)^2 \n\\leq \n(1 + O(\\varepsilon))D_{\\hat{K}}(\\Delta_2)^2.  \n\\label{eq:netgaperror}\n\\end{equation}\n\nBy the triangle inequality, equation (\\ref{eq:near-linear}), and  $t \\sqrt{d} \\log n > 1$, we have  \n\\begin{equation}  \\label{eq:trineterror}\n|D_K(\\Delta_1) - D_K(\\Delta_2)|\n\\leq \nD_K(\\Delta_1,\\Delta_2) \n\\leq \n\\|\\Delta_1 - \\Delta_2\\| \n\\leq  \n\\gamma  \n\\leq\n\\varepsilon \\cdot \\frac{\\sqrt{\\varepsilon}}{\\log n} \\cdot \\frac{1}{t \\sqrt{d} \\log n} \n\\leq \n\\varepsilon \\cdot O(D_K(\\Delta_1)).  \n\\end{equation}\n\n\nWe will choose $t = \\Omega(\\frac{1}{\\varepsilon^2} \\log |\\Gamma_\\gamma|)$ such that the following holds over $\\Gamma_\\gamma$ with high probability\n\\begin{equation}\n(1 - O(\\varepsilon)) D_{K}(\\Delta_2)^2 \\leq D_{\\hat{K}}(\\Delta_2)^2 \\leq (1 + O(\\varepsilon))D_{K}(\\Delta_2)^2.  \n\\label{eq:neterror}\n\\end{equation}\n\n\nThese equations (\\ref{eq:netgaperror}), (\\ref{eq:trineterror}), and (\\ref{eq:neterror}) show, respectively that the ratios\n$\\frac{D_{\\hat K}(\\Delta_1)}{D_{\\hat K}(\\Delta_2)}$,\n$\\frac{D_{\\hat K}(\\Delta_2)}{D_{K}(\\Delta_2)}$, and\n$\\frac{D_{K}(\\Delta_2)}{D_{K}(\\Delta_1)}$\nare all in $[1+O(\\varepsilon), 1-O(\\varepsilon)]$; hence we can conclude\n\\begin{equation}\n\\label{eq:error-on-sphere}\n|D_K(\\Delta_1) - D_{\\hat{K}}(\\Delta_1)| \\leq O(\\varepsilon) \\cdot D_{K}(\\Delta_1).  \n\\end{equation}\nWhich are in turn $1 \\pm O(\\varepsilon)$ relative error bounds for the kernel distance, over $\\{ \\Delta : \\|\\Delta\\| = \\frac{\\sqrt{\\varepsilon}}{\\log n} \\}$.\n\n\\noindent\\textbf{All distances case:  }\nFor the region $\\{ \\Delta : \\|\\Delta\\| < \\frac{\\sqrt{\\varepsilon}}{\\log n} \\}$, consider again $\\Gamma_\\gamma$. For each net point $p \\in \\Gamma_\\gamma$ we draw a line segment from $p$ to the origin, producing the set of line segments $\\bar{\\Gamma}_\\gamma$, that we call the \\emph{$\\gamma$-urchin}.\nBy Lemma~\\ref{lem:SpikePreservation}, and $t = \\Omega(\\frac{1}{\\varepsilon^2} \\log |\\Gamma_\\gamma|)$, we have relative error bounds for the Gaussian kernel distance over the $\\gamma$-urchin.\n\nNow for any $\\lambda \\in (0,1)$, consider the intersection $\\{ \\Delta : \\|\\Delta\\| = \\lambda \\frac{\\sqrt{\\varepsilon}}{\\log n} \\} \\cap \\bar{\\Gamma}_\\gamma$. We see that the $\\gamma$-urchin induces a net over $\\{ \\Delta : \\|\\Delta\\|= \\lambda \\frac{\\sqrt{\\varepsilon}}{\\log n}\\}$. Due to scaling we can see that, in fact, it is a $(\\lambda \\gamma)$-net.   \nSo the distance between any point in $\\{ \\Delta : \\|\\Delta\\| = \\lambda \\frac{\\sqrt{\\varepsilon}}{\\log n}\\}$ and the closest net point is bounded above by  \n$\\frac{ \\lambda \\cdot \\varepsilon^{1.5}}{t\\sqrt{d}\\log^2 n}$. \nFrom Corollary \\ref{corollary:lipschitz}, the Lipschitz constant is now $O(t \\lambda  \\sqrt{\\varepsilon d})$.\n\nBy arguments similar to those leading to (\\ref{eq:error-on-sphere}) we obtain, for any $\\Delta_1 \\in \\{ \\Delta : \\|\\Delta\\| = \\lambda \\frac{\\sqrt{\\varepsilon}}{\\log n}\\}$\n\\begin{equation}\n|D_K(\\Delta_1) - D_{\\hat{K}}(\\Delta_1)| \\leq O(\\varepsilon) \\cdot \\lambda \\cdot \\frac{\\sqrt{\\varepsilon}}{\\log n} \\leq O(\\varepsilon) \\cdot D_{K}(\\Delta_1).    \n\\label{eq:errorlambdasphere}\n\\end{equation}\nSince this holds for all $\\lambda \\in [0,1]$, we obtain relative error bounds over $\\{ \\Delta : \\frac{\\|\\Delta\\|}{\\sigma} \\leq \\frac{\\sqrt{\\varepsilon}}{\\log n}\\}$.  \n\nThe size of $\\Gamma_\\gamma$ is bounded above by $O(\\left(\\frac{t\\sqrt{d}\\log n}{ \\varepsilon}\\right)^d)$. It is sufficient to set $\\log n = O(d \\log(d\/\\varepsilon))$ and thus $t = O(\\frac{1}{\\varepsilon^2} d \\log (\\frac{d}{ \\varepsilon}\\frac{1}{\\delta}))$ so that relative error holds over the $\\gamma$-net and the $\\gamma$-urchin simultaneously, which imply (\\ref{eq:errorlambdasphere}) and (\\ref{eq:error-on-sphere}), with probability at least $1 - \\delta$.\n\\end{proof}\n\n\n\n\\begin{corollary} \\label{cor:rel-small-dist}\n If $t = \\Omega(\\frac{1}{\\varepsilon^2}d \\log (\\frac{d}{\\varepsilon} \\frac{1}{\\delta}))$, then for all $\\Delta$ such that $\\|\\Delta\\| \\leq 1$, $\\frac{D_{\\hat{K}}(\\Delta)}{D_K(\\Delta)} \\in [1 - \\varepsilon, 1 + \\varepsilon]$ with probability at least $1 - \\delta$.\n\\end{corollary}\n\\begin{proof}\nConsider the region $1 \\geq \\|\\Delta\\| > \\frac{\\sqrt{\\varepsilon}}{\\log n}$. The Lipschitz constant is bounded above by $O(t \\sqrt{d} \\log n)$ by Corollary \\ref{corollary:lipschitz}, so we only need a $\\gamma$-net where $\\gamma \\leq \\frac{\\varepsilon^2}{t\\sqrt{d}\\log n}$ to give relative error by standard net arguments. The size of this net is at most $\\left( \\frac{t\\sqrt{d}\\log n}{ \\varepsilon^2} \\right)^d$, so again it suffices to set $\\log n = O(d \\log (\\frac{d}{\\varepsilon}))$ and $t = O(\\frac{1}{\\varepsilon^2} d \\log(\\frac{d}{\\varepsilon} \\frac{1}{\\delta}) )$ for our embeddings as above.\n\\end{proof}\nCombined with Lemma \\ref{lem:bigD-rel} for $\\|\\Delta\\| > 1$ we obtain:\n\n\n\n\n\\begin{theorem}\n If $t = \\Omega\\left(\\frac{d}{\\varepsilon^2} \\log \\left( \\frac{d}{\\varepsilon} \\frac{\\mathcal{M}}{\\delta}\\right)\\right)$, then for any $\\mathcal{M} \\geq 0$, $\\frac{D_{\\hat{K}}(x,y)^2}{D_K(x,y)^2} \\in [1 - \\varepsilon, 1 + \\varepsilon]$ holds for all $x,y \\in \\ensuremath{\\mathbb{R}}^d$ such that $\\|x - y\\|\/\\sigma\\leq \\mathcal{M}$ with probability at least $1 - \\delta$.\n\\end{theorem}\n\\begin{proof}\nSet \n$t = \\Omega(\\frac{1}{\\varepsilon^2}d \\log (\\frac{d}{\\varepsilon}  \\frac{1}{\\delta})) \n+ \n\\Omega(\\frac{1}{\\varepsilon^2}d \\log (\\frac{d}{\\varepsilon} \\frac{\\mathcal{M}}{\\delta})) \n= \n\\Omega\\left(\\frac{d}{\\varepsilon^2} \\log \\left( \\frac{d}{\\varepsilon} \\frac{\\mathcal{M}}{\\delta}\\right)\\right)$ \nto account for both cases $\\|\\Delta\\| = \\frac{\\|x-y\\|}{\\sigma} \\leq 1$ and $1 \\leq \\frac{\\|x-y\\|}{\\sigma} \\leq \\mathcal{M}$, respectively.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Lower Bounds for Low Dimensions}\n\\label{sec:LowDimLB}\n\nWhen is $n$ is unbounded, a recent paper~\\cite{SS15} implies that, even for small $d$, $D_{\\hat K}$ cannot $(1 + \\varepsilon)$-approximate $D_K$ unless $\\mathcal{M}$ is bounded.\nHere we provide an explicit and \\emph{general} lower bound depending on $\\mathcal{M}$ and $d$ that matches the our upper bound up to a $O( \\log \\frac{1}{\\varepsilon})$ factor.  \n\nFirst we need the following general result (\\cite{Alon:2003:PRE:2651487.2651714} Theorem 9.3) related to embedding to $\\ell_2$.\nLet $B$ be an $n \\times n$ real matrix with $b_{i,i}=1$ for all $i$ and $|b_{i,j}| \\leq \\varepsilon$ for all $i \\neq j$. If the rank of $B$ is $r$, and $\\frac{1}{\\sqrt{n}} < \\varepsilon < 1\/2$, then\n$\nr \\geq \\Omega(\\frac{1}{\\varepsilon^2 \\log (1\/\\varepsilon)}  \\log n).  \n$\nGeometrically, $r$ is the minimum number of dimensions that can contain a set of $n$ near-orthogonal vectors. Indeed, any set $S$ of $n$ near-orthogonal vectors can be rotated to form the rows of a matrix of the form of $B$, and the rank is then the lowest number of dimensions that contain $S$.\n\n\\begin{lemma}\t\\label{lem:subspaceKernelLB}\nGiven $\\mathcal{M} \\geq 0$, let $B_\\mathcal{M}(0)$ be the ball in $\\mathbb{R}^d$ centered at the origin with radius $\\mathcal{M}$. Let $h : \\mathbb{R}^d \\rightarrow \\mathbb{R}^t$ be a mapping such that for any $x \\neq y \\in B_\\mathcal{M}(0)$  we have $|K(x,y) - h(x)\\cdot h(y)| \\leq \\varepsilon \\leq \\frac{1}{4}$.\nThen with sufficiently large $\\mathcal{M}$, $t = \\Omega(\\frac{d}{\\varepsilon^2 \\log  (1\/\\varepsilon)} \\log (\\frac{\\mathcal{M}}{ \\log (1\/\\varepsilon)}))$.\n\\end{lemma}\n\\begin{proof}\nConsider a subset $S \\subset \\mathbb{R}^d$ in $B_{\\mathcal{M}}(0)$ so for all $x,y \\in S$, with $x\\neq y$, we have $\\|x-y\\| \\geq  \\sigma \\sqrt{2 \\log \\frac{1}{\\varepsilon}}$. \nThen for any $x,y \\in S$, $K(x,y)=\\exp(-\\frac{\\|x - y\\|^2}{2\\sigma^2}) \\leq \\varepsilon$.  In particular, define $S$ as the intersection of $B_{\\mathcal{M}}(0)$ with an orthogonal grid of side length $\\sigma \\sqrt{2 \\log(1\/\\varepsilon)}$; it has size $\\Omega\\left( \\left(\\frac{\\mathcal{M}}{\\log (1\/\\varepsilon)} \\right)^d\\right)$.\n\t\nFor any $x,y \\in S$, $|h(x) \\cdot h(y)| \\leq 2\\varepsilon$, and also $|\\{h(s) \\mid s \\in S\\}| = |S|$. Then \\cite{Alon:2003:PRE:2651487.2651714} Theorem 9.3 implies the dimension of $h$ must be $t = \\Omega(\\frac{1}{\\varepsilon^2 \\log (1\/\\varepsilon)}  \\log |S|) = \\Omega(\\frac{d}{\\varepsilon^2 \\log  (1\/\\varepsilon)} \\log (\\frac{\\mathcal{M}}{\\log (1\/\\varepsilon)}))$.\n\\end{proof}\n\n\n\n\n\\begin{theorem}  \t\\label{thm:subspaceLB}\nGiven $\\mathcal{M} \\geq 0$, let $B_\\mathcal{M}(0)$ be the ball in $\\ensuremath{\\mathbb{R}}^d$ centered at the origin with radius $\\mathcal{M}$. Let $h : \\ensuremath{\\mathbb{R}}^d \\rightarrow \\mathbb{R}^t$ be a mapping such that for any $x,y \\in B_\\mathcal{M}(0)$  we have $1-\\varepsilon \\leq \\frac{D_K(x,y)}{\\|h(x) - h(y)\\|}  \\leq 1+\\varepsilon$ with $\\varepsilon \\leq \\frac{1}{4}$.  Restrict that for any $x \\in \\ensuremath{\\mathbb{R}}^d$ that $\\|h(x)\\| = 1$.  If $\\mathcal{M}$ is sufficiently large, $t = \\Omega(\\frac{d}{\\varepsilon^2 \\log  (1\/\\varepsilon)} \\log (\\frac{\\mathcal{M}}{ \\log (1\/\\varepsilon)}))$.\n\\end{theorem}\n\\begin{proof}\nConsider a set (as in proof of Lemma \\ref{lem:subspaceKernelLB}) $S \\subset B_{\\mathcal{M}}(0)$.  \nIf for all $x,y \\in S$ we have $1-\\varepsilon \\leq \\frac{D_K(x,y)}{\\|h(x) - h(y)\\|}  \\leq 1+\\varepsilon$, then it implies \n\\[\n|D_K(x,y)^2 - \\|h(x) - h(y)\\|^2| \\leq \\Theta(\\varepsilon) D_K(x,y)^2 \\leq \\Theta(\\varepsilon),\n\\] \nsince $D_K(x,y) < 2$.  \nExpanding $D_K(x,y)^2 = 2 - 2 K(x,y)$ and $\\|h(x) - h(y)\\|^2 = 2 - 2 \\langle h(x), h(y) \\rangle$ implies that \n$|K(x,y) - \\langle h(x), h(y) \\rangle| \\leq \\Theta(\\varepsilon)$ as well.  However, Lemma \\ref{lem:subspaceKernelLB} implies that for sufficiently small $\\varepsilon$ (adjusting the constant in $\\Theta(\\varepsilon)$) that we require the $t = \\Omega(\\frac{d}{\\varepsilon^2 \\log  (1\/\\varepsilon)} \\log (\\frac{\\mathcal{M}}{ \\log (1\/\\varepsilon)}))$.  \n\\end{proof}\n\n\n\n\nThis implies the impossibility of fully embedding into $\\ell_2$ the Gaussian kernel distance over the \\emph{entire} $\\ensuremath{\\mathbb{R}}^d$, i.e. for an infinite number of points, answering a question raised by ~\\cite{SS15}.\nThis argument can also extend to show a dependency on $d \\log\\mathcal{M}$ is inevitable when we do not have a bound on $n$.  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\n\\begin{minipage}{0.51\\textwidth}\n  \\centering\n\\includegraphics[width=\\textwidth]{usps.eps}  \n\\end{minipage}\n\\begin{minipage}{0.43\\textwidth}\n  \\centering\n\\begin{tabular}{cr||r|r|r}\n\t\t\\hline\n\t\t&    & $\\sigma=4$ &$\\sigma=8$ & $\\sigma=16$\\\\\n\t\t\\hline\n\\multicolumn{2}{c||}{Baseline} & 1667.1 & 882.5 & 206.1\\\\\n\t\t\\hline\n\t\t& 50  & 897.6  & 489.2 & 96.9 \\\\\n\t\t& 100 & 1257.6 & 666.5 & 152.2 \\\\\n\t$t$ & 200 & 1453.6 & 776.4 & 178.7 \\\\\n\t\t& 400 & 1554.7 & 831.5 & 189.2 \\\\\n\t\t& 800 & 1606.8 & 857.3 & 197.6 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{minipage}\n\n  \\caption{Relative error $|\\frac{\\hat{R}k}{R_k}-1|$ in \\% , against $t$, with $n=2000$, $k=40$ and different bandwidths. Relative error is roughly stable across different values of $\\sigma$, and consistently reduced by increasing $t$. \\label{tab:1}}\n\\end{figure}\n\n\n\n\n\\section{Empirical Demonstration of Relative Error}\n\\label{sec:exp}\n\nWe demonstrate that relative error actually results from the $\\hat \\phi$ kernel embeddings in two ways.  First we demonstrate relative error bounds for kernel PCA.  Second we show this explicitly for pairwise distances in the embedding.  \n\n\n\\subsection{Relative Error for Kernel PCA}\n\nWe consider two ways of running kernel PCA on the USPS data.  By default we use the first $n=2000$ data points in $\\ensuremath{\\mathbb{R}}^d$ for $d=256$, the first $n\/10$ data points of each digit.  \nIn the first way, we create the $n \\times n$ (centered) gram matrix $G$ of all inner products, and then use the top $k$ eigenvectors to describe the best subspace of RKHS to represent the data; this is treated as a baseline.  Second we embed each point into $\\ensuremath{\\mathbb{R}}^m$ using $\\ensuremath{\\hat{\\phi}}$, generating an $n \\times m$ matrix $Q$ (after centering).  The top $k$ right singular values $V_k$ of $Q$ describe the kernel PCA subspace.\n\nError in PCA is typically measured as the sum of squared residuals, that is for each point $q \\in Q \\subset \\ensuremath{\\mathbb{R}}^m$, its projection onto $V_k$ is $V_k^T V_k q$, and its residual is $r_q = \\|q - V_k^T V_k q\\|^2$.  Thus $r_q$ is precisely the squared kernel distance between $q$ and its projection.  And then the full error is $\\hat R_k = \\|Q - V_k^T V_k Q\\|_F^2 = \\sum_{q \\in Q} \\|q - V_k^T V_k q\\|^2$.  \nFor the non-approximate case, it can be calculated as the sum of eigenvalues in the tail $R_k = \\sum_{i=k+1}^n \\lambda_{i}$.  \n\nGiven $R_k$ and $\\hat R_k$ we can measure the relative error as $\\hat R_k \/ R_k$.  Our analysis indicates this should be in $[1-\\varepsilon, 1+\\varepsilon]$ using roughly $t = C\/\\varepsilon^2$ features, where $C$ depends on $n$ or $d \\log \\mathcal{M}$.  To isolate $\\varepsilon$ we calculate $|\\frac{\\hat R_k}{R_k} - 1|$ averaged over $10$ trials in the randomness in $\\ensuremath{\\hat{\\phi}}$.  \nThis is shown in Figure \\ref{tab:1} using $k = 40$, with $\\sigma \\in \\{4,8,16\\}$ and varying $t \\in \\{50, 100, 200, 400, 800\\}$.  \nWe observe that our measured error decreases quadratically in $t$ as expected.  Moreover, this rate is stable as a function of $\\sigma$ as would be expected where the correct way to quantify error is the relative error we measure.  \n\n\n\n\n\n\n\n\n\n\\paragraph{Pairwise Demonstrations of Relative Error}\n\nHere we provide simulations that confirm our theoretical findings.\nWe randomly generate pairs of points $(x_1,y_1)$ \\ldots $(x_n,y_n)$ with varying $\\ell_2$ distance $\\|x_i - y_i\\|$; in particular, $x_i$ is a random point in a ball or radius $500$ and $y_i$ is generated to be a random point in the sphere $\\|x-y\\| = r_i$ where $r_1, ..., r_n$ follow a geometric distribution, ranging from approximately $10^{-4}$ to $10^4$.\n\nIn Figure~\\ref{fig:VaryT+Dist}(left), for different values of $t$ (the number of features) we generate a fresh sequence of $2000$ random pairs, and record the maximum relative error $\\varepsilon_{\\mathrm{max}} = \\max_i \\frac{D_K(x_i,y_i)}{\\|\\phi(x_i) - \\phi(y_i)\\|}$. The graph shows that $t$ is roughly proportional to $\\varepsilon^{-2}_{\\mathrm{max}}$.\n\nIn Figure~\\ref{fig:VaryT+Dist}(right), we examine the relative errors for all the random pairs at a wide range of $\\ell_2$ norms, for $t=100$ and $t=1000$. A slight change in the error profile occurs within $\\|x_i-y_i\\|\/\\sigma \\in [10^0,10^1]$, coinciding with the separation of cases $\\|x-y\\| \\leq \\sigma$ and $\\|x-y\\| > \\sigma$ i.e. whether $\\frac{\\|x-y\\|}{\\sigma} = \\Theta(1)$ in the analyses.\n\nIn either case, the relative error is bounded by a small constant value, even when $\\|x_i-y_i\\|$ is several magnitudes smaller than $1$, demonstrating that the extremely high concentration of the RFF for very small $\\|x_i-y_i\\|$ results in relative error approximation for the Gaussian kernel distance.\n\n\n\\begin{figure}  \n\\includegraphics[width=0.45\\textwidth]{varyt.eps}  \n\\hspace{0.09 \\textwidth}\n\\includegraphics[width=0.45\\textwidth]{varydist.eps} \n\n\n\\caption{\\label{fig:VaryT+Dist}  \n(left) Inverse squared relative errors.  \n(right) Relative errors with varying distance.}\n\\end{figure}\n\n\n\\paragraph{Conclusion:}  \nWe demonstrate theoretically and empirically tight relative error for kernel distance using random Fourier features, indicating tighter approximations for important learning applications.  \n\n\n\n\\newpage\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\n\n\nThe idea of the collapse as a trend to catastrophic shrinkage of a\nself-attracting system has proved its relevance in many branches of\nnonlinear physics \\cite{Sulem1999,Fibich}.\nThe realization of the ultracold atomic matter has greatly increased the variety of possible\nnonlinear phenomena \\cite{Carr2009}. In atomic Bose-Einstein condensates (BECs), \nthe self-attraction driving collapse either occurs naturally, or\ncan be achieved by means of the engineered Feshbach resonance \\cite{Cornish2000}. \nRecently developed techniques which\nmake it possible to produce synthetic gauge fields \\cite{spielman2009,\nDalibard2011} and spin-orbit coupling (SOC) \\cite{wang2010, spielman2011,\nZhai2012,Spielman2013,Zhang2016} greatly expands the versatility of the\nself-interacting quantum matter and variety of collapse-related phenomena\n\\cite{Konotop2005,Dias2016}, as well as general properties of nonlinear two-component systems \\cite{WangKdv}.\nBy introducing coupled spin-mass-density\ndynamics with a spin-dependent velocity, SOC can affect the collapse \\cite%\n{Mardonov2015,Yu2017} and produce soliton-like stable states \\cite%\n{Sakaguchi2014,Sakaguchi2016}, which would be unstable, or would not exists,\nin the absence of SOC.\n\nIn effectively one-dimensional (1D) settings, the collapse is driven by\nthree-body attractive interactions, which correspond to the quintic\nnonlinearity in the respective Gross-Pitaevskii equations (GPEs), as the 1D\ncollapse cannot be caused by two-body attraction (represented by cubic terms\nin the GPEs) \\cite{Konler2002}. Indeed, an elementary estimate demonstrates\nthat the absolute values of the negative three-body energy, which drives the\nself-compression of the condensate, exceeds its kinetic energy, which\nimpedes the collapse, by a factor $\\sim N^{2},$ where $N$ is the\ncondensate's norm. Thus, the collapse may set in if $N$ is large enough.\n\nHere we consider effects of SOC and Zeeman splitting (ZS) on the collapse in\na 1D system with various forms of the quintic self-attraction \\cite%\n{Xi2016,Astrakharchik2005,Chiquillo2017}. In 1D settings these effects can\nbe presented in simple and transparent form as a competition between the\nvelocity caused by self-attraction, which is generated in the collapse\nprocess, and an anomalous SOC-induced spin-dependent velocity.\n\nThis paper is organized as follows. In Sec. \\ref{oneb}, the model of the\nspin-orbit and Zeeman-coupled BEC with three-body interactions is introduced. Then, in\nSec. \\ref{sec:no-soc} we address the dynamics of the system with different initial\nspin states in the absence of spin-related effects, by means of the\nvariational approximation. Section \\ref{Sec:with-soc} deals with combined\nSOC-ZS effects, by considering several different realizations of the model\ndemonstrating qualitatively different behaviors. The paper is concluded by\nSec. \\ref{end}.\n\n\\section{The model and self-interaction}\n\n\\label{oneb}\n\nWe consider a quasi-1D BEC with pseudospin 1\/2, subject to the action of the\nartificial SOC and ZS and extended along the $x$ direction. The\ncorresponding two-component wave function is%\n\\begin{equation}\n{\\bm\\psi }\\left( x,t\\right) =\\left(\n\\begin{array}{c}\nu(x,t) \\\\\nv(x,t)%\n\\end{array}%\n\\right) \\,,\n\\end{equation}%\n{with total norm}\n\\begin{equation}\nN\\equiv \\int_{-\\infty }^{+\\infty }\\left( |u|^{2}+|v|^{2}\\right) dx\\,.\n\\end{equation}%\nFor brevity, the explicit $(x,t)$ dependence is written only when it is\nnecessary. The combination of same- and cross-spin interaction energies is\ndefined as (cf. Refs. \\cite{Maim,Abdullaev2005})\n\\begin{equation}\nE_{g}=-\\frac{1}{3}\\int_{-\\infty }^{+\\infty }\\left[ g_{1}\\left( \\left\\vert\nu\\right\\vert ^{6}+\\left\\vert v\\right\\vert ^{6}\\right) +3g_{2}\\left\\vert\nu\\right\\vert ^{2}\\left\\vert v\\right\\vert ^{2}\\left( \\left\\vert u\\right\\vert\n^{2}+\\left\\vert v\\right\\vert ^{2}\\right) \\right] dx\\,,  \\label{Eg}\n\\end{equation}%\nwhere positive coupling constants $g_{1}$ and $g_{2}$ represent the\nthree-particle attraction in the same- and cross-spin channels, respectively.\n\nThe evolution of the system is governed by the equation\n\\begin{equation}\ni\\hbar \\frac{\\partial {\\bm\\psi }}{\\partial t}=H{\\bm\\psi }\\,,  \\label{evolution}\n\\end{equation}\nwith Hamiltonian%\n\\begin{equation}\nH=\\frac{p^{2}}{2M}+H_{\\mathrm{so}}+H_{Z}+H_{g}\\,,  \\label{eq:Hspin}\n\\end{equation}%\nwhere $p=-i\\hbar \\partial \/\\partial x$ is the momentum and $M$ the atomic\nmass, while the SOC and ZS terms are%\n\\begin{equation}\nH_{\\mathrm{so}}=\\frac{\\alpha }{\\hbar }p\\,\\sigma _{x},\\qquad H_{Z}=\\frac{\\Delta }{2}%\n\\sigma _{z}\\,.  \\label{eq:HsoHZ}\n\\end{equation}%\nHere $\\alpha $ and $\\Delta $ are the SOC and ZS strengths, respectively, with $%\n\\sigma _{x}$ and $\\sigma_{z}$ being the Pauli matrices.\nThe synthetic spin-orbit interaction in cold atoms can be generated by Raman\ncoupling schemes which simultaneously flip atomic pseudospin and transfer momentum  \\cite{spielman2009}.\nAs a result, a variety of synthetic fields can be engineered using highly coherent laser beams.\nThe synthetic Zeeman fields can be produced with the same experimental setup \\cite{Dalibard2011,Zhai2012,Spielman2013}. \nThe self-interaction term has the form\n\\begin{equation}\nH_{g}=\\left(\n\\begin{matrix}\n{\\delta E_{g}}\/{\\delta u^{\\ast }} & 0 \\\\\n0 & {\\delta E_{g}}\/{\\delta v^{\\ast }}%\n\\end{matrix}%\n\\right) ,  \\label{Hg}\n\\end{equation}%\nwith variations\n\\begin{eqnarray}\n&&\\frac{\\delta E_{g}}{\\delta u^{\\ast }}=-\\left[ g_{1}\\left\\vert u\\right\\vert\n^{4}+g_{2}\\left( 2\\left\\vert uv\\right\\vert ^{2}+\\left\\vert v\\right\\vert\n^{4}\\right) \\right] ,  \\notag \\\\\n&&  \\label{d\/d} \\\\\n&&\\frac{\\delta E_{g}}{\\delta v^{\\ast }}=-\\left[ g_{1}\\left\\vert v\\right\\vert\n^{4}+g_{2}\\left( 2\\left\\vert uv\\right\\vert ^{2}+\\left\\vert u\\right\\vert\n^{4}\\right) \\right] .  \\notag\n\\end{eqnarray}\n\n\nThe total energy of the system is\n\\begin{equation}\nE=E_{k}+E_{\\mathrm{so}}+E_{Z}+E_{g}\\,,\n\\end{equation}%\nwhere\n\\begin{equation}\nE_{k}=\\frac{1}{2M}\\int_{-\\infty }^{+\\infty }\\left( \\left\\vert \\frac{\\partial u}{%\n\\partial x}\\right\\vert ^{2}+\\left\\vert \\frac{\\partial v}{\\partial x}%\n\\right\\vert ^{2}\\right) dx  \\label{Ek}\n\\end{equation}%\nis the kinetic energy, with the SOC and ZS\nterms being\n\\begin{eqnarray}\nE_{\\mathrm{so}} &=&-i\\alpha \\int_{-\\infty }^{+\\infty }\\,{\\bm\\psi }^{\\dag\n}\\,\\sigma _{x}\\,\\frac{\\partial {\\bm\\psi }}{\\partial x}dx\\,,  \\label{SO} \\\\\nE_{Z} &=&\\frac{\\Delta }{2}\\langle \\sigma _{z}\\rangle \\,N\\,,\n\\label{eq:Zeeman1}\n\\end{eqnarray}%\nand\n\\begin{equation}\n\\langle \\sigma _{z}\\rangle \\equiv \\frac{1}{N}\\int_{-\\infty }^{+\\infty }\\,{\\bm%\n\\psi }^{\\dag }\\sigma _{z}\\,{\\bm\\psi }dx\\,.  \\label{sigma}\n\\end{equation}\n\nBy means of obvious rescaling, we set $\\hbar =M=1$ and thus cast Eqs. (\\ref%\n{evolution})-(\\ref{d\/d}) into a coupled GPE system:\n\\begin{eqnarray}\ni\\frac{\\partial }{\\partial t}u &=&\\left( -\\frac{1}{2}\\frac{\\partial ^{2}}{%\n\\partial x^{2}}+\\frac{\\Delta }{2}+\\frac{\\delta E_{g}}{\\delta u^{\\ast }}%\n\\right) u-i\\alpha \\frac{\\partial }{\\partial x}v\\,,  \\label{eq:eq1} \\\\\ni\\frac{\\partial }{\\partial t}v &=&\\left( -\\frac{1}{2}\\frac{\\partial ^{2}}{%\n\\partial x^{2}}-\\frac{\\Delta }{2}+\\frac{\\delta E_{g}}{\\delta v^{\\ast }}%\n\\right) v-i\\alpha \\frac{\\partial }{\\partial x}u\\,.  \\label{eq:eq2}\n\\end{eqnarray}%\nNote that the SOC\\ Hamiltonian introduces a spin-dependent velocity defined\nby commutator\n\\begin{equation}\nV=i\\left[ \\frac{p^{2}}{2}+H_{\\mathrm{so}},x\\right] =p+V_{\\mathrm{so}}\\,,\n\\label{eq:vso}\n\\end{equation}%\nwhere $V_{\\mathrm{so}}=\\alpha\\, \\sigma _{x}$ becomes a non-diagonal $2\\times 2$ matrix.\nAs shown below, in the course of the BEC evolution, this anomalous\nvelocity makes its density spatially split in two spin-projected species. Depending on the\n$E_{g}$ energy in Eq. \\eqref{Eg}, this splitting can either prevent the\ncollapse, suppressing the same-spin self-interaction due to the $g_{1}$ term,\nor drive the collapse by enhancing cross-spin coupling caused by the $g_{2}-$related contribution.\n\nIt is also relevant to mention that, in the absence of ZS ($\\Delta =0$),\nEqs. (\\ref{eq:eq1}) and (\\ref{eq:eq2}) admit a reduction to the single GPE\nwith the quintic self-attractive term and no SOC. Indeed, the substitution\nof\n\\begin{equation}\nv\\left( x,t\\right) =\\pm u\\left( x,t\\right) =U\\left( x,t\\right) \\exp \\left(\n\\mp i\\alpha x+i\\alpha ^{2}t\/2\\right)   \\label{+-}\n\\end{equation}%\ntransforms both equations into one:%\n\\begin{equation}\ni\\frac{\\partial }{\\partial t}U=\\left( -\\frac{1}{2}\\frac{\\partial ^{2}}{%\n\\partial x^{2}}-g|U|^{4}\\right) U,~g\\equiv g_{1}+3g_{2}.  \\label{U}\n\\end{equation}%\nIt is commonly known that Eq. (\\ref{U}) admits a family of \\textit{%\nTownes-soliton} solutions \\cite{Abdullaev2005} with arbitrary chemical\npotential $\\mu <0$:%\n\\begin{equation}\nU=e^{-i\\mu t}\\left( -\\frac{3\\mu }{g}\\right) ^{1\/4}\\sqrt{\\mathrm{sech}\\left( 2%\n\\sqrt{-2\\mu }x\\right) }\\,.  \\label{Townes}\n\\end{equation}%\nThe solution family is degenerate, as the soliton's norm does not depend on $%\n\\mu $:%\n\\begin{equation}\nN_{\\mathrm{Townes}}=\\frac{\\pi }{2}\\sqrt{\\frac{3}{2g}}\\,,  \\label{NT}\n\\end{equation}%\nand all the solitons are unstable against the onset of the collapse.\nHowever, the possibility of the reduction, based on Eq. (\\ref{+-}), to the\nsingle GPE (\\ref{U}) does not mean that the dynamics governed by Eqs. (\\ref%\n{eq:eq1}) and (\\ref{eq:eq2}) with $\\Delta =0$ reduces to that of the\nsingle-GPE in the generic situation. In particular, we will show that the SOC suppresses\nthe onset of the collapse in solutions stemming from the single-component input, given by Eq. (\\ref%\n{ux0vx0}), if the spin-orbit coupling $\\alpha$ is larger than a critical value $\\alpha_{\\mathrm{cr}},$ dependent\non the system parameters $N,g_{1},g_{2},\\Delta,$ and the initial conditions.\n\n\\section{Dynamics without spin-orbit coupling: the role of the spin\nstate}\n\n\\label{sec:no-soc}\nAs a starting point, we present here results for the collapse in the absence\nof SOC, taking the input as an eigenstate of $\\sigma _{z}$,\n\\begin{equation}\nu(x,0)=A(0)\\exp \\left( -\\frac{x^{2}}{2a^{2}(0)}\\right) ,\\quad v(x,0)=0,\n\\label{ux0vx0}\n\\end{equation}%\nwhere $A(0)$ and $a(0)$ are the initial amplitude and width of the wave\npacket, with norm\n\\begin{equation}\nN=\\sqrt{\\pi }\\,A^{2}(0)\\,a(0)\\,.  \\label{eq:norm-z}\n\\end{equation}%\nThe corresponding interaction and kinetic energies, (\\ref{Eg}) and (\\ref{Ek}%\n),\\ are%\n\\begin{equation}\nE_{g}=-\\frac{1}{3}\\int_{-\\infty }^{+\\infty }g_{1}|u(x,0)|^{6}dx=-\\frac{%\ng_{1}\\,N^{3}}{3\\sqrt{3}\\,\\pi \\,a^{2}(0)}\\,,\n\\end{equation}%\n\\begin{equation}\nE_{k}=\\frac{1}{2}\\int_{-\\infty }^{+\\infty }\\left\\vert \\frac{\\partial u(x,0)}{%\n\\partial x}\\right\\vert ^{2}dx=\\frac{N}{4a^{2}\\left( 0\\right) }\\,,\n\\label{eq:Ekin}\n\\end{equation}%\nwhose ratio $|E_{g}\/E_{k}|$ \\ scales $\\sim N^{2}$, as mentioned above,\nimplying the domination of the the interaction energy for large norms. To\nstudy the evolution of the first component of ${\\bm\\psi }(x,t)$, we use the\nvariational ansatz,\n\\begin{equation}\n{\\bm\\psi }(x,t)=A(t)\\exp \\left[ -\\frac{x^{2}}{2a^{2}(t)}+ib(t)x^{2}\\right]\n\\left(\n\\begin{array}{c}\n1 \\\\\n0%\n\\end{array}%\n\\right) \\,,\n\\end{equation}%\nwhere $a(t)$ and $b(t)$ are the time-dependent width and chirp,\nrespectively. The corresponding Euler-Lagrange equations, corresponding to\nEqs. (\\ref{eq:eq1}) and (\\ref{eq:eq2}), are \\cite%\n{Anderson,Progress}%\n\\begin{equation}\n\\frac{da}{dt}=2ab,\\qquad\\frac{db}{dt}=\\frac{1}{2a^{4}}-2b^{2}-\\frac{2g_{1}N^{2}}{3%\n\\sqrt{3}\\,\\pi a^{4}}\\,.  \\label{ab}\n\\end{equation}%\nFurther, after the elimination of $b$ from Eq. (\\ref{ab}), one arrives at the\nErmakov equation \\cite{Ermakov,Ermakov2},%\n\\begin{equation}\n\\ddot{a}=-\\frac{\\Lambda }{a^{3}},\\qquad \\Lambda \\equiv \\frac{4g_{1}N^{2}}{3\\sqrt{3}%\n\\,\\pi }-1\\,,  \\label{eq:Gaussian}\n\\end{equation}%\nwhich admits well-known analytical solutions with two integration\nconstants, $C_{1,2}$:\n\\begin{equation}\na(t)=\\pm \\sqrt{-\\Lambda \/C_{1}+C_{1}\\,t^{2}+2C_{1}C_{2}\\,t+C_{1}C_{2}^{2}}\\,.\n\\label{a}\n\\end{equation}%\nFor the initial condition $(da\/dt)(0)=0$, the positive branch in Eq. (\\ref{a})\nyields\n\\begin{equation}\na(t)=\\sqrt{-\\Lambda \/C_{1}+C_{1}\\,t^{2}}\\,.  \\label{a2}\n\\end{equation}%\nThe critical norm for the Gaussian ansatz (\\ref{ux0vx0}) with initial $%\n\\langle \\sigma _{z}\\rangle =1$ [see Eq. (\\ref{sigma})], determined by\nsetting $\\Lambda =0$ in Eq. (\\ref{eq:Gaussian}), is\n\\begin{equation}\nN_{c}^{\\langle \\sigma _{z}\\rangle =1}=\\sqrt{3\\sqrt{3}\\pi \/4}\\,%\ng_{1}^{-1\/2}\\approx 2.02\\,g_{1}^{-1\/2}.\n\\label{crit}\n\\end{equation}%\nThis value is only slightly larger than the exact value of the norm of the\ncorresponding Townes soliton, $(\\pi \/2)\\sqrt{3\/2}\\,g_{1}^{-1\/2}\\approx\n1.924\\,g_{1}^{-1\/2}$, see Eq. (\\ref{NT}), demonstrating that the\nGaussian ansatz is suitable for predicting the corresponding critical norm.\n\nFor $\\Lambda >0$, solution (\\ref{a2}) with $C_{1}<0$ describes the collapse\nprocess \\cite{Abdullaev2003}:\n\\begin{equation}\na(t)=a(0)\\sqrt{1-\\frac{t^{2}}{t_{c}^{2}}}\\,,  \\label{eq:at}\n\\end{equation}%\nwhere $a^{2}(0)\\equiv \\Lambda \/\\left\\vert C_{1}\\right\\vert $, and the\ncollapse time is $t_{c}=\\sqrt{\\Lambda }\/\\left\\vert C_{1}\\right\\vert \\equiv\na^{2}(0)\/\\sqrt{\\Lambda }.$ The characteristic velocity developed in the\ncourse of collapsing is\n\\begin{equation}\nV_{c}=a(0)\/t_{c}=\\sqrt{\\Lambda }\/a(0).  \\label{Vc}\n\\end{equation}\n\nTo estimate an effect of {the interplay of two spin components on the onset\nof the collapse}, we now take {the eigenstate of $\\sigma _{x}$ as the\ninitial state} [cf. Eq. (\\ref{ux0vx0})]:%\n\\begin{equation}\nu(x,0)=v(x,0)=\\frac{A(0)}{\\sqrt{2}}\\exp \\left( -\\frac{x^{2}}{2a^{2}(0)}\\right)\n\\label{ux0vx01}\n\\end{equation}%\nwith the same norm as in Eq. \\eqref{eq:norm-z}. In this case, the kinetic\nenergy is again given by Eq. \\eqref{eq:Ekin}, while the interaction energy\nis\n\\begin{equation}\nE_{g}=-\\frac{1}{4}\\,(g_{1}+3g_{2})\\,\\frac{\\,N^{3}}{3\\sqrt{3}\\,\\pi \\,a^{2}(0)}%\n\\,.\n\\label{eq:g1g2}\n\\end{equation}%\nSince at $g_{2}=0,$ the self-interaction energy for the state in Eq. \\eqref{ux0vx01} is factor\nof 2 smaller than that in Eq. \\eqref{ux0vx0} while their kinetic energies are\nequal, the critical value $N_{c}^{\\langle \\sigma\n_{x}\\rangle =1}$ is larger by a factor of $2$ than one given by Eq. (\\ref%\n{crit}). For $g_{2}=g_{1}$\n(the Manakov's spin-isotropic form of the interaction \\cite{Manakov1974}),\nthe critical norm is independent of the spin orientation.\n\n\\section{Collapse in the presence of the spin-orbit coupling and Zeeman\ncouplings}\n\n\\label{Sec:with-soc}\n\nHere we focus on how the SOC and ZS strengths, $\\alpha $ and $\\Delta $,\nrespectively, affect the collapse dynamics. To this end, we perform a\nqualitative analysis and numerically solve the GPE system of Eqs. (\\ref%\n{eq:eq1}) and (\\ref{eq:eq2}), using the split-operator technique \\cite%\n{Chaves2015} for SOC systems.\n\nTo characterize the evolution of the width of the wave packet, we use the\ntime-dependent inverse participation ratio (IPR) \\cite{IPR2000}, defined as\n\\begin{equation}\n\\zeta (t)\\equiv \\int_{-\\infty }^{+\\infty }\\left( |u|^{2}+|v|^{2}\\right)\n^{2}dx.  \\label{zeta}\n\\end{equation}%\nIn turn, the IPR-related width is defined as\n\\begin{equation}\na_{\\zeta }(t)=N^{2}\/\\sqrt{2\\pi }\\,\\zeta .  \\label{azeta}\n\\end{equation}%\nAnother definition of the width is determined by the total spread of the\nwave packet,\n\\begin{equation}\na_{s}^{2}(t)\\equiv \\langle x^{2}(t)\\rangle \\equiv \\frac{1}{N}\\int_{-\\infty }^{+\\infty\n}x^{2}\\left( |u|^{2}+|v|^{2}\\right) dx.  \\label{x2}\n\\end{equation}%\nIt is shown below that these widths feature qualitatively different time\ndependences, due to a non-Gaussian actual shape of the collapsing wave\npacket.\n\n\\subsection{Effect of the spin-orbit coupling at short time}\n\nHere we take an eigenstate of $\\sigma _{z}$ as the initial state, i.e. ${\\bm%\n\\psi }(x,0)=[u_{0}(x),0]^{\\mathrm{T}}$ ($\\mathrm{T}$ stands for\ntransposition), and begin with an analytical consideration for small $t$,\nwhich provides a good insight into the collapse dynamics. To single out the\nqualitative SOC\\ effect, we, for the time being, omit the nonlinearity (and\nZS) in Eqs. (\\ref{eq:eq1}) and (\\ref{eq:eq2}), thus observing that SOC\nsplits the input into eigenstates of $\\sigma _{x}$, and these spin-projected\ncomponents evolve independently:\n\\begin{equation}\n{\\bm\\psi (x,t)}=\\frac{1}{2}\\left(\n\\begin{array}{c}\nu_{0}\\left( x-\\alpha t\\right) \\\\\nu_{0}\\left( x-\\alpha t\\right)%\n\\end{array}%\n\\right) +\\frac{1}{2}\\left(\n\\begin{array}{c}\nu_{0}\\left( x+\\alpha t\\right) \\\\\n-u_{0}\\left( x+\\alpha t\\right)%\n\\end{array}%\n\\right) \\,.\n\\label{eq:splitting}\n\\end{equation}\n\nAt small $t,$ using expansion\n\\begin{equation}\nu_{0}(x\\pm \\alpha t)\\approx u_{0}(x)\\pm \\alpha tu_{0}^{\\prime }(x)+\\frac{1}{2%\n}\\left( \\alpha t\\right) ^{2}u_{0}^{\\prime \\prime }(x),  \\label{prime}\n\\end{equation}%\nwe identify the leading terms,\n\\begin{equation}\nu(x,t)=u_{0}(x)+\\frac{1}{2}u_{0}^{\\prime \\prime }(x)\\left( \\alpha\nt\\right) ^{2},\n\\quad\nv(x,t)=-u_{0}^{\\prime }(x)\\alpha t.  \\label{prime-prime}\n\\end{equation}%\nThe evolution produces a spin dipole moment, defined as\n\\begin{equation}\nd_{\\sigma }\\equiv \\frac{1}{N}\\int_{-\\infty }^{+\\infty }\\left\\langle {\\bm\\psi\n}|x\\,\\sigma _{x}|{\\bm\\psi }\\,\\right\\rangle dx\\,,\n\\end{equation}%\nwhich represents separation of the spin components, and may be considered as\nan additional dynamical variable. For the spinor wave function taken as per\nEqs. (\\ref{eq:splitting})-(\\ref{prime-prime}), one has\n\\begin{equation}\nd_{\\sigma }=\\alpha \\,t,\n\\end{equation}%\ndemonstrating the separation of spin contributions with opposite $\\langle\n\\sigma _{x}\\rangle $. Accordingly, the width of the wave packet, defined as\nper Eq. (\\ref{zeta}), is\n\\begin{equation}\na_{\\zeta }(t)=a_{\\zeta }(0)\\left[ 1+4\\frac{d_{\\sigma }^{2}}{\\zeta (0)}%\n\\int_{-\\infty }^{+\\infty }\\left( u_{0}(x)u_{0}^{\\prime }(x)\\right) ^{2}dx%\n\\right] .\n\\end{equation}\n\nNow one can calculate the evolution of the $z$ component of the spin,\n\\begin{equation}\n\\langle \\sigma _{z}\\rangle =1-4d_{\\sigma }^{2}\\frac{E_{k}}{N}\\,.\n\\label{eq:Zeeman2}\n\\end{equation}%\nIf the ZS term is restored, the respective Zeeman energy shift is\n\\begin{equation}\nE_{Z}=-2\\,\\Delta\\, d_{\\sigma }^{2}\\,E_{k}.  \\label{Zeeman}\n\\end{equation}\nIt depends on the sign of the ZS strength, $\\Delta $, demonstrating the\nsignificance of the direction of the Zeeman field. Further, the correction\nto interaction energy (\\ref{Eg}) is\n\\begin{equation}\n\\Delta E_{g}=\\left( 5g_{1}-g_{2}\\right) d_{\\sigma }^{2}\\int_{-\\infty\n}^{\\infty }\\left[ u_{0}^{\\prime }(x)\\right] ^{2}u_{0}^{4}(x)dx,\n\\end{equation}%\ndemonstrating that the effect of self-interaction nearly cancels at $%\n5g_{1}=g_{2}$.\n\nAs an example, we take the initial state in the form of Eq. (\\ref{ux0vx0}),\nwhere $a_{\\zeta }(0)=a(0)$. For the interaction-energy correction we obtain\n\\begin{equation}\n\\Delta E_{g}=\\left( 5g_{1}-g_{2}\\right) d_{\\sigma }^{2}\\frac{\\sqrt{3}}{18\\pi\n}\\frac{N^{3}}{a^{4}(0)},\n\\label{eq:DeltaEg1g2}\n\\end{equation}%\nand the spin projection becomes%\n\\begin{equation}\n\\langle \\sigma _{z}\\rangle =1-\\frac{d_{\\sigma }^{2}}{a^{2}(0)}.\n\\end{equation}%\nAccordingly, the IPR-determined packet's width increases as\n\\begin{equation}\na_{\\zeta }(t)=a(0)\\left[ 1+\\frac{d_{\\sigma }^{2}}{a^{2}(0)}\\right] .\n\\end{equation}%\nComparing this expression to Eq. \\eqref{eq:at}, we see that the effect of\nSOC on the BEC width dominates for\n\\begin{equation}\n\\alpha >V_{c},  \\label{>}\n\\end{equation}\ni.e., if the anomalous velocity exceeds the characteristic velocity\ndeveloped in the course of collapsing.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics*[width=0.45\\textwidth]{figure_1n.eps}\n\\caption{(Color online) Widths $a_{\\zeta }(t)$, calculated as per\nEq. (\\ref{zeta}) (the main plot), and $a_{s}(t)$, defined as per Eq.\n(\\ref{x2}) (the inset), for different values of the SOC strength, $%\n\\alpha $, shown near the plots. Here $\\alpha _{\\mathrm{cr}%\n}=1.16$ for $N=3$, $g_{1}=1$, $g_{2}=0$, and $\\Delta =0$. For comparison,\nthe dashed line displays the width evolution for the Gaussian-ansatz\nsolution, given by Eq. \\eqref{eq:at}. Note that for this choice of\nparameters $\\Lambda \\approx 1.2$ [see Eq. (\\ref{eq:Gaussian})], and\nthe ansatz predicts the collapse at $t_{c}\\approx 0.9$ and $V_{c}\\approx 1.1$%\n, very close to the numerically obtained value, $\\alpha _{\\mathrm{cr}%\n}=1.16$, as seen here. Here and in the following figures we use $a(0)=1$ for the initial Gaussian\nstates.}\n\\label{fig:collapse_1}\n\\end{figure}\n\n\\subsection{Zero cross-interaction: $g_{1}\\neq 0,g_{2}=0,\\Delta =0$}\n\nWe begin systematic analysis of the dynamics, neglecting the nonlinear\ninteraction between the components, i.e., setting $g_{2}=0$ in Eqs. (\\ref%\n{eq:eq1}) and (\\ref{eq:eq2}). In Fig. \\ref{fig:collapse_1}, we display\nnumerical results for three different values of the SOC strength, $\\alpha $.\nNote that the packet's shape is strongly non-Gaussian even in the case of $%\n\\alpha =0$. As a result, width $a_{s}(t)$, defined by Eq. (\\ref{x2}), does\nnot display the collapse, and one needs to examine the IPR as in Eqs. \\eqref{zeta} and \\eqref{azeta}.\nThe reason for this peculiarity is discussed below.\n\nNaturally, the collapse time strongly depends on $\\alpha $, diverging when\nthe SOC strength is approaching a critical value, $\\alpha _{\\mathrm{cr}}.$\nThis value is determined by condition (\\ref{>}), which implies that a\ntypical anomalous velocity induced by SOC, $V_{\\mathrm{so}}\\sim \\alpha $,\nexceed the collapse velocity $V_{c}$, see Eq. (\\ref{Vc}). If $\\alpha =\\alpha\n_{\\mathrm{cr}}$, the simulations demonstrate that the BEC starts its\nevolution by compressing in the beginning, but then slightly expands (not\nshown in detail). As $\\alpha$ approaches $\\alpha_{\\mathrm{cr}},$ the\ncollapse time diverges as $\\sim (\\alpha _{\\mathrm{cr}}-\\alpha )^{-1\/2}$.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics*[width=0.45\\textwidth]{figure_2n.eps}\n\\caption{(Color online) Density profiles of the collapsing wave packet, $%\n\\rho (x)=\\left\\vert v\\right\\vert ^{2}+\\left\\vert u\\right\\vert ^{2}$,\nproduced by numerical simulations of Eqs. (\\ref{eq:eq1}) and (%\n\\ref{eq:eq2}) with initial conditions (\\ref{ux0vx0}), at\ntime $t$ marked near the plots. Wings are clearly seen in the figure at $\\left\\vert\nx\\right\\vert>1\/2.$ Here $\\alpha =0.8\\alpha _{\\mathrm{cr}}$\nfor $N=3$, $g_{1}=1$, $g_{2}=0$, and $\\Delta =0$.}\n\\label{fig:density}\n\\end{figure}\n\nTo better understand the difference between the evolution of the IPR- and\nthe $\\sqrt{\\langle x^{2}\\rangle }$-based widths, which are defined by Eqs. (\\ref{eq:at})\nand  (\\ref{azeta}), we display, in Fig. \\ref{fig:density},\ndensity profiles at different times, which show well-developed wings.\nTherefore, while the core part of the wave packet collapses, its shell\nextends and leads to finite $\\langle x^{2}\\rangle .$ The evolution of the\nwings, initially observed in Ref. \\cite{Abdullaev2005}, is attributed to\nvery fast decrease in the interaction-energy density.\nThis is confirmed by comparing the numerical and variational solutions.\nThe collapse occurs faster than predicted by Eq. \\eqref{eq:at}, as only a\nfraction of the BEC density in the vicinity of the origin undergoes the\ncollapse. Thus, although the Gaussian ansatz accurately predicts the\ncritical norm, it is not suitable for modeling dynamical features of the\ncollapse.\n\nAs follows from condition $\\alpha _{\\mathrm{cr}}\\sim V_{c}$, the critical\nvalue of the SOC strongly depends on the norm of the\nwave packet. For sufficiently large $N$, no value of $\\alpha $ is sufficient\nto arrest the collapse, as shown in Fig. \\ref{fig:alphaN}, since a fully\nseparated packet still has a sufficient norm to drive the collapse. Taking\ninto account the fact that $N_{c}^{\\left\\langle \\sigma _{x}\\right\\rangle\n=1}=2N_{c}^{\\left\\langle \\sigma _{z}\\right\\rangle =1}$, and that the fast\nseparation reduces the norm per component by a factor of two, we conclude\nthat, at $N>4N_{c}^{\\left\\langle \\sigma _{z}\\right\\rangle =1}$, the critical\nvalue $\\alpha _{\\mathrm{cr}}$ diverges, as can be seen in the more rapid increase in $%\n\\alpha _{\\mathrm{cr}}$ in Fig. \\ref{fig:alphaN} as $N$ approaches\n$4N_{c}^{\\left\\langle \\sigma _{z}\\right\\rangle =1}.$\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics*[width=0.45\\textwidth]{figure_3n.eps}\n\\caption{(Color online) Critical value $\\alpha _{\\mathrm{cr}}$ of\nthe SOC strength, such that the collapse in the system with the\nself-attraction ($g_{1}=1$, $g_{2}=0,\\Delta =0$) does not occur at $%\n\\alpha >\\alpha _{\\mathrm{cr}}$, vs. the total norm, $N$. The red\nline is an interpolation connecting all data points. Note that, as clearly\nseen in the figure, the condition $\\alpha _{\\mathrm{cr}}\\sim %\n\\sqrt{\\Lambda }\/a(0)$ predicts linear dependence of $\\alpha _{%\n\\mathrm{cr}}(N)$ in the interval of $N_{\\mathrm{cr}}^{\\left\\langle %\n\\sigma _{z}\\right\\rangle =1}\\alt N\\alt4N_{\\mathrm{cr}}^{\\left\\langle %\n\\sigma _{z}\\right\\rangle =1}$, where $N_{c}^{\\left\\langle \\sigma %\n_{z}\\right\\rangle =1}\\approx 2.0$ is the critical value either in the variational approximation (\\ref{crit}) or\nthe exact one in Eq. \\eqref{NT}.}\n\\label{fig:alphaN}\n\\end{figure}\n\n\n\\subsection{Effect of the Zeeman splitting: $g_{1}\\neq 0,g_{2}=0,\\Delta \\neq\n0$}\n\nRegarding the effect of ZS on the collapse, there may be either a\ncompetition or mutual enhancement of the SOC and ZS terms. The dependence of\nthe critical value, $\\alpha _{\\mathrm{cr}}$, on the ZS strength, $\\Delta $,\nis plotted in Fig. \\ref{fig:boundary}, indicating a strong $\\Delta\n\\leftrightarrow -\\Delta $ asymmetry, related to the $\\Delta $-dependence of\nthe Zeeman energy in Eq. \\eqref{Zeeman}. In particular, the total energy\ndecreases at $\\Delta <0$, which facilitates the onset of the collapse, hence\n$\\alpha _{\\mathrm{cr}}$ is larger in this case. On the other hand, at $%\n\\Delta >0$ both the SOC\\ and ZS terms resist the collapse, as seen in Fig. %\n\\ref{fig:spin}.\n\nAnother characteristic feature of the collapse-expansion dichotomy is the\ntime dependence of the spin presented in the inset of Fig. \\ref{fig:spin},\nwhich shows relatively slow non-decaying evolution if the collapse occurs,\nand decays to zero otherwise (if the condensate expands, instead of blowing\nup). The spin separation caused by the anomalous velocity leads to decrease\nof the total spin, by producing a mixed spin state with $\\sum_{j=x,y,z}%\n\\langle \\sigma _{j}\\rangle ^{2}<1.$ In particular, the state approximately\ngiven by Eq. \\eqref{eq:splitting} shows that, in the limit of $\\alpha t\\gg\na(0),$ all expectation components of the spin vanish, corresponding to the\nmaximally mixed spin state, where $\\sum_{j=x,y,z}\\langle \\sigma _{j}\\rangle\n^{2}=0.$\n\nAt a sufficiently large $\\Delta >0$, ZS can dominate in the spin evolution, making it\ndifficult to spatially separate the spin components.\nThis is shown in Fig. \\ref{fig:strong_D}, where it is seen that,\nin the limit of very large $\\Delta$, the collapse occurs as in the absence\nof SOC. The suppression of the SOC effect by large $\\Delta $ can be\nexplained as follows \\cite{Sherman2014}. Consider a Fourier component of the BEC\nwave function with momentum $p.$ The corresponding spin precession rate, as\nfollows from Eq. \\eqref{eq:Hspin}, is $\\sqrt{\\Delta ^{2}+4\\alpha ^{2}p^{2}}$%\n, and the orientation of the spin-precession axis is $\\mathbf{n}_{s}=\\left(\n\\alpha p,0,\\Delta \/2\\right).$ In the limit of $\\Delta \\gg \\alpha p,$ equivalent to\nthe condition $\\Delta \\gg \\alpha\/a(0),$ the\ndirection of $\\mathbf{n}_{s}$ is close to the $z$-axis and the maximum\nvalue of the anomalous velocity $\\alpha\\sigma_{x},$ achieved in the course of the BEC\nevolution, corresponds to the maximally achieved $\\sigma_{x}\\sim 4\\left(\\alpha\/a(0)\\right)\/\\Delta,$\nand becomes $\\sim 4\\alpha\\left(\\alpha\/a(0)\\right)\/\\Delta,$\ndecreasing as $1\/\\Delta.$ Therefore, in this limit, imposed on the value of $\\Delta,$ the additional\ncondition, dependent on the BEC self-interaction, for the occurrence of the collapse can be rewritten as $4\\alpha\n\\left(\\alpha\/a(0)\\Delta\\right)\\ll V_{c},$ or, equivalently, $\\Delta\\gg 4\\alpha^{2}\/(a(0)V_{c}),$\nwhere $V_{c}=\\sqrt{\\Lambda}\/a(0)$ is given by Eq. \\eqref{Vc}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics*[width=0.45\\textwidth]{figure_4n.eps}\n\\caption{(Color online) The stability diagram in the $(\\alpha _{%\n\\mathrm{cr}},\\Delta )$ plane for solutions stemming from the\nsingle-component input based on Eq. (\\ref{ux0vx0}). Other parameters\nare $N=3$, $g_{1}=1,$ $g_{2}=0$.}\n\\label{fig:boundary}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics*[width=0.45\\textwidth]{figure_5n.eps}\n\\caption{(Color online) The time dependence of the width and mean value of the spin component $\\langle\\sigma_{z}\\rangle$\nof the collapsing and expanding wave packets, generated by input (\\ref%\n{ux0vx0}), at different values of the ZS strength, $\\Delta $ (shown near the\ncurves). Other parameters are $N=3$, $g_{1}=1,$ $g_{2}=0$. }\n\\label{fig:spin}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics*[width=0.45\\textwidth]{figure_6n.eps}\n\\caption{(Color online) The time dependence of the width of the collapsing\nor expanding wave packets, defined as per Eq. (\\ref{zeta}), at\nvalues of parameters shown near the plots. The simulations were performed\nwith initial conditions given by Eq. (\\ref{ux0vx0}). At sufficiently\nlarge $\\Delta $ the collapse is enforced, the evolution of the width being\nclose to that in the absence of SOC. Note that $\\alpha _{\\mathrm{cr}%\n}=1.16$ at $\\Delta =0$. The norm fixed here is $N=3$. }\n\\label{fig:strong_D}\n\\end{figure}\n\n\\subsection{The collapse driven by the cross-spin attraction: $%\ng_{1}=0,g_{2}\\neq 0,\\Delta =0$}\n\nTo complete the analysis of the effects of SOC on the BEC collapse,\nwe consider the system of GPEs (\\ref{eq:eq1}) and (\\ref{eq:eq2}),\nwhich includes solely the cross-nonlinearity, \\textit{viz}., $g_{1}=0$ and $%\ng_{2}=1$ at the initial condition $\\langle\\sigma_{z}\\rangle=1,$ as\npresented in Eq. \\eqref{ux0vx0}. Here the dynamics is qualitatively different from the\nevolution for the same-spin interaction since in the absence of the spin-orbit coupling\nthe initial state spreads for any norm. Therefore, the\ncollapse can happen only as a result of the SOC-driven spatial\nsplitting of the wave packet into spin-polarized complexes, similar to that\npredicted by Eq. (\\ref{eq:splitting}) resulting in the cross-spin attraction\nbetween the spin components. Thus, in contrast to the above\ncase with $g_{2}=0$, here the collapse becomes possible when the SOC\nstrength \\textit{exceeds} a certain critical value: $\\alpha >\\alpha _{%\n\\mathrm{cr}}$. It is worth noting that at $g_{1}=0$ in a state with $\\langle\\sigma_{x}\\rangle=1,$\ndefined in Eq. \\eqref{ux0vx01}, the critical norm $\\widetilde{N}_{\\rm cr}$ is\ngiven by $\\widetilde{N}_{\\rm cr}=(3\\pi)^{1\/4}g_{2}^{-1\/2},$\nas can be seen from Eqs. \\eqref{eq:Ekin},\\eqref{crit}, and \\eqref{eq:g1g2}.\n\nAlthough the resulting evolution is very complex, the critical value can be estimated here from the following scaling\nargument based on presentation of the wavefunction similar to that in Eq. \\eqref{eq:splitting}.\nFirst, let us assume that the spin-orbit coupling is sufficiently strong to ensure that\nthe initial wavepacket with the norm $N$ and $\\langle\\sigma_{z}\\rangle=1$ rapidly splits in two\nGaussian wavepackets, having the same density profiles as the initial one, albeit with the norm $N\/2$\nand spin projections $\\langle\\sigma_{x}\\rangle=\\pm1.$ Then, if $N>2\\widetilde{N}_{\\rm cr},$ each spatial\ncomponent of the BEC will collapse independently. This argument, consistent with Eq. \\eqref{eq:DeltaEg1g2},\nshows that the minimal time required to switch on the cross-spin interaction is the time of the\nessential spatial separation of the spin components, that is $\\sim a(0)\/\\alpha$ \\cite{Sherman2014}.\nOn the other hand, in the absence of the self-interaction, at time $t>a^{2},$ the wavepacket spreads\nwith the rate of the order of $1\/a(0),$ due to the Heisenberg momentum-position\nuncertainty ratio. Therefore, the required separation can be achieved at $\\alpha\\sim 1\/a(0),$\nestablishing the low boundary for the critical $\\alpha$ for the norm of the order of $\\widetilde{N}_{\\rm cr}.$\nNote that for the norm $N\\gg\\widetilde{N}_{\\rm cr},$ a strong separation is not needed for the collapse,\nleading to a decrease in the $\\alpha_{\\rm cr}$ with $N.$ This realization requires a separate analysis.\nThe results of numerical calculations shown in Fig. \\ref{fig:crosscoupling} confirm that this realization\nself-interaction is indeed  qualitatively different from the one with the same-spin interaction. The numerical value\n$\\alpha _{\\mathrm{cr}}=0.76$ agrees well with the above presented scaling argument $\\alpha_{\\mathrm{cr}}\\sim 1.$\n\nFor experimental manifestations of the effects considered here, one might follow proposal of Ref. \\cite{Altin2011}\nby  adding the realizations of SOC and ZS terms. Since it is possible to tune the atomic\nscattering length to change the interatomic interaction by Feshbach resonance \\cite{Cornish2000},\none might reduce the strength of the repulsive two-body interaction and amplify\nthe three-body attractive interaction, making the three-body collisions dominant.\nIn addition, the three-body losses could be ignored in the system since the collapse\ntime scale is short.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics*[width=0.45\\textwidth]{figure_7n.eps}\n\\caption{(Color online) The widths of the collapsing wave packets,\ncalculated as per Eq. (\\ref{zeta}), in the system which includes\nsolely the cross-attraction between the components, i.e., $g_{1}=0$ and $%\ng_{2}=1$ in Eqs. (\\ref{eq:eq1}) and (\\ref{eq:eq2}), other\nparameters being $N=6$ and $\\Delta =0$, which corresponds to $%\n\\alpha _{\\mathrm{cr}}=0.76$. The simulations were initiated by input (%\n\\ref{ux0vx0}).}\n\\label{fig:crosscoupling}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\\label{end}\n\nWe have studied the dynamics of density and spin patterns in the\none-dimensional BEC collapse driven by the quintic same- and\ncross-spin interaction in the presence of artificial spin-orbit coupling (SOC) and Zeeman splitting\n(ZS). The dynamics is determined by the interplay of the\nnonlinear attraction and the splitting of the spinor wave function into\nspin-polarized complexes, driven by the SOC-induced anomalous spin-dependent velocity [see\nEq. (\\ref{eq:splitting})]. We performed\ninvestigation of the qualitative features of the collapse dynamics, varying the SOC and ZS strengths. In\nparticular, we found that the Zeeman term can either support or suppress the collapse,\ndepending on the initial conditions and direction (sign) of the Zeeman\nfield. The effect of the SOC critically depends on the form of the quintic attraction\nterms. Namely, it tends to suppress the collapse which is driven by the\nself-attraction in each BEC component, and, on the other hand, SOC supports the\ncollapse in systems with the cross-spin attraction. We numerically obtained\ncritical values of the spin-orbit coupling, establishing the boundary between collapsing and\nexpanding states. In addition, our calculations demonstrated that due to a non-Gaussian evolution\nof the density, to see the collapse in these systems,\none needs to study the inverse participation ratio rather than expectation values of the coordinate.\nTaking into account continuous Lie symmetries \\cite{VVLieS} can be useful for the future studies of the spin effects. \nAlthough topological arguments\nrelated to the formation of vortex-like\nstructures, which appear due to the spin-orbit coupling \\cite{Sakaguchi2014,Sakaguchi2016}\nor without it \\cite{WangPRA},\ncannot be directly applied here, studies of possible stationary\npseudo-spinor structures in two dimensions, including relaxation effects,\nare of interest and will be a topic of further research.\n\n\n\n\n\\section{Acknowledgement}\n\nThis work was partially supported by NSFC (11474193), SMSTC (18010500400 and\n18ZR1415500), and the Program for Eastern Scholar in China. J.L. and W.L.\nacknowledge support by Okinawa Institute of Science and Technology Graduate\nUniversity. E.Y.S. acknowledges support by the Spanish Ministry of Science and the European Regional\nDevelopment Fund through PGC2018-101355-B-I00 (MCIU\/AEI\/FEDER,UE), and the Basque Government through\nGrant No. IT986-16. The work of B.A.M. is supported, in part, by Israel Science Foundation through grant No. 1287\/17. X.C. thanks the Ram\\'{o}n y Cajal grant (RYC-2017-22482).\nWe are grateful to Th. Busch for discussion and valuable comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nThe privacy of the contents of the downloaded information from curious public databases has attracted considerable research within the computer science community \\cite{ChorPIR, yekhanin2010private, PIRsurvey2004, ostrovsky2007survey}. The problem is motivated by practical examples such as: ensuring privacy of investors as they download records in a stock market, since revealing the interest in a certain record may influence its value; ensuring the privacy of an inventor as they look up existing patents in a database, since revealing what they are looking at leaks some information about the current invention they are working on; and protecting activists in oppressive regimes as they browse restricted content on the internet \\cite{yekhanin2010private}. In the classical private information retrieval (PIR) problem, a user wishes to download a certain message (or file) from $N$ non-communicating databases without leaking any information about the identity of the downloaded message. The contents of the databases are identical. The user performs this operation by preparing and submitting queries to all databases. The databases respond truthfully with answer strings which are functions of the queries and the messages. The user needs to reconstruct the desired message from these answer strings. A trivial solution for this seemingly difficult problem is for the user to download the contents of all databases. This solution however is extremely inefficient. The efficiency is measured by the retrieval rate which is the ratio of the number of retrieved desired message symbols to the number of total downloaded symbols. The capacity of PIR is the maximum retrieval rate over all possible PIR schemes.\n\nThe computer science formulation of this problem assumes that the messages are of length one. The metrics in this case are the download cost, i.e., the sum of lengths of the answer strings, and the upload cost, i.e., the size of the queries. Most of this work is computational PIR as it ensures only that a server cannot get any information about user intent unless it solves a certain computationally hard problem \\cite{yekhanin2010private, cachin1999computationally}. The information-theoretic re-formulation of the problem considers arbitrarily large message sizes, and ignores the upload cost. This formulation provides an absolute, i.e., information-theoretic, guarantee that no server participating in the protocol gets any information about the user intent. Towards that end, recently, \\cite{JafarPIRBlind} has drawn a connection between the PIR problem and the blind interference alignment scheme proposed in \\cite{BIA}. Then, \\cite{JafarPIR} has determined the exact capacity of the classical PIR problem. The retrieval scheme in \\cite{JafarPIR} is based on three principles: message symmetry, symmetry across databases, and exploiting side information from the undesired messages through alignment.\n\nThe basic PIR setting has been extended in several interesting directions. The first extension is the coded PIR (CPIR) problem \\cite{RamchandranPIR,RazanPIR,YamamotoPIR}. The contents of the databases in this problem are coded by an $(N,K)$ storage code instead of being replicated. This is a natural extension since most storage systems nowadays are in fact coded to achieve reliability against node failures and erasures with manageable storage cost. In \\cite{KarimCoded}, the exact capacity of the MDS-coded PIR is determined. Another interesting extension is PIR with colluding databases (TPIR). In this setting, $T$ databases can communicate and exchange the queries to identify the desired message. The exact capacity of colluded PIR is determined in \\cite{JafarColluding}. The case of coded colluded PIR is investigated in \\cite{codedcolluded}. The robust PIR problem (RPIR) extension considers the case when some databases are not responsive \\cite{JafarColluding}. Lastly, in the symmetric PIR problem (SPIR) the privacy of the remaining records should be maintained against the user in addition to the usual privacy constraint on the databases, i.e., the user should not learn any other messages other than the one it wished to retrieve. The exact capacity of symmetric PIR is determined in \\cite{symmetricPIR}; and the exact capacity of symmetric PIR from coded databases is determined in \\cite{codedsymmetric}.\n\nIn some applications, the user may be interested in retrieving multiple messages from the databases without revealing the identities of these messages. Returning to the examples presented earlier: the investor may be interested in comparing the values of multiple records at the same time, and the inventor may be looking up several patents that are closely related to their work. One possible solution to this problem is to use single-message retrieval scheme in \\cite{JafarPIR} successively. We show in this work that multiple messages can be retrieved more efficiently than retrieving them one-by-one in a sequence. This resembles superiority of joint decoding in multiple access channels over multiple simultaneous single-user transmissions \\cite{cover}. To motivate the multi-message private information retrieval problem (MPIR), consider the example in \\cite[Section~4.3]{JafarPIR} where the number of messages is $M=3$, number of databases is $N=2$, and the user is interested in retrieving only $P=1$ message. Here the optimal scheme retrieves 8 desired bits in 14 downloads, hence with a rate $4\/7$. When the user wishes to retrieve $P=2$ messages, if we use the scheme in \\cite{JafarPIR} twice in a row, we retrieve 16 bits in 28 downloads, hence again a \\emph{sum rate} of $4\/7$. Even considering the fact that the scheme in \\cite{JafarPIR} retrieves 2 bits of the second message \\emph{for free} in downloading the first message, i.e., it actually retrieves 10 bits in 14 downloads, hence a sum rate of $5\/7$, we show in this paper that a better sum rate of $4\/5$ can be achieved by joint retrieval of the messages.\n\nAlthough there is a vast literature on classical PIR in the computer science literature, only a few works exist in MPIR, such as:  \\cite{OneblockFits} which proposes a multi-block (multi-message) scheme and observes that if the user requests multiple blocks (messages), it is possible to reuse randomly mixed data blocks (answer strings) across multiple requests (queries). \\cite{RAID} develops a multi-block scheme which further reduces the communication overhead. \\cite{multiblock} develops an achievable scheme for the multi-block PIR by designing $k$-safe binary matrices that uses XOR operations. \\cite{multiblock} extends the scheme in \\cite{ChorPIR} to multiple blocks. \\cite{multiquery} designs an efficient non-trivial multi-query computational PIR protocol and gives a lower bound on the communication of any multi-query information retrieval protocol. These works do not consider determining the information-theoretic capacity.\n\nIn this paper, we formulate the MPIR problem with non-colluding repeated databases from an information-theoretic perspective. Our goal is to characterize the sum capacity of the MPIR problem $C_s^P$, which is defined as the maximum ratio of the number of retrieved symbols from the $P$ desired messages to the number of total downloaded symbols. When the number of desired messages $P$ is at least half of the total number of messages $M$, i.e., $P \\geq \\frac{M}{2}$, we determine the exact sum capacity of MPIR as $C_s^P=\\frac{1}{1+\\frac{M-P}{PN}}$. We use a novel achievable scheme which downloads MDS-coded mixtures of all messages. We show that joint retrieving of the desired messages strictly outperforms successive use of single-message retrieval for $P$ times. Additionally, we present an achievable rate region to characterize the trade-off between the retrieval rates of the desired $P$ messages.\n\nFor the case of $P \\leq \\frac{M}{2}$, we derive lower and upper bounds that match if the total number of messages $M$ is an integer multiple of the number of desired messages $P$, i.e., $\\frac{M}{P} \\in \\mathbb{N}$. In this case, the sum capacity is $C_s^P=\\frac{1-\\frac{1}{N}}{1-(\\frac{1}{N})^{M\/P}}$. The result resembles the single-message capacity with the number of messages equal to $\\frac{M}{P}$. In other cases, although the exact capacity is still an open problem, we show numerically that the gap between the lower and upper bounds is monotonically decreasing in $N$ and is upper bounded by $0.0082$. The achievable scheme when $P \\leq \\frac{M}{2}$ is inspired by the greedy algorithm in \\cite{JafarPIR}, which retrieves all possible combinations of messages. The main difference of our scheme from the scheme in \\cite{JafarPIR} is the number of stages required in each download round. For example, round $M-P+1$ to round $M-1$, which correspond to retrieving the sum of $M-P+1$ to sum of $M-1$ messages, respectively, are suppressed in our scheme. This is because, they do not generate any useful side information for our purposes here, in contrast to \\cite{JafarPIR}. Interestingly, the number of stages for each round is related to the output of a $P$-order IIR filter \\cite{oppenheim}. Our converse proof generalizes the proof in \\cite{JafarPIR} for $P \\geq 1$. The essence of the proof is captured in two lemmas: the first lemma lower bounds the uncertainty of the interference for the case $P \\geq \\frac{M}{2}$, and the second lemma upper bounds the remaining uncertainty after conditioning on $P$ interfering messages.\n\n\\section{Problem Formulation}\n\nConsider a classical PIR setting storing $M$ messages (or files). Each message is a vector $W_i \\in \\mathbb{F}_q^L,\\: i \\in \\{1, \\cdots, M\\} $, whose elements are picked uniformly and independently from sufficiently large field $\\mathbb{F}_q$. Denote the contents of message $W_m$ by the vector $[w_m(1), w_m(2), \\cdots, w_m(L)]^T$. The messages are independent and identically distributed, and thus,\n\\begin{align}\nH(W_i)&=L, \\quad i \\in \\{1, \\cdots, M\\} \\\\\n\\label{msg_indep}H\\left(W_{1:M}\\right)&=ML\n\\end{align}\nwhere $W_{1:M}=(W_1,W_2, \\cdots, W_M)$. The messages are stored in $N$ non-colluding (non-communicating) databases. Each database stores an identical copy of all $M$ messages, i.e., the databases encode the messages via $(N,1)$ repetition storage code \\cite{KarimCoded}.\n\nIn the MPIR problem, the user aims to retrieve a subset of messages indexed by the index set $\\mathcal{P}=\\{i_1, \\cdots, i_P\\} \\subseteq \\{1, \\cdots, M\\}$ out of the available messages, where $|\\mathcal{P}|=P$, without leaking the identity of the subset $\\mathcal{P}$. We assume that the cardinality of the potential message set, $P$, is known to all databases. To retrieve $W_\\mathcal{P}=(W_{i_1}, W_{i_2}, \\cdots, W_{i_P})$, the user generates a query $Q_n^{[\\mathcal{P}]}$ and sends it to the $n$th database. The user does not have any knowledge about the messages in advance, hence the messages and the queries are statistically independent,\n\\begin{align}\\label{msg_query_indep}\nI\\left(W_1, \\cdots, W_M;Q_1^{[\\mathcal{P}]}, \\cdots, Q_N^{[\\mathcal{P}]} \\right)=I\\left(W_{1:M};Q_{1:N}^{[\\mathcal{P}]}\\right)=0\n\\end{align}\nThe privacy is satisfied by ensuring statistical independence between the queries and the message index set $\\mathcal{P}=\\{i_1, \\cdots, i_P\\}$, i.e., the privacy constraint is given by,\n\\begin{align}\\label{privacy_constraint}\nI\\left(Q_n^{[i_1, \\cdots, i_P]};i_1, \\cdots, i_P\\right)=I\\left(Q_n^{[\\mathcal{P}]};\\mathcal{P}\\right)=0, \\quad n \\in \\{1, \\cdots, N\\}\n\\end{align}\nThe $n$th database responds with an answer string $A_n^{[\\mathcal{P}]}$, which is a deterministic function of the queries and the messages, hence\n\\begin{align}\nH(A_n^{[\\mathcal{P}]}|Q_n^{[\\mathcal{P}]},W_{1:M})=0\n\\end{align}\nWe further note that by the data processing inequality and (\\ref{privacy_constraint}),\n\\begin{align}\nI\\left(A_n^{[\\mathcal{P}]};\\mathcal{P}\\right)=0, \\quad n \\in \\{1, \\cdots, N\\}\n\\end{align}\nIn addition, the user should be able to reconstruct the messages $W_\\mathcal{P}$ reliably from the collected answers from all databases given the knowledge of the queries. Thus, we write the reliability constraint as,\n\\begin{align}\\label{reliability}\nH\\left(W_{i_1}, \\cdots, W_{i_P}|A_1^{[\\mathcal{P}]}, \\cdots, A_N^{[\\mathcal{P}]}, Q_1^{[\\mathcal{P}]}, \\cdots, Q_N^{[\\mathcal{P}]}\\right)=H\\left(W_\\mathcal{P}|A_{1:N}^{[\\mathcal{P}]},Q_{1:N}^{[\\mathcal{P}]}\\right)=0\n\\end{align}\n\nWe denote the retrieval rate of the $i$th message by $R_i$, where $i \\in \\mathcal{P}$. The retrieval rate of the $i$th message is the ratio between the length of message $i$ and the total download cost of the message set $\\mathcal{P}$ that includes $W_i$. Hence,\n\\begin{align}\nR_i=\\frac{H(W_i)}{\\sum_{n=1}^N H\\left(A_n^{[\\mathcal{P}]}\\right)}\n\\end{align}\nThe sum retrieval rate of $W_\\mathcal{P}$ is given by,\n\\begin{align}\n\\sum_{i=1}^PR_i=\\frac{H(W_\\mathcal{P})}{\\sum_{n=1}^N H\\left(A_n^{[\\mathcal{P}]}\\right)}=\\frac{PL}{\\sum_{n=1}^N H\\left(A_n^{[\\mathcal{P}]}\\right)}\n\\end{align}\nThe sum capacity of the MPIR problem is given by\n\\begin{align}\nC_s^P=\\sup \\: \\sum_{i=1}^PR_i\n\\end{align}\nwhere the $\\sup$ is over all private retrieval schemes.\n\nIn this paper, we follow the information-theoretic assumptions of large enough message size, large enough field size, and ignore the upload cost as  in \\cite{KarimCoded,JafarPIR,JafarColluding,YamamotoPIR}. A formal treatment of the capacity under message and field size constraints for $P=1$ can be found in \\cite{arbmsgPIR}. We note that the MPIR problem described here reduces to the classical PIR problem when $P=1$, whose capacity is characterized in \\cite{JafarPIR}.\n\n\\section{Main Results and Discussions}\n\nOur first result is the exact characterization of the sum capacity for the case $P \\geq \\frac{M}{2}$, i.e., when the user wishes to privately retrieve at least half of the messages stored in the databases.\n\\begin{theorem}\\label{thm1}\nFor the MPIR problem with non-colluding and replicated databases, if the number of desired messages $P$ is at least half of the number of overall stored messages $M$, i.e., if $P \\geq \\frac{M}{2}$, then the sum capacity is given by,\n\\begin{align}\\label{result1}\nC_s^P=\\frac{1}{1+\\frac{M-P}{PN}}\n\\end{align}\n\\end{theorem}\n\nThe achievability proof for Theorem~\\ref{thm1} is given in Section~\\ref{sec:achievable1}, and the converse proof is given in Section~\\ref{converse1}. We note that when $P=1$, the constraint of Theorem~\\ref{thm1} is equivalent to $M=2$, and the result in  (\\ref{result1}) reduces to the known result of \\cite{JafarPIR} for $P=1$, $M=2$, which is $\\frac{1}{1+\\frac{1}{N}}$. We observe that the sum capacity in (\\ref{result1}) is a strictly increasing function of $N$, and $C_s^P \\rightarrow 1$ as $N \\rightarrow \\infty$. We also observe that the sum capacity in this regime is a strictly increasing function of $P$, and approaches $1$ as $P \\rightarrow M$.\n\nThe following corollary compares our result and the rate corresponding to the repeated use of single-message retrieval scheme \\cite{JafarPIR}.\n\\begin{corollary}\\label{corrollary1}\nFor the MPIR problem with $P \\geq \\frac{M}{2}$, the repetition of the single-message retrieval scheme of \\cite{JafarPIR} $P$ times in a row, which achieves a sum rate of,\n\\begin{align}\nR_s^{rep}=\\frac{(N-1)(N^{M-1}+P-1)}{N^M-1}\n\\label{repeat}\n\\end{align}\nis strictly sub-optimal with respect to the exact capacity in (\\ref{result1}).\n\\end{corollary}\n\n\\begin{Proof}\nIn order to use the single-message capacity achieving PIR scheme as an MPIR scheme, the user repeats the single-message achievable scheme for each individual message that belongs to $\\cp$. We note that at each repetition, the scheme downloads extra decodable symbols from other messages. By this argument, the following rate $R_s^{rep}$ is achievable using a repetition of the single-message scheme,\n\\begin{align}\nR_s^{rep}=C+\\Delta(M,P,N) \\label{eqn-delta}\n\\end{align}\nwhere $C$ is the single-message capacity which is given by $C=\\frac{1-\\frac{1}{N}}{1-(\\frac{1}{N})^M}$ \\cite{JafarPIR}, and $\\Delta(M,P,N)$ is the rate of the extra decodable symbols that belong to $\\cp$. To calculate $\\Delta(M,P,N)$, we note that the total download cost $D$ is given by $D=\\frac{L}{C}$ by definition. Since $L=N^M$ in the single-message scheme, $D=\\frac{N^M(1-(\\frac{1}{N})^M)}{1-\\frac{1}{N}}=\\frac{N^{M+1}-N}{N-1}$. The single-message scheme downloads one symbol from every message from every database, i.e., the scheme downloads extra $(P-1)N$ symbols from the remaining desired messages that belong to $\\cp$, thus,\n\\begin{align}\n\\Delta(M,P,N)=\\frac{(P-1)N(N-1)}{N^{M+1}-N}=\\frac{(P-1)(N-1)}{N^M-1}\n\\end{align}\nUsing this in (\\ref{eqn-delta}) gives the $R_s^{rep}$ expression in (\\ref{repeat}).\n\nNow, the difference between the capacity in (\\ref{result1}) and achievable rate in (\\ref{repeat}) is,\n\\begin{align}\nC_s^P-R_s^{rep}&=\\frac{PN}{P(N-1)+M}-\\frac{(N-1)(N^{M-1}+P-1)}{N^M-1}\\\\\n& =\\frac{\\eta(P,M,N)}{(N^M-1)(P(N-1)+M)}\n\\end{align}\nIt suffices to prove that $\\eta(P,M,N) \\geq 0$ for all $P$, $M$, $N$ when $P \\geq \\frac{M}{2}$ and $N \\geq 2$. Note,\n\\begin{align}\\label{corr0}\n\\eta(P,M,N)=&(2P-M)N^M+(M-P)N^{M-1} -P(P-1)N^2\\notag\\\\\n            &+((P-1)(2P-M)-P)N+(M-P)(P-1)\n\\end{align}\nIn the regime $P \\geq \\frac{M}{2}$, coefficients of $N^M,N^{M-1},N^0$ are non-negative. Denote the negative terms in $\\eta(\\cdot)$ by $\\nu(P,N)$ which is $\\nu(P,N)=P(P-1)N^2+PN$. We note $\\nu(P,N) < P^2N^2$ when $N>1$, which is the case here. Thus,\n\\begin{align}\n\\eta(P,M,N) \\geq &(2P-M)N^M+(M-P)N^{M-1} \\notag\\\\\n&+(P-1)(2P-M)N+(M-P)(P-1)-P^2N^2\\\\\n> &(2P-M)N^M+(M-P)N^{M-1}-P^2N^2\\\\\n= &N^2\\left((2P-M)N^{M-2}+(M-P)N^{M-3}-P^2\\right)\\\\\n\\label{corr1}\\geq & N^2\\left((2P-M)2^{M-2}+(M-P)2^{M-3}-P^2\\right)\\\\\n=& N^2\\left(2^{M-3}(3P-M)-P^2\\right) \\\\\n\\label{corr2} \\geq & N^2\\left(2^{M-3}\\cdot\\frac{M}{2}-M^2\\right)\\\\\n\\label{corr3} =&MN^2\\left(2^{M-4}-M\\right)\n\\end{align}\nwhere (\\ref{corr1}) follows from the fact that $(2P-M)N^{M-2}+(M-P)N^{M-3}-P^2$ is monotone increasing in $N \\geq 2$ for $M\\geq 3$, and (\\ref{corr2}) follows from $\\frac{M}{2} \\leq P \\leq M$. From (\\ref{corr3}), we conclude that $\\eta(M,P,N) > 0$ for all $M \\geq 7$, $P \\geq \\frac{M}{2}$ and $N \\geq 2$. Examining the expression in (\\ref{corr0}) for the remaining cases manually, i.e., when $M \\leq 6$, we note that $\\eta(M,P,N) > 0$ in these cases as well. Therefore, $\\eta(M,P,N) > 0$ for all possible cases, and the MPIR capacity is strictly larger than the rate achieved by repeating the optimum single-message PIR scheme.\n\\end{Proof}\n\nFor the example in the introduction, where $M=3$, $P=2$, $N=2$, our MPIR scheme achieves a sum capacity of $\\frac{4}{5}$ in (\\ref{result1}), which is strictly larger than the repeating-based achievable sum rate of $\\frac{5}{7}$ in (\\ref{repeat}).\n\nThe following corollary gives an achievable rate region for the MPIR problem.\n\n\\begin{corollary}\\label{corollary2}\nFor the MPIR problem, for the case $P \\geq \\frac{M}{2}$, the following rate region is achievable,\n\\begin{align}\\label{achievable_region}\n\\mathcal{C}=\\emph{conv} & \\left\\{ (C,\\delta, \\cdots, \\delta), (\\delta, C, \\cdots, \\delta), \\cdots, (\\delta, \\cdots, \\delta, C) , (C,0,0,\\cdots,0),\\right.\\notag\\\\\n&\\:\\:\\left. (0,C,0,\\cdots,0), \\cdots, (0,0,\\cdots,C), (0,0, \\cdots, 0), \\left(C^P,C^P, \\cdots, C^P\\right) \\right\\}\n\\end{align}\nwhere\n\\begin{align}\nC=\\frac{1-\\frac{1}{N}}{1-(\\frac{1}{N})^M}, \\qquad\nC^P=\\frac{C_s^P}{P}=\\frac{N}{PN+(M-P)}, \\qquad\n\\delta=\\frac{\\Delta(M,P,N)}{P-1}=\\frac{N-1}{N^M-1}\n\\end{align}\nand where $\\emph{conv}$ denotes the convex hull, and all corner points lie in the $P$-dimensional space.\n\\end{corollary}\n\n\\begin{Proof}\nThis is a direct consequence of Theorem~\\ref{thm1} and Corollary~\\ref{corrollary1}. The corner point  $\\left(C,\\frac{\\Delta(M,P,N)}{P-1},\\frac{\\Delta(M,P,N)}{P-1}, \\cdots, \\frac{\\Delta(M,P,N)}{P-1}\\right)=\\left(\\frac{1-\\frac{1}{N}}{1-(\\frac{1}{N})^M}, \\frac{N-1}{N^M-1},\\frac{N-1}{N^M-1}, \\cdots, \\frac{N-1}{N^M-1}\\right)$ is achievable from the single-message achievable scheme. Due to the symmetry of the problem any other permutation for the coordinates of this corner point is also achievable by changing the roles of the desired messages. Theorem~\\ref{thm1} gives the symmetric sum capacity corner point for the case of $P \\geq \\frac{M}{2}$, namely $\\left(\\frac{C_s^P}{P}, \\frac{C_s^P}{P}, \\cdots, \\frac{C_s^P}{P}\\right) =\\left(\\frac{N}{PN+(M-P)}, \\frac{N}{PN+(M-P)}, \\cdots, \\frac{N}{PN+(M-P)}\\right)$. By time sharing of these corner points along with the origin, the  region in (\\ref{achievable_region}) is achievable.\n\\end{Proof}\n\nAs an example for this achievable region, consider again the example in the introduction, where $M=3$, $P=2$, $N=2$. In this case, we have a two-dimensional rate region with three corner points: $(\\frac{4}{7},\\frac{1}{7})$, which corresponds to the single-message capacity achieving point that aims at retrieving $W_1$;  $(\\frac{1}{7},\\frac{4}{7})$, which corresponds to single-message capacity achieving point that aims at retrieving $W_2$; and $(\\frac{2}{5},\\frac{2}{5})$, which corresponds to the symmetric sum capacity point. The convex hull of these corner points together with the points on the axes gives the achievable region in Fig.~\\ref{fig:region322}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{MPIR_region322.eps}\n\\caption{The achievable rate region of $M=3$, $P=2$, $N=2$.}\n\\label{fig:region322}\n\\end{figure}\n\nFor the case $P \\leq \\frac{M}{2}$, we have the following result, where the lower and upper bound match if $\\frac{M}{P} \\in \\mathbb{N}$.\n\n\\begin{theorem}\\label{thm2}\nFor the MPIR problem with non-colluding and replicated databases, when $P \\leq \\frac{M}{2}$, the sum capacity is lower and upper bounded as,\n\\begin{align}\\label{bounds}\n\\underaccent{\\bar}{R}_s \\leq C_s^P \\leq \\bar{R}_s\n\\end{align}\nwhere the upper bound $\\bar{R}_s$ is given by,\n\\begin{align}\\label{upper_bound}\n\\bar{R}_s &=\\frac{1}{1+\\frac{1}{N}+\\cdots+\\frac{1}{N^{\\lfloor\\frac{M}{P}\\rfloor-1}}+\\left(\\frac{M}{P}-\\lfloor\\frac{M}{P}\\rfloor\\right)\\frac{1}{N^{\\lfloor \\frac{M}{P}\\rfloor}}} \\\\ &=\\frac{1}{\\frac{1-(\\frac{1}{N})^{\\lfloor\\frac{M}{P}\\rfloor}}{1-\\frac{1}{N}}+\\left(\\frac{M}{P}-\\left\\lfloor\\frac{M}{P}\\right\\rfloor\\right)\\frac{1}{N^{\\left\\lfloor \\frac{M}{P}\\right\\rfloor}}}\n\\end{align}\nFor the lower bound, define $r_i$ as,\n\\begin{align}\\label{root}\nr_i=\\frac{e^{j2\\pi(i-1)\/P}}{N^{1\/P}-e^{j2\\pi(i-1)\/P}}, \\quad i=1,\\cdots, P\n\\end{align}\nwhere $j=\\sqrt{-1}$, and denote $\\gamma_i, \\: i=1, \\cdots, P$, to be the solutions of the linear equations $\\sum_{i=1}^P \\gamma_i r_i^{-P}=(N-1)^{M-P}$, and $\\sum_{i=1}^P \\gamma_i r_i^{-k}=0, \\: k=1, \\cdots, P-1$, then $\\underaccent{\\bar}{R}_s$ is given by,\n\\begin{align} \\label{ach-rate}\n\\underaccent{\\bar}{R}_s=\\frac{\\sum_{i=1}^{P}\\gamma_i r_i^{M-P}\\left[\\left(1+\\frac{1}{r_i}\\right)^M-\\left(1+\\frac{1}{r_i}\\right)^{M-P}\\right]}{\\sum_{i=1}^{P}\\gamma_i r_i^{M-P}\\left[\\left(1+\\frac{1}{r_i}\\right)^M-1\\right]}\n\\end{align}\n\\end{theorem}\n\nThe achievability lower bound in Theorem~\\ref{thm2} is shown in Section~\\ref{sec:achievable2} and the upper bound is derived in Section~\\ref{converse2}. The following corollary states that the bounds in Theorem~\\ref{thm2} match if the total number of messages is an integer multiple of the number of desired messages.\n\n\\begin{corollary}\\label{corollary3}\nFor the MPIR problem  with non-colluding and replicated databases, if $\\frac{M}{P}$ is an integer, then the bounds in (\\ref{bounds}) match, and hence,\n\\begin{align}\nC_s^P=\\frac{1-\\frac{1}{N}}{1-(\\frac{1}{N})^{\\frac{M}{P}}}, \\quad \\frac{M}{P} \\in \\mathbb{N}\n\\end{align}\n\\end{corollary}\n\n\\begin{Proof}\nFor the upper bound, observe that if $\\frac{M}{P} \\in \\mathbb{N}$, then $\\frac{M}{P}=\\left\\lfloor\\frac{M}{P}\\right\\rfloor$. Hence, (\\ref{upper_bound}) becomes\n\\begin{align}\n\\bar{R}_s=\\frac{1-\\frac{1}{N}}{1-(\\frac{1}{N})^{\\frac{M}{P}}}\n\\end{align}\nFor the lower bound, consider the case $\\frac{M}{P} \\in \\mathbb{N}$. From (\\ref{root}),\n\\begin{align}\n\\left(1+\\frac{1}{r_i}\\right)^M=\\left(\\frac{N^{1\/P}}{e^{j2\\pi(i-1)\/P}}\\right)^M=N^{\\frac{M}{P}}\n\\end{align}\nsince $e^{j2\\pi(i-1)M\/P}=1$ for $\\frac{M}{P} \\in \\mathbb{N}$. Similarly, $\\left(1+\\frac{1}{r_i}\\right)^{M-P}=N^{\\frac{M}{P}-1}$. Hence, if  $\\frac{M}{P} \\in \\mathbb{N}$,\n\\begin{align}\n\\underaccent{\\bar}{R}_s&=\\frac{\\sum_{i=1}^{P}\\gamma_i r_i^{M-P}\\left[N^{\\frac{M}{P}}-N^{\\frac{M}{P}-1}\\right]}{\\sum_{i=1}^{P}\\gamma_i r_i^{M-P}\\left[N^{\\frac{M}{P}}-1\\right]}\\\\\n&=\\frac{N^{\\frac{M}{P}}-N^{\\frac{M}{P}-1}}{N^{\\frac{M}{P}}-1} \\\\\n&=\\frac{1-\\frac{1}{N}}{1-(\\frac{1}{N})^{\\frac{M}{P}}}\n\\end{align}\nThus, $\\underaccent{\\bar}{R}_s=C_s^P=\\bar{R}_s$ if $\\frac{M}{P} \\in \\mathbb{N}$, and we have an exact capacity result in this case.\n\\end{Proof}\n\nExamining the result, we observe that when the total number of messages is an integer multiple of the number of  desired messages, the sum capacity of the MPIR is the same as the capacity of the single-message PIR with the number of messages equal to $\\frac{M}{P}$. Note that, although at first the result may seem as if every $P$ messages can be lumped together as a single message, and the achievable scheme in \\cite{JafarPIR} can be used, this is not the case. The reason for this is that, we need to ensure the privacy constraint for \\emph{every subset} of messages of size $P$. That is why, in this paper, we develop a new achievable scheme.\n\nThe state of the results is summarized in Fig.~\\ref{fig:main_result}: Consider the $(M,P)$ plane, where naturally $M\\geq P$. The valid part of the plane is divided into two regions. The first region is confined between the lines $P=\\frac{M}{2}$ and $P=M$; the sum capacity in this region is exactly characterized (Theorem~\\ref{thm1}). The second region is confined between the lines $P=1$ and $P=\\frac{M}{2}$; the sum capacity in this region is characterized only for the cases when $\\frac{M}{P} \\in \\mathbb{N}$ (Corollary~\\ref{corollary3}). The line $P=1$ corresponds to the previously known result for the single-message PIR \\cite{JafarPIR}. The exact capacity for the rest of the cases is still an open problem; however, the achievable scheme in Theorem~\\ref{thm2} yields near-optimal sum rates for all the remaining cases with the largest difference of $0.0082$ from the upper bound, as discussed next.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{MPIR_result.eps}\n\\caption{Summary of the state of the results.}\n\\label{fig:main_result}\n\\end{figure}\n\nFig.~\\ref{fig:3D_error} shows the difference of the achievable rate $\\underaccent{\\bar}{R}_s$ and the upper bound $\\bar{R}_s$ in Theorem~\\ref{thm2}. The figure shows that the difference decreases as $N$ increases. This difference in all cases is small and is upper bounded by $0.0082$, which occurs when $M=5$, $P=2$, $N=2$. In addition, the difference is zero for the cases $P \\geq \\frac{M}{2}$ (Theorem~\\ref{thm1}) or $\\frac{M}{P} \\in \\mathbb{N}$ (Corollary~\\ref{corollary3}).\n\n\\begin{figure}[t]\n\\centering\n\\begin{subfigure}[b]{0.48\\textwidth}\n\\includegraphics[width=\\textwidth]{deviation2.eps}\n\\caption{$N=2$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.48\\textwidth}\n\\includegraphics[width=\\textwidth]{deviation5.eps}\n\\caption{$N=5$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.48\\textwidth}\n\\includegraphics[width=\\textwidth]{deviation10.eps}\n\\caption{$N=10$}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.48\\textwidth}\n\\includegraphics[width=\\textwidth]{deviation20.eps}\n\\caption{$N=20$}\n\\end{subfigure}\n\\caption{Deviation of the achievable sum rate from the upper bound.}\n\\label{fig:3D_error}\n\\end{figure}\n\nFig.~\\ref{fig:M_slice} shows the effect of changing $M$ for fixed $(P,N)$. We observe that as $M$ increases, the sum rate monotonically decreases and has a limit of $1-\\frac{1}{N}$. In addition, Fig.~\\ref{fig:N_slice} shows the effect of changing $N$ for fixed $(P,M)$. We observe that as $N$ increases, the sum rate increases and approaches $1$, as expected.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{effectM.eps}\n\\caption{Effect of changing $M$ for fixed $P=5,6,10$ and fixed $N=2$.}\n\\label{fig:M_slice}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{effectN.eps}\n\\caption{Effect of changing $N$ for fixed $(M,P)=(5,2),(10,5),(20,3)$.}\n\\label{fig:N_slice}\n\\end{figure}\n\n\\section{Achievability Proof for the Case $P \\geq \\frac{M}{2}$} \\label{sec:achievable1}\n\nIn this section, we present the general achievable scheme that attains the upper bound for the case $P \\geq \\frac{M}{2}$. The scheme applies the concepts of message symmetry, database symmetry, and exploiting side information as in \\cite{JafarPIR}. However, our scheme requires the extra ingredient of MDS coding of the desired symbols and the side information in its second stage.\n\n\\subsection{Motivating Example: $M=3$, $P=2$ Messages, $N=2$ Databases} \\label{mot1}\n\nWe start with a simple motivating example in this sub-section. The scheme operates over message size $N^2=4$. For sake of clarity, we assume that the three messages after interleaving their indices are $W_1=(a_1, \\cdots, a_4)^T$, $W_2=(b_1, \\cdots, b_4)^T$, and $W_3=(c_1, \\cdots, c_4)^T$. We use $\\bg_{2 \\times 3}$ Reed-Solomon generator matrix over $\\mathbb{F}_3$ as\n\\begin{align}\n\\bg_{2 \\times 3}=\\begin{bmatrix}\n1 & 1 & 1 \\\\\n1 & 2 & 3\n\\end{bmatrix}\n\\end{align}\nThe user picks a random permutation for the columns of $\\bg_{2 \\times 3}$ from the 6 possible permutations, e.g., in this example we use the permutation $2,1,3$. In the first round, the user starts by downloading one symbol from each database and each message, i.e., the user downloads $(a_1,b_1,c_1)$ from the first database, and $(a_2,b_2,c_2)$ from the second database. In the second round, the user encodes the side information from database 2 which is $c_2$ with two new symbols from $W_1,W_2$ which are $(a_3,b_3)$ using the permuted generator matrix, i.e., the user downloads two equations from database 1 in the second round,\n\\begin{align}\n\\bg\\bs_1\\begin{bmatrix}\na_3\\\\b_3\\\\c_2\n\\end{bmatrix}=\n\\begin{bmatrix}\n1 & 1 & 1 \\\\\n1 & 2 & 3\n\\end{bmatrix}\n\\begin{bmatrix}\n0 & 1 & 0 \\\\\n1 & 0 & 0 \\\\\n0 & 0 & 1\n\\end{bmatrix}\n\\begin{bmatrix}\na_3\\\\b_3\\\\c_2\n\\end{bmatrix}\n=\\begin{bmatrix}\na_3+b_3+c_2 \\\\\n2a_3+b_3+3c_2\n\\end{bmatrix}\n\\end{align}\nThe user repeats this operation for the second database with $(a_4,b_4)$ as desired symbols and $c_1$ as the side information from the first database.\n\nFor the decodability: The user subtracts out $c_2$ from round two in the first database, then the user can decode $(a_3,b_3)$ from $a_3+b_3$ and $2a_3+b_3$. Similarly, by subtracting out $c_1$ from round two in the second database, the user can decode $(a_4,b_4)$ from $a_4+b_4$ and $2a_4+b_4$.\n\nFor the privacy: Single bit retrievals of $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ from the two databases in the first round satisfy message symmetry and database symmetry, and do not leak any information. In addition, due to the private shuffling of bit indices, the different coefficients of 1, 2 and 3 in front of the bits in the MDS-coded summations in the second round do not leak any information either; see a formal proof in Section~\\ref{dec-pri}. To see the privacy constraint intuitively from another angle, we note that the user can alter the queries for the second database when the queries for the first database are fixed, when the user wishes to retrieve another set of two messages. For instance, if the user wishes to retrieve $(W_1,W_3)$ instead of $(W_1,W_2)$, it can alter the queries for the second database by changing every $c_2$ in the queries of the second database with $c_3$, $c_1$ with $c_4$, $b_2$ with $b_3$, and $b_4$ with $b_1$.\n\nThe query table for this case is shown in Table~\\ref{table(3,2,2)} below. The scheme retrieves $a_1, \\cdots, a_4$ and $b_1,\\cdots, b_4$, i.e., 8 bits in 10 downloads (5 from each database). Thus, the achievable sum rate for this scheme is $\\frac{8}{10}=\\frac{4}{5}=\\frac{1}{1+\\frac{M-P}{PN}}$. If we use the single-message optimal scheme in \\cite{JafarPIR}, which is given in \\cite[Example~4.3]{JafarPIR} for this specific case, twice in a row to retrieve two messages, we achieve a sum rate of $\\frac{20}{28}=\\frac{5}{7} < \\frac{4}{5}$ as discussed in the introduction.\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{The query table for the case $M=3,P=2,N=2$.}\n\t\\label{table(3,2,2)}\n\t\\begin{tabular}{|c|c|}\n\t\t\\hline\n\t\tDatabase 1 & Database 2 \\\\\n\t\t\\hline\n\t\t$a_1,b_1,c_1$ & $a_2,b_2,c_2$ \\\\\n\t\t\\hline\n\t\t$a_3+b_3+c_2$ & $a_4+b_4+c_1$ \\\\\n\t\t$2a_3+b_3+3c_2$& $2a_4+b_4+3c_1$\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsection{General Achievable Scheme}\n\nThe scheme requires $L=N^2$, and is completed in two rounds. The main ingredient of the scheme is MDS coding of the desired symbols and side information in the second round. The details of the scheme are as follows.\n\n\\begin{enumerate}\n\\item \\textit{Index preparation:} The user interleaves the contents of each message randomly and independently from the remaining messages using a random interleaver $\\pi_m(.)$ which is known privately to the user only, i.e.,\n\t\\begin{align}\n\tx_m(i)=w_m(\\pi_m(i)), \\quad i \\in \\{1, \\cdots, L\\}\n\t\\end{align}\n\twhere $X_m=[x_m(1), \\cdots, x_m(L)]^T$ is the interleaved message. Thus, the downloaded symbol $x_m(i)$ at any database appears to be chosen at random and independent from the desired message subset $\\cp$.\n\t\n\t\\item \\textit{Round one:} As in \\cite{JafarPIR}, the user downloads one symbol from every message from every database, i.e., the user downloads $(x_{1}(n), x_{2}(n), \\cdots, x_{M}(n))$ from the $n$th database. This implements \\textit{message symmetry}, \\textit{symmetry across databases} and satisfies the privacy constraint.\n\t\n\t\\item \\textit{Round two:} The user downloads a coded mixture of new symbols from the desired messages and the undesired symbols downloaded from the other databases. Specifically,\n\t\n    \\begin{enumerate}\n\t\t\\item The user picks an MDS generator matrix $\\bg \\in \\mathbb{F}_q^{P \\times M}$, which has the property that every $P \\times P$ submatrix is full-rank. This implies that if the user can cancel out any $M-P$ symbols from the mixture, the remaining symbols can be decoded. One explicit MDS generator matrix is the Reed-Solomon generator matrix over $\\mathbb{F}_q$, where $q>M$, \\cite{Reed_Solomon,wicker1999reed}\n\t\t\\begin{align}\n\t\t\\bg=\\begin{bmatrix}\n\t\t1 & 1 & 1 & \\cdots & 1 \\\\\n\t\t1 & 2 & 3 & \\cdots & M \\\\\n\t\t1^2&2^2&3^2& \\cdots & M^2 \\\\\n\t\t\\vdots&\\vdots&\\vdots& \\vdots&\\vdots \\\\\n\t\t1^{P-1}&2^{P-1}&3^{P-1}& \\cdots &M^{P-1}\n\t\t\\end{bmatrix}_{P \\times M}\n\t\t\\end{align}\n\t\t\n\t\t\\item The user picks uniformly and independently at random the permutation matrices $\\bs_1, \\bs_2, \\cdots, \\bs_{N-1}$ of size $M \\times M$. These matrices shuffle the order of columns of $\\bg$ to be independent of $\\cp$.\n\t\t\n\t\t\\item At the first database, the user downloads an MDS-coded version of $P$ new symbols from the desired set $\\cp$ and $M-P$ undesired symbols that are already decoded from the second database in the first round, i.e., the user downloads $P$ equations of the form\n\t\t\\begin{align}\n\t\t\\bg\\bs_1 \t\\begin{bmatrix}\n\t\tx_{i_1}(n+1) & x_{i_2}(n+1) & \\cdots & x_{i_P} (n+1) & x_{j_1}(2) & x_{j_2}(2) & \\cdots & x_{j_{M-P}}(2)\n\t\t\\end{bmatrix}^T\n\t\t\\end{align}\n\t\twhere $\\cp=\\{i_1,i_2, \\cdots, i_P\\}$ are the indices of the desired messages and $\\bar{\\cp}=\\{j_1,j_2, \\cdots, j_{M-P}\\}$ are the indices of the undesired messages. In this case, the user can cancel out the undesired messages and be left with a $P \\times P$ invertible system of equations that it can solve to get $[x_{i_1}(n+1), x_{i_2}(n+1),\\cdots,x_{i_P} (n+1)]$. This implements \\textit{exploiting side information} as in \\cite{JafarPIR}.\n\t\t\n\t\t\\item The user repeats the last step for each set of side information from database 3 to database $N$, each with different permutation matrix.\n\t\t\n\t\t\\item By \\textit{database symmetry}, the user repeats all steps of round two at all other databases.\n\t\\end{enumerate}\n\\end{enumerate}\n\n\\subsection{Decodability, Privacy, and Calculation of the Achievable Rate} \\label{dec-pri}\n\nNow, we verify that this achievable scheme satisfies the reliability and privacy constraints.\n\nFor the reliability: The user gets individual symbols from all databases in the first round, and hence they are all decodable by definition. In the second round, the user can subtract out all the undesired message symbols using the undesired symbols downloaded from all other databases during the first round. Consequently, the user is left with a $P \\times P$ system of equations which is guaranteed to be invertible by the MDS property, hence all symbols that belong to $W_\\cp$ are decodable.\n\nFor the privacy: At each database, for every message subset $\\cp$ of size $P$, the achievable scheme retrieves randomly interleaved symbols which are encoded by the following matrix:\n\\begin{align}\n\\mathbf{H}_\\cp=\\begin{bmatrix}\n\\mathbf{I}_P & \\mathbf{0}_P & \\mathbf{0}_P & \\cdots & \\mathbf{0}_P \\\\\n\\mathbf{0}_P   & \\bg^1_\\cp  & \\mathbf{0}_P & \\cdots & \\mathbf{0}_P \\\\\n\\mathbf{0}_P   & \\mathbf{0}_P & \\bg^2_\\cp  & \\cdots & \\mathbf{0}_P \\\\\n\\vdots     &  \\vdots    &   \\vdots&   \\vdots & \\vdots \\\\\n\\mathbf{0}_P& \\mathbf{0}_P& \\mathbf{0}_P& \\cdots& \\bg^{N-1}_\\cp\n\\end{bmatrix}\n\\end{align}\nwhere $\\bg^n_\\cp=\\bg\\bs_n(:,\\cp)$ are the columns of the encoding matrix that correspond to the message subset $\\cp$ after applying the random permutation $\\bs_n$. Since the permutation matrices are chosen uniformly and independently from each other, the probability distribution of $\\mathbf{H}_\\cp$ is uniform irrespective to $\\cp$ (the probability of realizing such a matrix is $\\left(\\frac{(M-P)!}{M!}\\right)^{N-1}$). Furthermore, the symbols are chosen randomly and uniformly by applying the random interleaver. Hence, the retrieval scheme is private.\n\nTo calculate the achievable rate: We note that at each database, the user downloads $M$ individual symbols in the first round that includes $P$ desired symbols. The user exploits the side information from the remaining $(N-1)$ databases to generate $P$ equations for each side information set. Each set of $P$ equations in turn generates $P$ desired symbols. Hence, the achievable rate is calculated as,\n\\begin{align}\n\\sum_{i=1}^P R_i&=\\frac{\\text{total number of desired symbols}}{\\text{total downloaded equations}} \\\\\n                &=\\frac{N(P+P(N-1))}{N(M+P(N-1))} \\\\\n                &=\\frac{PN}{(M-P)+PN} \\\\\n                &=\\frac{1}{1+\\frac{M-P}{PN}}\n\\end{align}\n\n\\subsection{Further Examples for the Case $P \\geq \\frac{M}{2}$}\n\nIn this section, we illustrate our achievable scheme with two more basic examples. In Section~\\ref{mot1}, we considered the case $M=3$, $P=2$, $N=2$. In the next two sub-sections, we will consider examples with larger $M$, $P$ (Section~\\ref{ex1}), and larger $N$ (Section~\\ref{ex2}).\n\n\\subsubsection{$M=5$ Messages, $P=3$ Messages, $N=2$ Databases} \\label{ex1}\n\nLet $\\cp=\\{1,2,3\\}$, and $a$ to $e$ denote the contents of $W_1$ to $W_5$, respectively. The achievable scheme is similar to the example in Section~\\ref{mot1}. The difference is the use $5 \\times 5$ permutation matrix for $\\bs_1$ and $\\bg_{3 \\times 5}$ Reed-Solomon generator matrix over $\\mathbb{F}_5$ as:\n\\begin{align}\n\\bg_{3 \\times 5}=\\begin{bmatrix}\n1 & 1 & 1  & 1 & 1\\\\\n1 & 2 & 3  & 4 & 5\\\\\n1 & 4 & 4  & 1 & 0\n\\end{bmatrix}\n\\end{align}\nThe query table is shown in Table~\\ref{table(5,3,2)} below with the following random permutation for the columns: $2,5,1,3,4$. The reliability and privacy constraints are satisfied due to the MDS property that implies that any subset of $3$ messages corresponds to a $3 \\times 3$ invertible submatrix if the remaining symbols are decodable from the other database. This scheme retrieves $a_1,\\cdots,a_4$, $b_1,\\cdots,b_4$ and $c_1,\\cdots,c_4$, hence 12 bits in 16 downloads (8 from each database). Thus, the achievable sum rate is $\\frac{12}{16}=\\frac{3}{4}$ which equals the sum capacity $\\frac{1}{1+\\frac{M-P}{PN}}$ in (\\ref{result1}). This strictly outperforms the repetition-based achievable sum rate $\\frac{18}{31}$ in (\\ref{repeat}).\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{The query table for the case $M=5,P=3,N=2$.}\n\t\\label{table(5,3,2)}\n\t\\begin{tabular}{|c|c|}\n\t\t\\hline\n\t\tDatabase 1 & Database 2 \\\\\n\t\t\\hline\n\t\t$a_1,b_1,c_1,d_1,e_1$ & $a_2,b_2,c_2,d_2,e_2$ \\\\\n\t\t\\hline\n\t\t$a_3+b_3+c_3+d_2+e_2$ & $a_4+b_4+c_4+d_1+e_1$ \\\\\n\t\t$2a_3+5b_3+c_3+3d_2+4e_2$& $2a_4+5b_4+c_4+3d_1+4e_1$\\\\\n\t\t$4a_3+c_3+4d_2+e_2$& $4a_4+c_4+4d_1+e_1$\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsubsection{$M=4$ Messages, $P=2$ Messages, $N=3$ Databases}\\label{ex2}\n\nNext, we give an example with a larger $N$. Here, the message size is $N^2=9$. With a generator matrix $\\bg_{2 \\times 4}=\\bg_{3 \\times 5}([1:2],[1:4])$ to be the upper left submatrix of the previous example and two set of random permutations (corresponding to $\\bs_1,\\bs_2$) as $1,3,2,4$, and $4,1,3,2$. The query table is shown in Table~\\ref{table(4,2,3)} below. This scheme retrieves $a_1,\\cdots,a_9$ and $b_1,\\cdots,b_9$, hence 18 bits in 24 downloads (8 from each database). Thus, the achievable rate is $\\frac{18}{24}=\\frac{3}{4}=\\frac{1}{1+\\frac{M-P}{PN}}$. This strictly outperforms the repetition-based achievable scheme sum rate $\\frac{7}{10}$ in (\\ref{repeat}).\n\n\\begin{table}\n\t\\centering\n\t\\caption{The query table for the case $M=4,P=2,N=3$.}\n\t\\label{table(4,2,3)}\n\t\\begin{tabular}{|c|c|c|}\n\t\t\\hline\n\t\tDatabase 1 & Database 2 & Database 3\\\\\n\t\t\\hline\n\t\t$a_1,b_1,c_1,d_1$ & $a_2,b_2,c_2,d_2$& $a_3,b_3,c_3,d_3$\\\\\n\t\t\\hline\n\t\t$a_4+b_4+c_2+d_2$ & $a_6+b_6+c_1+d_1$&$a_8+b_8+c_1+d_1$ \\\\\n\t\t$a_4+3b_4+2c_2+4d_2$ & $a_6+3b_6+2c_1+4d_1$&$a_8+3b_8+2c_1+4d_1$ \\\\\n\t\t$a_5+b_5+c_3+d_3$ & $a_7+b_7+c_3+d_3$&$a_9+b_9+c_2+d_2$ \\\\\n\t\t$4a_5+b_5+3c_3+2d_3$ & $4a_7+b_7+3c_3+2d_3$&$4a_9+b_9+3c_2+2d_2$ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\section{Achievability Proof for the Case $P \\leq \\frac{M}{2}$}\\label{sec:achievable2}\n\nIn this section, we describe an achievable scheme for the case $P \\leq \\frac{M}{2}$. We show that this scheme is optimal when the total number of messages $M$ is an integer multiple of the number of desired messages $P$. The scheme incurs a small loss from the upper bound for all other cases. The scheme generalizes the ideas in \\cite{JafarPIR}. Different than \\cite{JafarPIR}, our scheme uses unequal number of stages for each round of download. Interestingly, the number of stages at each round can be thought of as the output of an all-poles IIR filter. Our scheme reduces to \\cite{JafarPIR} if we let $P=1$. In the sequel, we define the $i$th round as the download queries that retrieve sum of $i$ different symbols. We define the stage as a block of queries that exhausts all $\\binom{M}{i}$ combinations of the sum of $i$ symbols in the $i$th round.\n\n\\subsection{Motivating Example: $M=5$, $P=2$ Messages, $N=2$ Databases} \\label{mot2}\n\nTo motivate our achievable scheme, consider the case of retrieving two messages denoted by letters $(a,b)$ from five stored messages denoted by letters $(a, b, c, d, e)$. Instead of designing the queries beginning from the top as usual, i.e., beginning by downloading individual symbols, we design the scheme backwards starting from the last round that corresponds to downloading sums of all five messages and trace back to identify the  side information needed at each round from the other database. Our steps described below can be followed through in the query table in Table~\\ref{table(5,2,2)}.\n\nNow, let us fix the number of stages in the 5th round to be 1 as in \\cite{JafarPIR} since $N=2$. Round 5 corresponds to downloading the sum of all five messages and contains one combination of symbols $a+b+c+d+e$; please see the last line in Table~\\ref{table(5,2,2)}. Since we wish to retrieve $(a,b)$, we need one side information equation in the form of $c+d+e$ from earlier rounds. The combination $c+d+e$ can be created directly from round 3 without using round 4. Hence, we suppress round 4, as it does not create any useful side information in our case, and download one stage from round 3 to generate one side information equation $c+d+e$.\n\nIn round 3, we download sums of $3$ messages. Each stage of round 3 consists of $\\binom{5}{3}=10$ equations. One of those 10 equations is in the desired $c+d+e$ form, and the remaining 9 of them have either $a$ or $b$ or both $a,b$ in them. In tabulating all these 9 combinations, we recognize two categories of side information equations needed from earlier rounds. The first category corresponds to equations of the form $a+b+(c,d,e)$, where $(c,d,e)$ means possible choices for the rest of the equation, i.e., these equations have both $a$ and $b$ in them and plus one more symbol in the form of $c$ or $d$ or $e$. This category requires downloading one stage of individual symbols (i.e., an individual $c$ or $d$ or $e$), that is, one stage of round 1. We note also that one of the symbols $(a,b)$ should be known as a side information from the second database in order to solve for the remaining new symbol. The second category corresponds to equations of the form $a+(c+d,c+e,d+e)$ and $b+(c+d,c+e,d+e)$, i.e., these equations have only one of $a$ or $b$ but not both. This category requires two stages of round 2, as we need different side information equations that contain sum of twos, e.g., $c+d$, $c+e$, $d+e$.\n\nIn round 2, we download sums of $2$ messages. Each stage of the second round contains $\\binom{5}{2}=10$ equations. In each stage, we need one category of side information equations, which is $a+(c,d,e)$ and $b+(c,d,e)$. This necessitates two different stages of individual symbols, i.e., two stages of round 1 for each stage of round 2.\n\nDenoting $\\alpha_i$ to be the number of stages needed for the $i$th round, we sum all the required stages for round 1 to be $\\alpha_1=2\\cdot2+1=5$ stages. Hence, the user identifies the number stages as $\\alpha_1=5, \\alpha_2=2, \\alpha_3=1, \\alpha_4=0, \\alpha_5=1$. These can be observed in the query table in Table~\\ref{table(5,2,2)}. Note that, we have $\\alpha_1=5$ stages in round 1 where we download individual bits; then we have $\\alpha_2=2$ stages in round 2 where we download sums of two symbols; then we have $\\alpha_3=1$ stage in round 3 where we download sums of three symbols; we skip round 4 as $\\alpha_4=0$; and we have $\\alpha_5=1$ stage of round 5 where we download sum of all five symbols.\n\nNow, after designing the structure of the queries and the number of stages needed for each round, we apply the rest of the scheme described in \\cite{JafarPIR}. The user randomly interleaves the messages as usual. In the first round, the user downloads one symbol from each message at each database. This is repeated $\\alpha_1=5$ times for each database. Hence, the user downloads $a_{1:10}, b_{1:10},c_{1:10},d_{1:10},e_{1:10}$ from the two databases. In the second round, the user downloads sums of two messages. Each stage contains $\\binom{5}{2}=10$ equations. This is repeated $\\alpha_2=2$ times. For example, in database 1, user exploits $c_6,d_6,e_6$ to get $a_{12},a_{13},a_{14}$ and $c_7,d_7,e_7$ to obtain $b_{11},b_{12},b_{13}$. These are from round 1. Round 2 generates $c_{11}+d_{11}$, $c_{12}+e_{11}$, $d_{12}+e_{12}$ from stage 1, and $c_{13}+d_{13}$, $c_{14}+e_{13}$, $d_{14}+e_{14}$ from stage 2 as side information for round 3. In round 3, the user downloads sum of three symbols. There are $\\binom{5}{3}=10$ of them. Symbols $c_{10},d_{10},e_{10}$ downloaded from round 1 in database 2 are used to be summed with mixtures of $a+b$. The two sets of side information generated in the second round are exploited in the equations that have one $a$ or $b$. Note that for each such equation, one of $a$ or $b$ is new and the other one is decoded from database 2. Round 3 generates one side information as $c_{19}+d_{19}+e_{19}$ that is used in round 5. This last round includes the sum of all five messages.\n\nTherefore, as seen in Table~\\ref{table(5,2,2)}, we have retrieved $a_1,\\cdots,a_{34}$ and $b_1\\cdots,b_{34}$, i.e., 68 bits in a total of 112 downloads (56 from each database). Thus, the achievable sum rate is $\\frac{68}{112}=\\frac{17}{28}$. This is $\\underaccent{\\bar}{R}_s$ in Theorem~\\ref{thm2}, whereas the upper bound $\\bar{R}_s$ in Theorem~\\ref{thm2} is   $\\frac{1}{1+\\frac{1}{N}+\\frac{1}{2N^2}}=\\frac{8}{13}$. The gap between $\\underaccent{\\bar}{R}_s$ and $\\bar{R}_s$ is equal to $\\frac{3}{364}\\simeq 0.0082$, which also is the largest possible gap between $\\underaccent{\\bar}{R}_s$ and $\\bar{R}_s$ over all possible values of $M$, $P$ and $N$.\n\n\\begin{table}[]\n\t\\centering\n\t\\caption{The query table for the case $M=5,P=2,N=2$.}\n\t\\label{table(5,2,2)}\n\t\\begin{tabular}{|l|l|c|c|}\n\t\t\\hline\n\t\t&                     & Database 1                          & Database 2                           \\\\ \\hline\n\t\t\\multirow{5}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering round 1}}}  & stg~1                        & $a_1,b_1,c_1,d_1,e_1$               & $a_6,b_6,c_6,d_6,e_6$                \\\\ \\cline{2-4}\n\t\t& stg~2                        & $a_2,b_2,c_2,d_2,e_2$               & $a_7,b_7,c_7,d_7,e_7$                \\\\ \\cline{2-4}\n\t\t& stg~3                        & $a_3,b_3,c_3,d_3,e_3$               & $a_8,b_8,c_8,d_8,e_8$                \\\\ \\cline{2-4}\n\t\t& stg~4                        & $a_4,b_4,c_4,d_4,e_4$               & $a_9,b_9,c_9,d_9,e_9$                \\\\ \\cline{2-4}\n\t\t& stg~5                        & $a_5,b_5,c_5,d_5,e_5$               & $a_{10},b_{10},c_{10},d_{10},e_{10}$ \\\\ \\hline\\hline\n\t\t\\multirow{20}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering round 2}}} & \\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 1}}} & $a_{11}+b_6$                        & $a_{18}+b_1$                         \\\\\n\t\t&                           & $a_{12}+c_6$                        & $a_{19}+c_1$                         \\\\\n\t\t&                           & $a_{13}+d_6$                        & $a_{20}+d_1$                         \\\\\n\t\t&                           & $a_{14}+e_6$                        & $a_{21}+e_1$                         \\\\\n\t\t&                           & $b_{11}+c_7$                        & $b_{18}+c_2$                         \\\\\n\t\t&                           & $b_{12}+d_7$                        & $b_{19}+d_2$                         \\\\\n\t\t&                           & $b_{13}+e_7$                        & $b_{20}+e_2$                         \\\\\n\t\t&                           & $c_{11}+d_{11}$                     & $c_{15}+d_{15}$                      \\\\\n\t\t&                           & $c_{12}+e_{11}$                     & $c_{16}+e_{15}$                      \\\\\n\t\t&                           & $d_{12}+e_{12}$                     & $d_{16}+e_{16}$                      \\\\ \\cline{2-4}\n\t\t& \\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 2}}} & $a_{6}+b_{14}$                      & $a_{1}+b_{21}$                       \\\\\n\t\t&                           & $a_{15}+c_8$                        & $a_{22}+c_3$                         \\\\\n\t\t&                           & $a_{16}+d_8$                        & $a_{23}+d_3$                         \\\\\n\t\t&                           & $a_{17}+e_8$                        & $a_{24}+e_3$                         \\\\\n\t\t&                           & $b_{15}+c_9$                        & $b_{22}+c_4$                         \\\\\n\t\t&                           & $b_{16}+d_9$                        & $b_{23}+d_4$                         \\\\\n\t\t&                           & $b_{17}+e_9$                        & $b_{24}+e_4$                         \\\\\n\t\t&                           & $c_{13}+d_{13}$                     & $c_{17}+d_{17}$                      \\\\\n\t\t&                           & $c_{14}+e_{13}$                     & $c_{18}+e_{17}$                      \\\\\n\t\t&                           & $d_{14}+e_{14}$                     & $d_{18}+e_{18}$                      \\\\ \\hline\\hline\n\t\t\\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering round 3}}} & \\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 1}}} & $a_{25}+b_{7}+c_{10}$               & $a_{2}+b_{29}+c_{5}$                 \\\\\n\t\t&                           & $a_{7}+b_{25}+d_{10}$               & $a_{30}+b_{2}+d_{5}$                 \\\\\n\t\t&                           & $a_{26}+b_{8}+e_{10}$               & $a_{3}+b_{30}+e_{5}$                 \\\\\n\t\t&                           & $a_{27}+c_{15}+d_{15}$              & $a_{31}+c_{11}+d_{11}$               \\\\\n\t\t&                           & $a_{28}+c_{16}+e_{15}$              & $a_{32}+c_{12}+e_{11}$               \\\\\n\t\t&                           & $a_{29}+d_{16}+e_{16}$              & $a_{33}+d_{12}+e_{12}$               \\\\\n\t\t&                           & $b_{26}+c_{17}+d_{17}$              & $b_{31}+c_{13}+d_{13}$               \\\\\n\t\t&                           & $b_{27}+c_{18}+e_{17}$              & $b_{32}+c_{14}+e_{13}$               \\\\\n\t\t&                           & $b_{28}+d_{18}+e_{18}$              & $b_{33}+d_{14}+e_{14}$               \\\\\n\t\t&                           & $c_{19}+d_{19}+e_{19}$              & $c_{20}+d_{20}+e_{20}$               \\\\ \\hline\\hline\n\t\t\\rotatebox[origin=c]{90}{\\parbox[c]{1cm}{\\centering rd.~5}}                 & stg~1                        & $a_{8}+b_{34}+c_{20}+d_{20}+e_{20}$ & $a_{34}+b_{3}+c_{19}+d_{19}+e_{19}$  \\\\ \\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsection{Calculation of the Number of Stages}\\label{sec:stages}\n\nThe main new ingredient of our scheme in comparison to the scheme in \\cite{JafarPIR} is the unequal number of stages in each round. In \\cite{JafarPIR}, the scheme is completed in $M$ rounds, and each round contains only 1 stage only when $N=2$. To generalize the ideas in Section~\\ref{mot2} and calculate the number of stages needed per round, we use Vandermonde's identity\n\\begin{align}\n\\binom{M}{i}=\\sum_{k=0}^{P}\\binom{P}{k}\\binom{M-P}{i-k} \\label{vander}\n\\end{align}\nThe relation in (\\ref{vander}) states that any combination of $i$ objects from a group of $M$ objects must have $k$ objects from a group of size $P$ and $i-k$ objects from a group of size $M-P$. In our context, the first group is the subset of the desired messages and the second group is the subset of the undesired messages. Then, the relation can be interpreted in our setting as follows: In the $i$th round, the $\\binom{M}{i}$ combinations of all possible sums of $i$ terms can be sorted into $P+1$ categories: The first category (i.e., $k=0$), contains no terms from the desired messages, the second category contains 1 term from the desired messages and $i-1$ terms from the undesired messages, etc. The relation gives also the number of query subgroups of each category $\\binom{P}{k}$ and the number of queries in each subgroup $\\binom{M-P}{i-k}$.\n\n\\textit{Let us consider the following concrete example for clarification:} Consider that we have $6$ messages denoted by $(a,b,c,d,e,f)$, and the desired group to be retrieved is $(a,b)$. Consider round 4 that consists of all combinations of sums of 4 symbols. From Vandermonde's identity, we know that $\\binom{6}{4}=\\binom{2}{0}\\binom{4}{4} + \\binom{2}{1}\\binom{4}{3} + \\binom{2}{2}\\binom{4}{2}$. Which means that there are three categories of sums: First category is with only undesired messages; we have $\\binom{2}{0}=1$ query subgroup of the form $c+d+e+f$. The second category is to have 1 term from the desired group and the remaining are undesired; we have $\\binom{2}{1}=2$ query subgroups, one corresponds to $a$ with combinations of 3 terms from $c,d,e,f$, and the other to $b$ with combinations of 3 terms from $c,d,e,f$. Each query subgroup contains $\\binom{4}{3}$ queries, i.e., the first query subgroup is of the form $a+(c+d+e, c+d+f, c+e+f , d+e+f)$ and the second query subgroup is of the form $b+(c+d+e, c+d+f, c+e+f , d+e+f)$. Third category is to have 2 terms from the desired group and 2 terms from the undesired group; we have $\\binom{2}{2}=1$ query subgroup of this category that takes the form $a+b+(c+d, c+e, \\cdots)$. The number of queries of this group is $\\binom{4}{2}$ corresponding to all combinations of 2 undesired symbols.\n\n\\textit{Back to the calculation of the number of stages:} To be able to cancel the undesired symbols from an $i$-term sum, the user needs to download these undesired symbols as side information in the previous rounds. Hence, round $i$ requires downloading $\\binom{P}{1}$ stages in round $(i-1)$, $\\binom{P}{2}$ stages in round $(i-2)$, etc. Note that these stages need to be downloaded from the remaining $(N-1)$ databases. Then, each database needs to download $\\frac{1}{N-1}\\binom{P}{1}$ stages in round $(i-1)$, $\\frac{1}{N-1}\\binom{P}{2}$ stages in round $(i-2)$, etc.\n\nFrom this observation, we can trace back the number of stages needed at each round. Denote $\\alpha_i$ to be the number of stages in round $i$. Fix the number of stages in the last round (round $M$) to be $\\alpha_M=(N-1)^{M-P}$ stages. This choice ensures that the number of stages in any round is an integer. Note that in round $M$, the user downloads a sum of all $M$ messages, this requires side information in the form of the sum of the undesired $M-P$ messages. Hence, we suppress the rounds $M-P+1$ through $M-1$ since they do not generate any useful side information. Note that the side information equations in round $M$ at each database are collected from the remaining $(N-1)$ databases. Then, the number of stages in round $(M-P)$ should be $(N-1)^{M-P-1}$. Therefore, we write\n\\begin{align}\n\\alpha_M&=(N-1)^{M-P} \\\\\n\\alpha_{M-1}&=\\cdots= \\alpha_{M-P+1}=0 \\\\\n\\alpha_{M-P}&=(N-1)^{M-P-1}=\\frac{1}{N-1} \\alpha_M=\\frac{1}{N-1}\\sum_{i=1}^{P} \\binom{P}{i} \\alpha_{M-P+i}\n\\end{align}\n\nNow, in round $(M-P)$, each stage requires $\\binom{P}{1}$ stages from round $(M-P-1)$, $\\binom{P}{2}$ stages from round $(M-P-2)$, and so on so forth, and these stages are divided across $(N-1)$ databases. Continuing with the same argument, for each round, we write\n\\begin{align}\n\\alpha_{M-P-1}&=\\frac{1}{N-1}\\binom{P}{1}\\alpha_{M-P}=\\frac{1}{N-1}\\sum_{i=1}^{P} \\binom{P}{i} \\alpha_{M-P-1+i} \\\\\n\\alpha_{M-P-2}&=\\frac{1}{N-1}\\binom{P}{1}\\alpha_{M-P-1}+\\frac{1}{N-1}\\binom{P}{2}\\alpha_{M-P}=\\frac{1}{N-1}\\sum_{i=1}^{P} \\binom{P}{i} \\alpha_{M-P-2+i}\\\\\n\\vdots \\nonumber \\\\\n\\alpha_k&=\\frac{1}{N-1}\\sum_{i=1}^{P} \\binom{P}{i} \\alpha_{k+i}\n\\end{align}\nInterestingly, this pattern closely resembles the output of an IIR filter $y[n]$ \\cite{oppenheim}, with the difference equation,\n\\begin{align}\\label{IIR}\ny[n]=\\frac{1}{N-1}\\sum_{i=1}^{P} \\binom{P}{i} y[n-i]\n\\end{align}\nand with the initial conditions $y[-P]=(N-1)^{M-P},\\: y[-P+1]= \\cdots=y[-1]=0$. Note that the only difference between the two seemingly different settings is the orientation of the time axis. The calculation of the number of stages is obtained backwards in contrast to the output of this IIR filter. Hence, we can systematically obtain the number of stages at each round by observing the output of the IIR filter characterized by (\\ref{IIR}), and mapping it to the number of stages via $\\alpha_k=y[(M-P)-k]$.\n\nWe note that for the special case $P=1$, the number of stages can be obtained from the first order filter $y[n]=\\frac{1}{N-1}y[n-1]$. The output of this filter is $y[n]=(N-1)^{M-2-n}$. Then, the number of stages in round $k$ is $\\alpha_k=y[M-1-k]=(N-1)^{k-1}$, which is exactly the number of stages used in \\cite{JafarPIR}; in particular if $N=2$, then $\\alpha_k=1$ for all $k$.\n\n\\subsection{General Achievable Scheme}\n\n\\begin{enumerate}\n\t\\item \\textit{Index preparation:} The user interleaves the contents of each message randomly and independently from the remaining messages using a random interleaver $\\pi_m(.)$ which is known privately to the user only, i.e.,\n\t\\begin{align}\n\tx_m(i)=w_m(\\pi_m(i)), \\quad i \\in \\{1, \\cdots, L\\}\n\t\\end{align}\n\t\n\t\\item \\textit{Number of stages:} We calculate the number of stages needed in each round. This can be done systematically by finding the output of the IIR filter characterized by,\n\t\\begin{align}\\label{filter}\n\ty[n]=\\frac{1}{N-1}\\sum_{i=1}^{P} \\binom{P}{i} y[n-i]\n\t\\end{align}\n\twith the initial conditions $y[-P]=(N-1)^{M-P},\\: y[-P+1]= \\cdots=y[-1]=0$. The number of stages in round $i$ is $\\alpha_i=y[(M-P)-i]$ as discussed in Section~\\ref{sec:stages}.\n\t\n\t\\item \\textit{Initialization:} From the first database, the user downloads one symbol from each message that belongs to the desired message set $\\cp$. The user sets the round index to $i=1$.\n\t\n\t\\item \\textit{Message symmetry:} In round $i$, the user downloads sums of $i$ terms from different symbols from the first database. To satisfy the privacy constraint, the user should download an equal amount of symbols from all messages. Therefore, the user downloads the remaining $\\binom{M-P}{i}$ combinations in round $i$ from the undesired symbol set $\\bar{\\cp}$. For example: In round 1, the user downloads one symbol from every undesired message with a total of $\\binom{M-P}{1}=M-P$ such symbols.\n\t\n\t\\item \\textit{Repetition of stages:} In the first database, the user repeats the operation in round $i$ according to the number of calculated stages $\\alpha_i$. This in total results in downloading $\\alpha_i\\binom{M-P}{i}$ undesired equations, and $\\alpha_i \\left(\\binom{M}{i}-\\binom{M-P}{i}\\right)$ desired equations.\n\t\n\t\\item \\textit{Symmetry across databases:} The user implements symmetry across databases by downloading $\\alpha_i\\binom{M-P}{i}$ new undesired equations, and $\\alpha_i\\left(\\binom{M}{i}-\\binom{M-P}{i}\\right)$ new desired equations from each database. These undesired equations will be used as side information in subsequent rounds. For example: In round 1, each database generates $\\alpha_1(M-P)$ undesired equations in the form of individual symbols. Hence, each database can exploit up to $\\alpha_1 (N-1)(M-P)$ side information equations from other $(N-1)$ databases.\n\t\n\t\\item \\textit{Exploiting side information:} Until now, we did not specify how the desired equations are constructed. Since each stage in round $i$ can be categorized using Vandermonde's identity as in the previous section, we form the desired equations as a sum of the desired symbols and the undesired symbols that can be decoded from other databases in the former $(i-1)$ rounds. If the user sums two or more symbols from $\\cp$, the user downloads one new symbol from one message only and the remaining symbols from $\\cp$ should be derived from other databases. Thus, in round $(i+1)$, the user mixes one symbol of $\\cp$ with the sum of $i$ undesired symbols from round $i$. This should be repeated for all $\\binom{P}{1}$ desired symbols. Then, the user mixes each sum of 2 desired symbols with the sum of $(i-1)$  undesired symbols generated in the $(i-1)$th round. This should be repeated for all the $\\binom{P}{2}$ combinations of the desired symbols, and so on.\n\t\n\t\\item \\textit{Repeating steps:} Repeat steps 4, 5, 6, 7 by setting $i=i+1$ until $i=M-P-1$.\n\t\n\t\\item \\textit{Last round:} We note that rounds $M-P+1$ to $M-1$ do not generate useful side information. Hence, $\\alpha_{M-P+1}= \\cdots=\\alpha_{M-1}=0$. In round $M$, which corresponds to summing all $M$ messages, the user mixes $P$ symbols from $\\cp$ (only one of them is new and the remaining are previously decoded from the other $(N-1)$ databases) and $M-P$ undesired symbol mixture that was generated in round $(M-P)$.\n\t\n\t\\item \\textit{Shuffling the order of queries:} After preparing the query table, the order of the queries are shuffled uniformly, so that all possible orders of queries are equally likely regardless of $\\cp$.\n\\end{enumerate}\n\n\\subsection{Decodability, Privacy, and Calculation of the Achievable Rate}\n\nNow, we verify that the proposed scheme satisfies the reliability and privacy constraints.\n\nFor the reliability: The scheme is designed to download the \\emph{exact} number of undesired equations that will be used as side information equation at subsequent rounds in other databases.\\footnote{Check for instance in Table~\\ref{table(5,2,2)} that all of the downloads (equations) involving undesired symbols from database 2 are used in database 1: singles $c_6, d_6, e_6$, $c_7, d_7, e_7$, $c_8, d_8, e_8$, $c_9, d_9, e_9$, $c_{10}, d_{10}, e_{10}$; sums of twos $c_{15}+d_{15}$, $c_{16}+e_{15}$, $d_{16}+e_{16}$, $c_{17}+d_{17}$, $c_{18}+e_{17}$, $d_{18}+e_{18}$; sum of threes $c_{20}+d_{20}+e_{20}$, all downloaded from database 2 are all used as side information in database 1.} Hence, each desired symbol at any round is mixed with a known mixture of symbols that can be decoded from other databases. Note that if the scheme encounters the case of having a mixture of desired symbols, one of them only is chosen to be new and the remaining symbols are downloaded previously from other databases. Thus, the reliability constraint is satisfied by canceling out the side information.\n\nFor the privacy: The randomized mapping of message bits and the randomization of the order of queries guarantees privacy as in \\cite{JafarPIR}. It can be checked that when we fix the queries for one database, we can adjust the queries for the remaining databases such that the user can decode any $\\cp$ subset of messages. This is true since all combinations of messages are generated by our scheme.\n\nTo calculate the achievable rate: From Vandermonde's identity $\\binom{M}{i}=\\sum_{p=0}^P \\binom{P}{p}\\binom{M-P}{i-p}$, round $i$ requires downloading $\\binom{P}{p}$ stages in round $(i-p)$. These stages should be downloaded from the remaining $(N-1)$ databases. Hence, as shown in the previous section, the number of stages at each round is calculated as the output of an IIR filter whose input-output relation is given in (\\ref{IIR}) with the initial conditions $y[-P]=(N-1)^{M-P},\\: y[-P+1]=\\cdots=y[-1]=0$, with the conversion of time index of the filter to the round index of the schemes as $\\alpha_i=y[(M-P)-i]$. These initial conditions imply that the user downloads $(N-1)^{M-P}$ stages in the last round that corresponds to downloading the sum of all messages. The $(P-1)$ rounds before the last round are suppressed because we only need to form sums of $(M-P)$ messages to be used in the last round.\n\nNow, to calculate the number of stages for round $i$, we first solve for the roots of the characteristic equation of (\\ref{IIR}) \\cite{oppenheim},\n\\begin{align}\nr^P-\\frac{1}{N-1}\\sum_{i=1}^P \\binom{P}{i} r^{P-i}=0\n\\end{align}\nwhich is equivalent to\n\\begin{align}\nr^P-\\frac{r^P}{N-1}\\sum_{i=1}^P \\binom{P}{i} r^{-i}=0\n\\end{align}\nwhich further reduces to\n\\begin{align} \\label{ccs_binom}\nr^P-\\frac{r^P}{N-1}\\left[\\left(1+\\frac{1}{r}\\right)^P-1\\right]=0\n\\end{align}\nusing the binomial theorem. Simplifying (\\ref{ccs_binom}), we have\n\\begin{align} \\label{ccs_simplified}\nNr^P-(r+1)^P=0\n\\end{align}\nBy applying the bijective mapping $t=N^{1\/P} \\cdot\\frac{r}{r+1}$, (\\ref{ccs_simplified}) is equivalent to $t^P=1$. The roots for this equation are the normal roots of unity, i.e., $t_k=e^{j2\\pi(k-1)\/P}, \\: k=1, \\cdots, P$, where $j=\\sqrt{-1}$. Hence, the roots of the characteristic equation are given by,\n\\begin{align}\nr_k=\\frac{t_k}{N^{1\/P}-t_k}=\\frac{e^{j2\\pi(k-1)\/P}}{N^{1\/P}-e^{j2\\pi(k-1)\/P}}, \\quad  k=1, \\cdots, P\n\\end{align}\nThus, the complete response of the IIR filter is given by $y[n]=\\sum_{i=1}^{P} \\gamma_i r_i^n$, where $\\gamma_i$ are constants that result from solving the initial conditions, i.e., $\\boldsymbol{\\gamma}=(\\gamma_1, \\cdots, \\gamma_P)^T$ is the solution of the system of equations,\n\\begin{align}\n\\begin{bmatrix}\nr_1^{-P} & r_2^{-P} & \\cdots & r_P^{-P} \\\\\nr_1^{-P+1} & r_2^{-P+1} & \\cdots & r_P^{-P+1} \\\\\n\\vdots   &  \\vdots  &  \\cdots& \\vdots \\\\\nr_1^{-1} & r_2^{-1} & \\cdots & r_P^{-1} \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\n\\gamma_1 \\\\\\gamma_2 \\\\ \\vdots\\\\ \\gamma_P\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n(N-1)^{M-P} \\\\ 0 \\\\ \\vdots \\\\ 0\n\\end{bmatrix}\n\\end{align}\n\nNow, we are ready to calculate the number of stages $\\alpha_k$ in round $k$. Since $\\alpha_k=y[(M-P)-k]$ by construction, then\n\\begin{align}\n\\alpha_k=\\sum_{i=1}^{P} \\gamma_i r_i^{M-P-k}\n\\end{align}\nIn round $k$, the user downloads sums of $k$ symbols. The user repeats this round for $\\alpha_k$ stages. Each stage contains all the combinations of any $k$ symbols which there are $\\binom{M}{k}$ of them. Hence, the total download cost $D$ is,\n\\begin{align}\nD&=\\sum_{k=1}^M \\binom{M}{k} \\alpha_k \\\\\n &=\\sum_{k=1}^M \\sum_{i=1}^P \\binom{M}{k} \\gamma_i r_i^{M-P-k}\\\\\n &=\\sum_{i=1}^P \\gamma_i r_i^{M-P} \\sum_{k=1}^{M} \\binom{M}{k} r_i^{-k}\\\\\n &=\\sum_{i=1}^P \\gamma_i r_i^{M-P}\\left[\\left(1+\\frac{1}{r_i}\\right)^M-1\\right]\n\\end{align}\nConsidering the undesired equations: in round $k$, the user downloads all combinations of the $(M-P)$ undesired messages which there are $\\binom{M-P}{k}$ of them. Therefore, similar to the above calculation, the total number of undesired equations $U$ is,\n\\begin{align}\nU=\\sum_{i=1}^P \\gamma_i r_i^{M-P}\\left[\\left(1+\\frac{1}{r_i}\\right)^{M-P}-1\\right]\n\\end{align}\nHence, the achievable rate $\\underaccent{\\bar}{R}_s$ is\n\\begin{align}\\label{rate1}\n\\underaccent{\\bar}{R}_s &=\\frac{D-U}{D} \\\\\n             &=\\frac{\\sum_{i=1}^{P}\\gamma_i r_i^{M-P}\\left[\\left(1+\\frac{1}{r_i}\\right)^M-\\left(1+\\frac{1}{r_i}\\right)^{M-P}\\right]}{\\sum_{i=1}^{P}\\gamma_i r_i^{M-P}\\left[\\left(1+\\frac{1}{r_i}\\right)^M-1\\right]}\n\\end{align}\nwhich is (\\ref{ach-rate}) in Theorem~\\ref{thm2}.\n\n\\subsection{Further Examples for the Case $P \\leq \\frac{M}{2}$}\n\nIn this section, we illustrate our proposed scheme with a few additional basic examples. In Section~\\ref{mot2}, we considered the case $M=5$, $P=2$, $N=2$. In the next three sub-sections, we consider three more examples. In the example in Section~\\ref{ex3}, the ratio $\\frac{M}{P}$ is exactly equal to 2, thus, both the achievable scheme here and the achievable scheme in Section~\\ref{sec:achievable1} can be used; we comment about the differences and advantages of both schemes. In the example in Section~\\ref{ex4}, we present the case of a larger $N$ for the example in Section~\\ref{mot2}. In the example in Section~\\ref{ex5}, we present a case with larger $M$, $P$ and $N$.\n\n\\subsubsection{$M=4$ Messages, $P=2$ Messages, $N=2$ Databases} \\label{ex3}\n\nThe first step of the achievable scheme is to identify the number of stages needed for each round of download. The IIR filter in (\\ref{IIR}) that determines the number of stages reduces in this case to\n\\begin{align}\ny[n]=2y[n-1]+y[n-2]\n\\end{align}\nwith the initial conditions $y[-2]=1,y[-1]=0$. The number of stages in round $k$ is $\\alpha_k=y[2-k]$. Since $M$ is small, we can calculate the output iteratively without using the canonical filter output as,\n\\begin{align}\n\\alpha_4&=y[-2]=1 \\\\\n\\alpha_3&=y[-1]=0 \\\\\n\\alpha_2&=y[0]=2y[-1]+y[-2]=1 \\\\\n\\alpha_1&=y[1]=2y[0]+y[-1]=2\n\\end{align}\nHence, we should download 2 stages of individual symbols (round 1), and 1 stage of sums of two symbols (round 2). We should suppress the round that retrieves sums of three symbols (round 3), and have 1 stage of sums of all four symbols (round 4).\n\nThe user initializes the scheme by randomly and independently interleaving the symbols of each message. The query table for this example is shown in Table~\\ref{table(4,2,2)}. In round 1, the user downloads individual symbols from all messages at each database. The user downloads $a_1,b_1,c_1,d_1$ and $a_2,b_2,c_2,d_2$ from database 1, as $\\alpha_1=2$. This is repeated for database 2. In round 2, the user downloads sums of two symbols. There are $\\binom{4}{2}=6$ such equations. At database 1, the undesired symbols from database 2 in the first round are exploited in some of these sums. These equations are either in the form $a+(c,d)$ or in the form $b+(c,d)$. This necessitates two sets of different individual symbols to be downloaded from database 2 in the first round, or otherwise the symbols are repeated and privacy is compromised. Moreover, we note that the user downloads $a_5+b_3$ which uses $b_3$ as side information even though $W_2$ is desired; this is reversed in database 2 to download $a_1+b_7$ with $a_1$ as a side information to have a symmetric scheme. Round 2 concludes with downloading $c_5+d_5$ and $c_6+d_6$ at the two databases, which will be used as side information in the last round. Round 3 is skipped and the user proceeds to round 4 (last round) directly. In round 4, the user downloads sum of four symbols, and uses the side information downloaded in round 2 and any decoded symbols for the other desired message. For example, in database 1, the user downloads $a_3+b_{10}+c_6+d_6$, hence, the side information $c_6+d_6$ is exploited in this round as well as $a_3$. The user finishes the scheme by shuffling the order of all queries randomly. The user retrieves $a_1,\\cdots,a_{10}$ and $b_1,\\cdots,b_{10}$ privately in 30 downloads (15 from each database) and achieves a sum rate of $\\frac{20}{30}=\\frac{2}{3}=\\frac{1}{1+\\frac{1}{N}}$, which matches the upper bound in Theorem~\\ref{thm2}. This sum rate outperforms the repetition-based achievable rate which is $\\frac{3}{5}$ in (\\ref{repeat}).\n\n\\begin{table}[t]\n\t\\centering\n\t\\caption{The query table for the case $M=4,P=2,N=2$.}\n\t\\label{table(4,2,2)}\n\t\\begin{tabular}{|l|l|c|c|}\n\t\t\\hline\n\t\t&                          & Database 1             & Database 2             \\\\ \\hline\n\t\t\\multirow{2}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{0.9cm}{\\centering rd.~1}}} & stg~1                     & $a_1,b_1,c_1,d_1$      & $a_3,b_3,c_3,d_3$      \\\\ \\cline{2-4}\n\t\t& stg~2                     & $a_2,b_2,c_2,d_2$      & $a_4,b_4,c_4,d_4$      \\\\ \\hline\\hline\n\t\t\\multirow{6}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering round 2}}} & \\multirow{6}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 1}}} & $a_5+b_3$              & $a_1+b_7$              \\\\\n\t\t&                          & $a_6+c_3$              & $a_8+c_1$              \\\\\n\t\t&                          & $a_7+d_3$              & $a_9+d_1$              \\\\\n\t\t&                          & $b_5+c_4$              & $b_8+c_2$              \\\\\n\t\t&                          & $b_6+d_4$              & $b_9+d_2$              \\\\\n\t\t&                          & $c_5+d_5$              & $c_6+d_6$              \\\\ \\hline\\hline\n\t\t\\rotatebox[origin=c]{90}{\\parbox[c]{1cm}{\\centering rd.~4}}                    & stg~1                     & $a_{3}+b_{10}+c_6+d_6$ & $a_{10}+b_{1}+c_5+d_5$ \\\\ \\hline\n\t\\end{tabular}\n\\end{table}\n\nWe note that this case can be solved using the achievable scheme presented in Section~\\ref{sec:achievable1} as well since $\\frac{M}{P}=2$ in this case. In fact, this is equivalent to the case considered in Section~\\ref{ex2}, if the number of databases is reduced from $N=3$ to $N=2$. Starting from Table~\\ref{table(4,2,3)} in Section~\\ref{ex2} and removing the downloads from database 3, we obtain the query table which uses MDS-coded queries shown in Table~\\ref{table(4,2,2)alt} below. Via the scheme in Table~\\ref{table(4,2,2)alt} below, the user retrieves $a_1,\\cdots,a_4$ and $b_1,\\cdots,b_4$ privately in 12 downloads (6 from each database), therefore achieving the same optimal sum rate of $\\frac{8}{12}=\\frac{2}{3}=\\frac{1}{1+\\frac{1}{N}}$.\n\n\\begin{table}[h]\n\t\\centering\n\t\\caption{Alternative query table for the case $M=4,P=2,N=2$.}\n\t\\label{table(4,2,2)alt}\n\t\\begin{tabular}{|c|c|}\n\t\t\\hline\n\t\tDatabase 1 & Database 2 \\\\\n\t\t\\hline\n\t\t$a_1,b_1,c_1,d_1$ & $a_2,b_2,c_2,d_2$ \\\\\n\t\t\\hline\n\t\t$a_3+b_3+c_2+d_2$ & $a_4+b_4+c_1+d_1$ \\\\\n\t\t$a_3+3b_3+2c_2+4d_2$ & $a_4+3b_4+2c_1+4d_1$ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nWe presented this case here even though it could be solved using the scheme in Section~\\ref{sec:achievable1}, in order to give an example where the second achievable scheme achieves the upper bound in Theorem~\\ref{thm2} and yields a capacity result since $\\frac{M}{P}$ is an integer. Interestingly, we observe that for all cases where $P=\\frac{M}{2}$, the two  achievable schemes are both optimal. The two schemes present an interesting trade-off between the field size and the upload cost: The first achievable scheme in Section~\\ref{sec:achievable1} requires using an MDS code with field size $q \\geq M$ but the number of queries for each database is limited to $M+P$. On the other hand, the second achievable scheme here in Section~\\ref{sec:achievable2} does not use any coding and can work with the storage field size, however, the number of queries increase exponentially since the number of stages for each round is related to an unstable IIR filter.\n\n\\subsubsection{$M=5$ Messages, $P=2$ Messages, $N=3$ Databases} \\label{ex4}\n\nIn this example, we show an explicit query structure for $N>2$. In this case the corresponding difference equation for the IIR filter is\n\\begin{align}\ny[n]=y[n-1]+\\frac{1}{2}y[n-2]\n\\end{align}\nwith the initial conditions $y[-1]=0,\\,y[-2]=(N-1)^{M-P}=8$. Thus, the number of stages in each round are: $\\alpha_1=6$, $\\alpha_2=4$, $\\alpha_3=4$, $\\alpha_4=0, \\alpha_5=8$. The query table is shown in Tables~\\ref{table(5,2,3)}, \\ref{table(5,2,3)_cont} and \\ref{table(5,2,3)_cont2}. This scheme retrieves $a_1,\\cdots,a_{126}$ and $b_1,\\cdots,b_{126}$ privately in 354 downloads (177 from each database), therefore, achieving a sum rate of $\\frac{252}{354}=\\frac{42}{59}<\\frac{1}{1+\\frac{1}{N}+\\frac{1}{2N^2}}=\\frac{18}{25}$. The gap is $\\frac{12}{1475} \\simeq 0.0081$.\n\n\\begin{table}[]\n\t\\centering\n\t\\caption{The query table for the case $M=5,P=2,N=3$.}\n\t\\label{table(5,2,3)}\n\t\\begin{tabular}{|l|l|c|c|c|}\n\t\t\\hline\n\t\t&                           & Database 1                      & Database 2                           & Database 3                           \\\\ \\hline\n\t\t\n\t\t\\multirow{6}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering round 1}}}  & stg~1                      & $a_1,b_1,c_1,d_1,e_1$           & $a_7,b_7,c_7,d_7,e_7$                & $a_{13},b_{13},c_{13},d_{13},e_{13}$ \\\\ \\cline{2-5}\n\t\t& stg~2                      & $a_2,b_2,c_2,d_2,e_2$           & $a_8,b_8,c_8,d_8,e_8$                & $a_{14},b_{14},c_{14},d_{14},e_{14}$ \\\\ \\cline{2-5}\n\t\t& stg~3                      & $a_3,b_3,c_3,d_3,e_3$           & $a_9,b_9,c_9,d_9,e_9$                & $a_{15},b_{15},c_{15},d_{15},e_{15}$ \\\\ \\cline{2-5}\n\t\t& stg~4                      & $a_4,b_4,c_4,d_4,e_4$           & $a_{10},b_{10},c_{10},d_{10},e_{10}$ & $a_{16},b_{16},c_{16},d_{16},e_{16}$ \\\\ \\cline{2-5}\n\t\t& stg~5                      & $a_5,b_5,c_5,d_5,e_5$           & $a_{11},b_{11},c_{11},d_{11},e_{11}$ & $a_{17},b_{17},c_{17},d_{17},e_{17}$ \\\\ \\cline{2-5}\n\t\t& stg~6                      & $a_{6},b_{6},c_{6},d_{6},e_{6}$ & $a_{12},b_{12},c_{12},d_{12},e_{12}$ & $a_{18},b_{18},c_{18},d_{18},e_{18}$ \\\\ \\hline\\hline\n\t\t\\multirow{30}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering round 2}}} & \\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 1}}} & $a_{19}+b_7$                    & $a_{33}+b_1$                         & $a_{47}+b_{1}$                       \\\\\n\t\t&                           & $a_{20}+c_7$                    & $a_{34}+c_1$                         & $a_{48}+c_1$                         \\\\\n\t\t&                           & $a_{21}+d_7$                    & $a_{35}+d_1$                         & $a_{49}+d_1$                         \\\\\n\t\t&                           & $a_{22}+e_7$                    & $a_{36}+e_1$                         & $a_{50}+e_1$                         \\\\\n\t\t&                           & $b_{19}+c_8$                    & $b_{33}+c_2$                         & $b_{47}+c_2$                         \\\\\n\t\t&                           & $b_{20}+d_8$                    & $b_{34}+d_2$                         & $b_{48}+d_2$                         \\\\\n\t\t&                           & $b_{21}+e_8$                    & $b_{35}+e_2$                         & $b_{49}+e_2$                         \\\\\n\t\t&                           & $c_{19}+d_{19}$                 & $c_{27}+d_{27}$                      & $c_{35}+d_{35}$                      \\\\\n\t\t&                           & $c_{20}+e_{19}$                 & $c_{28}+e_{27}$                      & $c_{36}+e_{35}$                      \\\\\n\t\t&                           & $d_{20}+e_{20}$                 & $d_{28}+e_{28}$                      & $d_{36}+e_{36}$                      \\\\ \\cline{2-5}\n\t\t& \\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 2}}} & $a_{7}+b_{22}$                  & $a_{1}+b_{36}$                       & $a_{1}+b_{50}$                       \\\\\n\t\t&                           & $a_{23}+c_9$                    & $a_{37}+c_3$                         & $a_{51}+c_3$                         \\\\\n\t\t&                           & $a_{24}+d_9$                    & $a_{38}+d_3$                         & $a_{52}+d_3$                         \\\\\n\t\t&                           & $a_{25}+e_9$                    & $a_{39}+e_3$                         & $a_{53}+e_3$                         \\\\\n\t\t&                           & $b_{23}+c_{10}$                 & $b_{37}+c_4$                         & $b_{51}+c_4$                         \\\\\n\t\t&                           & $b_{24}+d_{10}$                 & $b_{38}+d_4$                         & $b_{52}+d_4$                         \\\\\n\t\t&                           & $b_{25}+e_{10}$                 & $b_{39}+e_4$                         & $b_{53}+e_4$                         \\\\\n\t\t&                           & $c_{21}+d_{21}$                 & $c_{29}+d_{29}$                      & $c_{37}+d_{37}$                      \\\\\n\t\t&                           & $c_{22}+e_{21}$                 & $c_{30}+e_{29}$                      & $c_{38}+e_{37}$                      \\\\\n\t\t&                           & $d_{22}+e_{22}$                 & $d_{30}+e_{30}$                      & $d_{38}+e_{38}$                      \\\\ \\cline{2-5}\n\t\t& \\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 3}}} & $a_{26}+b_{13}$                 & $a_{40}+b_{13}$                      & $a_{54}+b_{7}$                       \\\\\n\t\t&                           & $a_{27}+c_{13}$                 & $a_{41}+c_{13}$                      & $a_{55}+c_7$                         \\\\\n\t\t&                           & $a_{28}+d_{13}$                 & $a_{42}+d_{13}$                      & $a_{56}+d_7$                         \\\\\n\t\t&                           & $a_{29}+e_{13}$                 & $a_{43}+e_{13}$                      & $a_{57}+e_7$                         \\\\\n\t\t&                           & $b_{26}+c_{14}$                 & $b_{40}+c_{14}$                      & $b_{54}+c_8$                         \\\\\n\t\t&                           & $b_{27}+d_{14}$                 & $b_{41}+d_{14}$                      & $b_{55}+d_8$                         \\\\\n\t\t&                           & $b_{28}+e_{14}$                 & $b_{42}+e_{14}$                      & $b_{56}+e_8$                         \\\\\n\t\t&                           & $c_{23}+d_{23}$                 & $c_{31}+d_{31}$                      & $c_{39}+d_{39}$                      \\\\\n\t\t&                           & $c_{24}+e_{23}$                 & $c_{32}+e_{31}$                      & $c_{40}+e_{39}$                      \\\\\n\t\t&                           & $d_{24}+e_{24}$                 & $d_{32}+e_{32}$                      & $d_{40}+e_{40}$                      \\\\ \\cline{1-5}\n\t\\end{tabular}\n\\end{table}\n\n\n\\begin{table}[]\n\t\\centering\n\t\\caption{The query table for the case $M=5,P=2,N=3$ (cont.).}\n\t\\label{table(5,2,3)_cont}\n\t\\begin{tabular}{|l|l|c|c|c|}\n\t\t\\hline\n\t\t&                           & Database 1                            & Database 2                            & Database 3                            \\\\ \\hline\n\t\t\\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering round 2}}}\t\t& \\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 4}}} & $a_{13}+b_{29}$                 & $a_{13}+b_{43}$                      & $a_{7}+b_{57}$                       \\\\\n\t\t&                           & $a_{30}+c_{15}$                 & $a_{44}+c_{15}$                      & $a_{58}+c_9$                         \\\\\n\t\t&                           & $a_{31}+d_{15}$                 & $a_{45}+d_{15}$                      & $a_{59}+d_9$                         \\\\\n\t\t&                           & $a_{32}+e_{15}$                 & $a_{46}+e_{15}$                      & $a_{60}+e_9$                         \\\\\n\t\t&                           & $b_{30}+c_{16}$                 & $b_{44}+c_{16}$                      & $b_{58}+c_{10}$                      \\\\\n\t\t&                           & $b_{31}+d_{16}$                 & $b_{45}+d_{16}$                      & $b_{59}+d_{10}$                      \\\\\n\t\t&                           & $b_{32}+e_{16}$                 & $b_{46}+e_{16}$                      & $b_{60}+e_{10}$                      \\\\\n\t\t&                           & $c_{25}+d_{25}$                 & $c_{33}+d_{33}$                      & $c_{41}+d_{41}$                      \\\\\n\t\t&                           & $c_{26}+e_{25}$                 & $c_{34}+e_{33}$                      & $c_{42}+e_{41}$                      \\\\\n\t\t&                           & $d_{26}+e_{26}$                 & $d_{34}+e_{34}$                      & $d_{42}+e_{42}$ \\\\ \\hline\\hline\n\t\t\\multirow{30}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering round 3}}} & \\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 1}}} & $a_{61}+b_{8}+c_{11}$                 & $a_{79}+b_{2}+c_{5}$                  & $a_{97}+b_{2}+c_{5}$                  \\\\\n\t\t&                           & $a_{8}+b_{61}+d_{11}$                 & $a_{2}+b_{79}+d_{5}$                  & $a_{2}+b_{97}+d_{5}$                  \\\\\n\t\t&                           & $a_{62}+b_{9}+e_{11}$                 & $a_{80}+b_{3}+e_{5}$                  & $a_{98}+b_{3}+e_{5}$                  \\\\\n\t\t&                           & $a_{63}+c_{27}+d_{27}$                & $a_{81}+c_{19}+d_{19}$                & $a_{99}+c_{19}+d_{19}$                \\\\\n\t\t&                           & $a_{64}+c_{28}+e_{27}$                & $a_{82}+c_{20}+e_{19}$                & $a_{100}+c_{20}+e_{19}$               \\\\\n\t\t&                           & $a_{65}+d_{28}+e_{28}$                & $a_{83}+d_{20}+e_{20}$                & $a_{101}+d_{20}+e_{20}$               \\\\\n\t\t&                           & $b_{62}+c_{29}+d_{29}$                & $b_{80}+c_{21}+d_{21}$                & $b_{98}+c_{21}+d_{21}$                \\\\\n\t\t&                           & $b_{63}+c_{30}+e_{29}$                & $b_{81}+c_{22}+e_{21}$                & $b_{99}+c_{22}+e_{21}$                \\\\\n\t\t&                           & $b_{64}+d_{30}+e_{30}$                & $b_{82}+d_{22}+e_{22}$                & $b_{100}+d_{22}+e_{22}$               \\\\\n\t\t&                           & $c_{43}+d_{43}+e_{43}$                & $c_{47}+d_{47}+e_{47}$                & $c_{51}+d_{51}+e_{51}$                \\\\ \\cline{2-5}\n\t\t& \\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 2}}} & $a_{9}+b_{65}+c_{12}$                 & $a_{3}+b_{83}+c_{6}$                  & $a_{3}+b_{101}+c_{6}$                 \\\\\n\t\t&                           & $a_{66}+b_{10}+d_{12}$                & $a_{84}+b_{4}+d_{6}$                  & $a_{102}+b_{4}+d_{6}$                 \\\\\n\t\t&                           & $a_{10}+b_{66}+e_{12}$                & $a_{4}+b_{84}+e_{6}$                  & $a_{4}+b_{102}+e_{6}$                 \\\\\n\t\t&                           & $a_{67}+c_{31}+d_{31}$                & $a_{85}+c_{23}+d_{23}$                & $a_{103}+c_{23}+d_{23}$               \\\\\n\t\t&                           & $a_{68}+c_{32}+e_{31}$                & $a_{86}+c_{24}+e_{23}$                & $a_{104}+c_{24}+e_{23}$               \\\\\n\t\t&                           & $a_{69}+d_{32}+e_{32}$                & $a_{87}+d_{24}+e_{24}$                & $a_{105}+d_{24}+e_{24}$               \\\\\n\t\t&                           & $b_{67}+c_{33}+d_{33}$                & $b_{85}+c_{25}+d_{25}$                & $b_{103}+c_{25}+d_{25}$               \\\\\n\t\t&                           & $b_{68}+c_{34}+e_{33}$                & $b_{86}+c_{26}+e_{25}$                & $b_{104}+c_{26}+e_{25}$               \\\\\n\t\t&                           & $b_{69}+d_{34}+e_{34}$                & $b_{87}+d_{26}+e_{26}$                & $b_{105}+d_{26}+e_{26}$               \\\\\n\t\t&                           & $c_{44}+d_{44}+e_{44}$                & $c_{48}+d_{48}+e_{48}$                & $c_{52}+d_{52}+e_{52}$                \\\\ \\cline{2-5}\n\t\t& \\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 3}}} & $a_{70}+b_{14}+c_{17}$                & $a_{88}+b_{14}+c_{17}$                & $a_{106}+b_{8}+c_{11}$                \\\\\n\t\t&                           & $a_{14}+b_{70}+d_{17}$                & $a_{14}+b_{88}+d_{17}$                & $a_{8}+b_{106}+d_{11}$                \\\\\n\t\t&                           & $a_{71}+b_{15}+e_{17}$                & $a_{89}+b_{15}+e_{17}$                & $a_{107}+b_{9}+e_{11}$                \\\\\n\t\t&                           & $a_{72}+c_{35}+d_{35}$                & $a_{90}+c_{35}+d_{35}$                & $a_{108}+c_{27}+d_{27}$               \\\\\n\t\t&                           & $a_{73}+c_{36}+e_{35}$                & $a_{91}+c_{36}+e_{35}$                & $a_{109}+c_{28}+e_{27}$               \\\\\n\t\t&                           & $a_{74}+d_{36}+e_{36}$                & $a_{92}+d_{36}+e_{36}$                & $a_{110}+d_{28}+e_{28}$               \\\\\n\t\t&                           & $b_{71}+c_{37}+d_{37}$                & $b_{89}+c_{37}+d_{37}$                & $b_{107}+c_{29}+d_{29}$               \\\\\n\t\t&                           & $b_{72}+c_{38}+e_{37}$                & $b_{90}+c_{38}+e_{37}$                & $b_{108}+c_{30}+e_{29}$               \\\\\n\t\t&                           & $b_{73}+d_{38}+e_{38}$                & $b_{91}+d_{38}+e_{38}$                & $b_{109}+d_{30}+e_{30}$               \\\\\n\t\t&                           & $c_{45}+d_{45}+e_{45}$                & $c_{49}+d_{49}+e_{49}$                & $c_{53}+d_{53}+e_{53}$                \\\\ \\cline{1-5}\n\t\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\t\\centering\n\t\\caption{The query table for the case $M=5,P=2,N=3$ (cont.).}\n\t\\label{table(5,2,3)_cont2}\n\t\\begin{tabular}{|l|l|c|c|c|}\n\t\t\\hline\n\t\t&                           & Database 1                            & Database 2                            & Database 3                            \\\\ \\hline\n\t\t\t\\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering round 3}}}\t\t\n\t\t\t& \\multirow{10}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering stage 4}}} & $a_{15}+b_{74}+c_{18}$                & $a_{15}+b_{92}+c_{18}$                & $a_{9}+b_{110}+c_{12}$                \\\\\n\t\t\t&                           & $a_{75}+b_{16}+d_{18}$                & $a_{93}+b_{16}+d_{18}$                & $a_{111}+b_{10}+d_{12}$               \\\\\n\t\t\t&                           & $a_{16}+b_{75}+e_{18}$                & $a_{16}+b_{93}+e_{18}$                & $a_{10}+b_{111}+e_{12}$               \\\\\n\t\t\t&                           & $a_{76}+c_{39}+d_{39}$                & $a_{94}+c_{39}+d_{39}$                & $a_{112}+c_{31}+d_{31}$               \\\\\n\t\t\t&                           & $a_{77}+c_{40}+e_{39}$                & $a_{95}+c_{40}+e_{39}$                & $a_{113}+c_{32}+e_{31}$               \\\\\n\t\t\t&                           & $a_{78}+d_{40}+e_{40}$                & $a_{96}+d_{40}+e_{40}$                & $a_{114}+d_{32}+e_{32}$               \\\\\n\t\t\t&                           & $b_{76}+c_{41}+d_{41}$                & $b_{94}+c_{41}+d_{41}$                & $b_{112}+c_{33}+d_{33}$               \\\\\n\t\t\t&                           & $b_{77}+c_{42}+e_{41}$                & $b_{95}+c_{42}+e_{41}$                & $b_{113}+c_{34}+e_{33}$               \\\\\n\t\t\t&                           & $b_{78}+d_{42}+e_{42}$                & $b_{96}+d_{42}+e_{42}$                & $b_{114}+d_{34}+e_{34}$               \\\\\n\t\t\t&                           & $c_{46}+d_{46}+e_{46}$                & $c_{50}+d_{50}+e_{50}$                & $c_{54}+d_{54}+e_{54}$                \\\\ \\hline\\hline\n\t\t\t\\multirow{8}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{2cm}{\\centering round 5}}}  & stg~1                      & $a_{115}+b_{11}+c_{47}+d_{47}+e_{47}$ & $a_{119}+b_{5}+c_{43}+d_{43}+e_{43}$  & $a_{123}+b_{5}+c_{43}+d_{43}+e_{43}$  \\\\ \\cline{2-5}\n\t\t\t& stg~2                      & $a_{11}+b_{115}+c_{48}+d_{48}+e_{48}$ & $a_{5}+b_{119}+c_{44}+d_{44}+e_{44}$  & $a_{5}+b_{123}+c_{44}+d_{44}+e_{44}$  \\\\ \\cline{2-5}\n\t\t\t& stg~3                      & $a_{116}+b_{12}+c_{49}+d_{49}+e_{49}$ & $a_{120}+b_{6}+c_{45}+d_{45}+e_{45}$  & $a_{124}+b_{6}+c_{45}+d_{45}+e_{45}$  \\\\ \\cline{2-5}\n\t\t\t& stg~4                      & $a_{12}+b_{116}+c_{50}+d_{50}+e_{50}$ & $a_{6}+b_{120}+c_{46}+d_{46}+e_{46}$  & $a_{6}+b_{124}+c_{46}+d_{46}+e_{46}$  \\\\ \\cline{2-5}\n\t\t\t& stg~5                      & $a_{117}+b_{17}+c_{51}+d_{51}+e_{51}$ & $a_{121}+b_{17}+c_{51}+d_{51}+e_{51}$ & $a_{125}+b_{11}+c_{47}+d_{47}+e_{47}$ \\\\ \\cline{2-5}\n\t\t\t& stg~6                      & $a_{17}+b_{117}+c_{52}+d_{52}+e_{52}$ & $a_{17}+b_{121}+c_{52}+d_{52}+e_{52}$ & $a_{11}+b_{125}+c_{48}+d_{48}+e_{48}$ \\\\ \\cline{2-5}\n\t\t\t& stg~7                      & $a_{118}+b_{18}+c_{53}+d_{53}+e_{53}$ & $a_{122}+b_{18}+c_{53}+d_{53}+e_{53}$ & $a_{126}+b_{12}+c_{49}+d_{49}+e_{49}$ \\\\ \\cline{2-5}\n\t\t\t& stg~8                      & $a_{18}+b_{118}+c_{54}+d_{54}+e_{54}$ & $a_{18}+b_{122}+c_{54}+d_{54}+e_{54}$ & $a_{12}+b_{126}+c_{50}+d_{50}+e_{50}$ \\\\ \\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsubsection{$M=7$ Messages, $P=3$ Messages, $N=3$ Databases} \\label{ex5}\n\nFinally, in this section, we consider an example with $N=3$ databases and larger $M$ and $P$ than in previous examples, where we describe the structure and the calculation of the number of queries without specifying the explicit query table as it grows quite large. We first calculate the number of stages at each round. The corresponding IIR filter is\n\\begin{align}\ny[n]=\\frac{1}{2}(3y[n-1]+3y[n-2]+y[n-3])\n\\end{align}\nwith the initial conditions $y[-3]=(N-1)^{M-P}=16$, $y[-2]=0$, $y[-1]=0$. Hence, the number of stages for each round $\\alpha_k=y[4-k]$,  $k=1,\\cdots, 7$, are calculated iteratively as $\\alpha_1=67$, $\\alpha_2=30$, $\\alpha_3=12$, $\\alpha_4=8$, $\\alpha_5=0$, $\\alpha_6=0$, $\\alpha_7=16$.\n\nIn round 1, the user downloads $67$ individual symbols from each message and from each database. Each database can use the side information generated by the other two databases. Hence, each database has $67\\cdot2=134$ side information equations in the form of single symbols from round 1 to exploit. In round 2, the user downloads sums of two symbols. Each stage in round 2 requires $3$ stages from round 1, since the user faces with $a+(d,e,f,g)$, $b+(d,e,f,g)$ or $c+(d,e,f,g)$ cases. Then, round 2 requires $30\\cdot3=90$ stages from the generated side information in round 1, and we are left with $134-90=44$ more stages of round 1. Each database can use the side information stages from the other two databases, i.e., each can use up to $2\\cdot30=60$ stages of side information in the form of sums of two.\n\nIn round 3, the user downloads sums of three symbols, which can be either of $a+b+(d,e,f,g)$, $a+c+(d,e,f,g)$, $b+c+(d,e,f,g)$, $a+(d+e,d+f,\\cdots)$, and similarly for $b,c$. Therefore, each stage in round 3 requires $3$ stages from round 2, and $3$ stages from round 1. This in total requires $12\\cdot3=36$ stages from round 1 and $36$ stages from round 2, and we will be left with $8$ stages from round 1 and $24$ stages from round 2. Round 3 generates $2\\cdot12=24$ stages of side information in the form of sums of threes. In round 4, the user downloads sums of 4 symbols, which can be either $a+b+(d+e, d+f, \\cdots)$, and similarly for $b+c$ and $a+c$, $a+(d+e+f,d+e+g, \\cdots)$ and similarly for $b$, $c$, or $a+b+c+(d,e,f,g)$. This means that for each stage of round 3, the user needs 1 stage of round 1, 3 stages of round 2, and 3 stages of round 3. This in total requires $8\\cdot3=24$ stages from round 2 and 3 and $8\\cdot1$ stages from round 1 and hence, we exhaust all the generated side information by round 4. Round 4 generates $8$ stages of side information in the form of sums of fours. This will be used in the last round to get $8\\cdot2$ new symbols from the desired messages.\n\nThe achievable sum rate in this case is $\\frac{3933}{5445} =\\frac{437}{605}< \\frac{1}{1+\\frac{1}{N}+\\frac{1}{3N^2}} =\\frac{27}{37}$. The gap is $\\frac{166}{22385} \\simeq 0.0074$.\n\n\\section{Converse Proof}\n\nIn this section, we derive an upper bound for the MPIR problem. The derived upper bound is tight when $P \\geq \\frac{M}{2}$ and when $\\frac{M}{P} \\in \\mathbb{N}$. We follow the notations and simplifications in \\cite{JafarPIR, KarimCoded}, and we define\n\\begin{align}\n\\cq&\\triangleq \\left\\{Q_n^{[\\cp]}: \\cp \\subseteq \\{1, \\cdots, M\\}, \\quad |\\cp|=P,\\: n \\in \\{1, \\cdots, N\\}\\right\\} \\\\\nA_{n_1:n_2}^{[\\cp]}&\\triangleq\\left\\{A_{n_1}^{[\\cp]}, A_{n_1+1}^{[\\cp]}, \\cdots, A_{n_2}^{[\\cp]} \\right\\}, \\quad n_1 \\leq n_2, \\: n_1,n_2 \\in \\{1, \\cdots, N\\}\n\\end{align}\nWithout loss of generality, the following simplifications hold for the MPIR problem:\n\\begin{enumerate}\n\\item We can assume that the MPIR scheme is symmetric. Since for every asymmetric scheme, there exists an equal rate symmetric scheme that can be constructed by replicating all permutations of databases and messages.\n\\item To invoke the privacy constraint, we fix the response of one database to be the same irrespective of the desired set of messages $\\cp$, i.e., $A_n^{[\\cp_i]}=A_n$, where $|\\cp_i|=P$ for every $i \\in \\left\\{1,2, \\cdots, \\beta\\right\\}$ for some $n \\in \\{1, \\cdots, N\\}$, and $\\beta=\\binom{M}{P}$. No loss of generality is incurred due to the fact that the queries and the answers are statistically independent from $\\cp$. In the sequel, we fix the answer string of the first database, i.e.,\n\\begin{align}\\label{fixed_answer}\nA_1^{[\\cp]}=A_1, \\: \\forall \\cp\n\\end{align}\n\\end{enumerate}\n\nThe following lemma is a consequence of the symmetry assumption; its proof can be found in \\cite{JafarPIR}.\n\n\\begin{lemma}[Symmetry \\cite{JafarPIR}] \\label{symmetry}\n\tFor any $W_\\mathcal{S}=\\{W_i:i \\in \\mathcal{S}\\}$\n\t\\begin{align}\n\tH(A_n^{[\\cp]}|W_\\mathcal{S},\\cq)&=H(A_1^{[\\cp]}|W_\\mathcal{S},\\cq), \\quad n \\in\\{1,\\cdots, N\\} \\\\\n\tH(A_1|\\cq)&=H(A_n^{[\\cp]}|\\cq), \\quad n \\in \\{1, \\cdots, N\\}, \\: \\forall \\cp\n\t\\end{align}\n\\end{lemma}\n\nWe construct the converse proof by induction over $\\lfloor\\frac{M}{P}\\rfloor$ in a similar way to \\cite{JafarPIR, KarimCoded}. The base induction step is obtained for $1 \\leq \\frac{M}{P} \\leq 2$ (this is the case $P\\geq\\frac{M}{2}$ as it was referred to so far, where the user wants to retrieve at least half of the messages). We obtain an inductive relation for the case $\\frac{M}{P}>2$. The converse proof extends the proof in \\cite{JafarPIR} for $P >1$.\n\n\\subsection{Converse Proof for the Case $1 \\leq \\frac{M}{P} \\leq 2$}\\label{converse1}\n\nTo prove the converse for the case $1 \\leq \\frac{M}{P} \\leq 2$, we need the following lemma which gives a lower bound on the interference within an answer string.\n\n\\begin{lemma}[Interference Lower Bound]\\label{interference}\nFor the MPIR problem with $P \\geq \\frac{M}{2}$, the uncertainty of the interfering messages $W_{P+1:M}$ within the answer string $A_1^{[1:P]}$ is lower bounded as,\n\\begin{align}\\label{lower_bound}\nH(A_1^{[1:P]}|W_{1:P},\\cq) \\geq \\frac{(M-P)L}{N}\n\\end{align}\nFurthermore, (\\ref{lower_bound}) is true for any set of desired messages $\\cp$ with $|\\cp|=P$, i.e.,\n\\begin{align}\nH(A_1^{[\\cp]}|W_{\\cp},\\cq) \\geq \\frac{(M-P)L}{N}\n\\end{align}\n\\end{lemma}\n\n\\begin{Proof}\nFor clarity of presentation, we assume that $\\cp=\\{1, \\cdots, P\\}$ without loss of generality. Hence,\n\t\\begin{align}\n  \t    (M-P)L&=H(W_{P+1:M}) \\\\\n\\label{lb1}   &=H(W_{P+1:M}|W_{1:P},\\cq) \\\\\n\\label{lb2}   &=H(W_{P+1:M}|W_{1:P},\\cq)-H(W_{P+1:M}|A_{1:N}^{[M-P+1:M]},W_{1:P},\\cq)\\\\\n              &=I(W_{P+1:M};A_{1:N}^{[M-P+1:M]}|W_{1:P},\\cq)\\\\\n\\label{lb3}   &=H(A_{1:N}^{[M-P+1:M]}|W_{1:P},\\cq) \\\\\n              &\\leq \\sum_{n=1}^N H(A_{n}^{[M-P+1:M]}|W_{1:P},\\cq) \\\\\n\\label{lb4}   &=NH(A_1|W_{1:P},\\cq)\n\t\\end{align}\nwhere (\\ref{lb1}) follows from the independence of the messages $W_{P+1:M}$ from the messages $W_{1:P}$ and the queries as in (\\ref{msg_indep}) and (\\ref{msg_query_indep}); (\\ref{lb2}) follows from the reliability constraint (\\ref{reliability}), since messages $W_{P+1:M}$ can be decoded correctly from the answer strings $A_{1:N}^{[M-P+1:M]}$ if $P \\geq \\frac{M}{2}$ as $\\{P+1, \\cdots, M\\} \\subseteq \\{M-P+1, \\cdots, M\\}$ in this regime; (\\ref{lb3}) follows from the fact that the answer strings are deterministic functions of all messages and queries $(\\cq,W_{1:M})$; and (\\ref{lb4}) follows from the independence bound and Lemma~\\ref{symmetry}.\n\nConsequently, $H(A_1|W_{1:P},\\cq) \\geq \\frac{(M-P)L}{N}$. The proof of the general statement can be done replacing $W_{1:P}$ by $W_{\\cp}$, $W_{P+1:M}$ by $W_{\\bar{\\cp}}$ which corresponds to the complement set of messages of $W_{\\cp}$, and the answer strings $A_{1:N}^{[M-P+1:M]}$ by $A_{1:N}^{[\\cp^*]}$, where $\\bar{\\cp}\\subseteq\\cp^*, \\: |\\cp^*|=P$.\n\\end{Proof}\n\nNow, we are ready to prove the converse of the case $P \\geq \\frac{M}{2}$. We use a similar converse technique to the case of $M=2, P=1$ in \\cite{JafarPIR},\n\\begin{align}\n\t\t  ML&=H(W_{1:M})\\\\\n\\label{c1}  &=H(W_{1:M}|\\cq) \\\\\n\\label{c2}  &=H(W_{1:M}|\\cq)-H(W_{1:M}|A_{1:N}^{[\\cp_1]},A_{1:N}^{[\\cp_2]}, \\cdots, A_{1:N}^{[\\cp_{\\beta}]},\\cq) \\\\\n            &=I(W_{1:M};A_{1:N}^{[\\cp_1]},A_{1:N}^{[\\cp_2]}, \\cdots, A_{1:N}^{[\\cp_{\\beta}]}|\\cq) \\\\\n\\label{c3}  &=H(A_{1:N}^{[\\cp_1]},A_{1:N}^{[\\cp_2]}, \\cdots, A_{1:N}^{[\\cp_{\\beta}]}|\\cq) \\\\\n\\label{c4}  &=H(A_1,A_{2:N}^{[\\cp_1]},A_{2:N}^{[\\cp_2]}, \\cdots, A_{2:N}^{[\\cp_\\beta]}|\\cq)\\\\\n            &=H(A_1,A_{2:N}^{[\\cp_1]}|\\cq)+H(A_{2:N}^{[\\cp_2]}, \\cdots, A_{2:N}^{[\\cp_\\beta]}|A_1,A_{2:N}^{[\\cp_1]},\\cq)\\\\\n\\label{c5}  &=H(A_1,A_{2:N}^{[\\cp_1]}|\\cq)+H(A_{2:N}^{[\\cp_2]}, \\cdots, A_{2:N}^{[\\cp_\\beta]}|A_1,A_{2:N}^{[\\cp_1]},W_\\cp,\\cq)\\\\\n\\label{c6}  &\\leq \\sum_{n=1}^N H(A_n^{[\\cp]}|\\cq)+H(A_{2:N}^{[\\cp_2]}, \\cdots, A_{2:N}^{[\\cp_\\beta]}|A_1,W_\\cp,\\cq)\\\\\n\\label{c7}  &=\\sum_{n=1}^N H(A_n^{[\\cp]}|\\cq)+H(A_{1:N}^{[\\cp_2]}, \\cdots, A_{1:N}^{[\\cp_\\beta]}|W_\\cp,\\cq)-H(A_1|W_\\cp,\\cq)\n\\end{align}\nwhere (\\ref{c1}) follows from the independence between the messages and the queries; (\\ref{c2}) follows from the reliability constraint in (\\ref{reliability}) with  $A_{1:N}^{[\\cp_1]},A_{1:N}^{[\\cp_2]}, \\cdots, A_{1:N}^{[\\cp_{\\beta}]}$ representing all answer strings from all databases to any subset of messages $\\cp \\subseteq \\{1, \\cdots, M\\}$; (\\ref{c3}) follows from the fact that answer strings are deterministic functions of the messages and the queries; (\\ref{c4}) follows from simplification (\\ref{fixed_answer}) without loss of generality; (\\ref{c5}) follows from the fact that the messages $W_\\mathcal{P}=(W_{i_1}, W_{i_2}, \\cdots, W_{i_P})$ can be reconstructed from $A_{1:N}^{[\\cp]}$; and (\\ref{c6}) is a consequence of the fact that conditioning does not increase entropy and Lemma~\\ref{symmetry}.\n\nNow, every message appears in $\\binom{M-1}{P-1}$ different message subsets of size $P$, therefore the answer strings $(A_{1:N}^{[\\cp_2]}, \\cdots, A_{1:N}^{[\\cp_\\beta]})$ are sufficient to construct all messages $W_{1:M}$ irrespective of $\\cp_1$. Therefore,\n\\begin{align}\nH(A_{1:N}^{[\\cp_2]}, \\cdots, A_{1:N}^{[\\cp_\\beta]}|W_\\cp,\\cq)=(M-P)L\n\\end{align}\nUsing this and Lemma~\\ref{interference} in (\\ref{c7}) yields\n\\begin{align}\nML &\\leq \\sum_{n=1}^N H(A_n^{[\\cp]}|\\cq)+(M-P)L-\\frac{(M-P)L}{N}\n\\end{align}\nwhich can be written as,\n\\begin{align}\nPL+\\frac{(M-P)L}{N} &\\leq \\sum_{n=1}^N H(A_n^{[\\cp]}|\\cq)\n\\end{align}\nwhich further can be written as,\n\\begin{align}\n\\left(1+\\frac{M-P}{PN}\\right)PL &\\leq \\sum_{n=1}^N H(A_n^{[\\cp]}|\\cq)\n\\end{align}\nwhich leads to the desired converse result,\n\\begin{align}\n\\sum_{i=1}^PR_i=\\frac{PL}{\\sum_{n=1}^N H\\left(A_n^{[\\mathcal{P}]}\\right)} \\leq \\frac{PL}{\\sum_{n=1}^N H\\left(A_n^{[\\mathcal{P}]}|\\cq\\right)} \\leq \\frac{1}{1+\\frac{M-P}{PN}}\n\\end{align}\n\n\\subsection{Converse Proof for the Case $\\frac{M}{P} > 2$}\\label{converse2}\n\nIn the sequel, we derive an inductive relation that can be used in addition to the base induction step of $1 \\leq \\frac{M}{P} \\leq 2$ derived in the previous sub-section to obtain an upper bound for the MPIR problem. The idea we pursue here is similar in spirit to the one in \\cite{JafarPIR}, where the authors developed a base converse step for $M=2$ messages, and developed an induction over the number of messages $M$ for the case $M>2$. Here, we have developed a base converse step for $1\\leq \\frac{M}{P}\\leq 2$, and now develop an induction over $\\left\\lfloor\\frac{M}{P}\\right\\rfloor$ for the case $\\frac{M}{P} > 2$.\n\nThe following lemma upper bounds the remaining uncertainty of the answer strings after knowing a subset of size $P$ of the interference messages.\n\n\\begin{lemma}[Interference Conditioning Lemma]\nThe remaining uncertainty in the answer strings $A_{2:N}^{[\\cp_2]}$ after conditioning on the messages indexed by $\\cp_1$, such that $\\cp_1 \\cap \\cp_2=\\phi$, $|\\cp_1|=|\\cp_2|=P$ is upper bounded by,\n\\begin{align}\nH(A_{2:N}^{[\\cp_2]}|W_{\\cp_1}, \\cq) \\leq (N-1)[NH(A_1|\\cq)-PL]\n\\end{align}\n\\end{lemma}\n\n\\begin{Proof}\nWe begin with\n\\begin{align}\nH(A_{2:N}^{[\\cp_2]}&|W_{\\cp_1}, \\cq)  \\notag \\\\\n\\label{cond1}&\\leq \\sum_{n=2}^N H(A_n^{[\\cp_2]}|W_{\\cp_1}, \\cq) \\\\\n\\label{cond2}&\\leq \\sum_{n=2}^N H(A_{1:n-1}^{[\\cp_1]},A_n^{[\\cp_2]},A_{n+1:N}^{[\\cp_1]}|W_{\\cp_1},\\cq) \\\\\n&= \\sum_{n=2}^N H(A_{1:n-1}^{[\\cp_1]},A_n^{[\\cp_2]},A_{n+1:N}^{[\\cp_1]},W_{\\cp_1}|\\cq)-H(W_{\\cp_1}|\\cq) \\\\\n\\label{cond3}&= \\sum_{n=2}^N H(A_{1:n-1}^{[\\cp_1]},A_n^{[\\cp_2]},A_{n+1:N}^{[\\cp_1]}|\\cq)+H(W_{\\cp_1}|A_{1:n-1}^{[\\cp_1]},A_n^{[\\cp_2]},A_{n+1:N}^{[\\cp_1]})-H(W_{\\cp_1}) \\\\\n\\label{cond4}&\\leq \\sum_{n=2}^N NH(A_1|\\cq)-H(W_{\\cp_1}) \\\\\n&=(N-1)[NH(A_1|\\cq)-PL]\n\\end{align}\nwhere (\\ref{cond1}) follows from the independence bound; (\\ref{cond2}) follows from the non-negativity of entropy; (\\ref{cond3}) follows from the statistical independence between the messages and the queries; and (\\ref{cond4}) follows from the decodability of $W_{\\cp_1}$ given the answer strings $(A_{1:n-1}^{[\\cp_1]},A_n^{[\\cp_2]}, \\break A_{n+1:N}^{[\\cp_1]})$, which is tantamount to the privacy constraint as in the second simplification.\n\\end{Proof}\n\nNow, we derive the inductive relation for $\\frac{M}{P} > 2$. Without loss of generality, let $\\cp_1=\\{1, \\cdots, P\\}$ and $\\cp_2=\\{P+1, \\cdots, 2P\\}$. Then, starting from (\\ref{c4}), we write\n\\begin{align}\nML=&H(A_1,A_{2:N}^{[\\cp_1]},A_{2:N}^{[\\cp_2]}, \\cdots, A_{2:N}^{[\\cp_\\beta]}|\\cq) \\\\\n  =&H(A_1,A_{2:N}^{[\\cp_1]}|\\cq)+H(A_{2:N}^{[\\cp_2]}|A_1,A_{2:N}^{[\\cp_1]},\\cq)+H(A_{2:N}^{[\\cp_3]}, \\cdots, A_{2:N}^{[\\cp_\\beta]}|A_1,A_{2:N}^{[\\cp_1]},A_{2:N}^{[\\cp_2]},\\cq) \\\\\n  \t\\label{ind1}  \\leq& NH(A_1|\\cq)+H(A_{2:N}^{[\\cp_2]}|A_1,A_{2:N}^{[\\cp_1]},W_{1:P},\\cq)\\notag\\\\\n  &+H(A_{2:N}^{[\\cp_3]}, \\cdots, A_{2:N}^{[\\cp_\\beta]}|A_1,A_{2:N}^{[\\cp_1]},A_{2:N}^{[\\cp_2]},W_{1:2P},\\cq) \\\\\n  \t\\label{ind2} \\leq & NH(A_1|\\cq)+H(A_{2:N}^{[\\cp_2]}|W_{1:P},\\cq)+H(A_{2:N}^{[\\cp_3]},\\! \\cdots\\!, A_{2:N}^{[\\cp_\\beta]}|A_1,W_{1:2P},\\cq) \\\\\n  \t=&NH(A_1|\\cq)+H(A_{2:N}^{[\\cp_2]}|W_{1:P},\\cq)\\!+\\!H(A_{1:N}^{[\\cp_3]},\\! \\cdots\\!, A_{1:N}^{[\\cp_\\beta]}|W_{1:2P},\\cq)\\!-\\!H(A_1|W_{1:2P},\\cq) \\\\\n\\label{ind3} =& NH(A_1|\\cq)+H(A_{2:N}^{[\\cp_2]}|W_{1:P},\\cq)+(M-2P)L-H(A_1|W_{1:2P},\\cq) \\\\\n\\label{ind4} \\leq& NH(A_1|\\cq)+(N-1)[NH(A_1|\\cq)-PL]+(M-2P)L-H(A_1|W_{1:2P},\\cq)\n\\end{align}\nwhere (\\ref{ind1}) follows from the decodability of $W_{1:2P}$ given $(A_1,A_{2:N}^{[\\cp_1]},A_{2:N}^{[\\cp_2]})$, the symmetry lemma and the independence bound; (\\ref{ind2}) follows from the fact that conditioning does not increase entropy. In (\\ref{ind3}), we note that subsets $(\\cp_3, \\cdots, \\cp_\\beta)$ include all messages $(W_1, \\cdots, W_M)$ because every message appears in $\\binom{M-1}{P-1}$ subsets. Hence, $H(A_{1:N}^{[\\cp_3]}, \\cdots, A_{1:N}^{[\\cp_\\beta]}| \\break W_{1:2P},\\cq)=(M-2P)L$ since $W_{2P+1:M}$ is decodable from $(A_{1:N}^{[\\cp_3]}, \\cdots, A_{1:N}^{[\\cp_\\beta]})$ after knowing $W_{1:2P}$. Finally, (\\ref{ind4}) follows from the interference conditioning lemma.\n\nConsequently, (\\ref{ind4}) can be written as\n\\begin{align}\nN^2H(A_1|\\cq) &\\geq (N+1)PL+H(A_1|W_{1:2P},\\cq)\n\\end{align}\nwhich is equivalent to\n\\begin{align}\n\\label{inductive_relation} NH(A_1|\\cq) &\\geq \\left(1+\\frac{1}{N}\\right)PL+\\frac{1}{N}H(A_1|W_{1:2P},\\cq)\n\\end{align}\nNow, (\\ref{inductive_relation}) constructs an inductive relation, since evaluating $NH(A_1|W_{1:2P},\\cq)$ is the same as $NH(A_1|\\cq)$ with $(M-2P)$ messages, i.e., the problem of MPIR with $M$ messages for fixed $P$ is reduced to an MPIR problem with $(M-2P)$ messages for the same fixed $P$. We note that (\\ref{inductive_relation}) generalizes the inductive relation in \\cite{JafarPIR} for $P=1$.\n\nWe can write the induction hypothesis for MPIR with $M$ messages as\n\\begin{align}\nNH(A_1|\\cq) \\geq PL \\left[\\sum_{i=0}^{\\lfloor \\frac{M}{P}\\rfloor-1} \\frac{1}{N^i}+\\left(\\frac{M}{P}-\\left\\lfloor\\frac{M}{P}\\right\\rfloor\\right)\\frac{1}{N^{\\left\\lfloor \\frac{M}{P}\\right\\rfloor}}\\right]\n\\end{align}\n\nNext, we proceed with proving this relation for $M+1$ messages. From the induction hypothesis, we have\n\\begin{align}\nNH(A_1|W_{1:2P},\\cq) &\\geq PL \\left[\\sum_{i=0}^{\\lfloor \\frac{M-2P+1}{P}\\rfloor-1} \\frac{1}{N^i}+\\left(\\frac{M-2P+1}{P}-\\left\\lfloor\\frac{M-2P+1}{P}\\right\\rfloor\\right)\\frac{1}{N^{\\left\\lfloor \\frac{M-2P+1}{P}\\right\\rfloor}}\\right]\\\\\n                     &=PL \\left[\\sum_{i=0}^{\\lfloor \\frac{M+1}{P}\\rfloor-3} \\frac{1}{N^i}+\\left(\\frac{M+1}{P}-\\left\\lfloor\\frac{M+1}{P}\\right\\rfloor\\right)\\frac{1}{N^{\\left\\lfloor \\frac{M+1}{P}\\right\\rfloor-2}}\\right]\n\\end{align}\nsubstituting this in (\\ref{inductive_relation}),\n\\begin{align}\nNH(A_1|\\cq) &\\geq \\left(1+\\frac{1}{N}\\right)PL+\\frac{PL}{N^2}\\left[\\sum_{i=0}^{\\lfloor \\frac{M+1}{P}\\rfloor-3} \\frac{1}{N^i}+\\left(\\frac{M+1}{P}-\\left\\lfloor\\frac{M+1}{P}\\right\\rfloor\\right)\\frac{1}{N^{\\left\\lfloor \\frac{M+1}{P}\\right\\rfloor-2}}\\right]\\\\\n           &=PL \\left[\\sum_{i=0}^{\\lfloor \\frac{M+1}{P}\\rfloor-1} \\frac{1}{N^i}+\\left(\\frac{M+1}{P}-\\left\\lfloor\\frac{M+1}{P}\\right\\rfloor\\right)\\frac{1}{N^{\\left\\lfloor \\frac{M+1}{P}\\right\\rfloor}}\\right] \\label{lastinduction}\n\\end{align}\nwhich concludes the induction argument.\n\nConsequently, the upper bound for the MPIR problem can be obtained as,\n\\begin{align}\n\\sum_{i=1}^P R_i &= \\frac{PL}{\\sum_{n=1}^N H\\left(A_n^{[\\mathcal{P}]}\\right)}\\\\\n                 &\\leq \\frac{PL}{NH(A_1|\\cq)}\\\\\n                 &=\\frac{1}{\\sum_{i=0}^{\\lfloor \\frac{M}{P}\\rfloor-1} \\frac{1}{N^i}+\\left(\\frac{M}{P}-\\left\\lfloor\\frac{M}{P}\\right\\rfloor\\right)\\frac{1}{N^{\\left\\lfloor \\frac{M}{P}\\right\\rfloor}}} \\label{interstep}\\\\\n                 &=\\left(\\frac{1-(\\frac{1}{N})^{\\lfloor \\frac{M}{P}\\rfloor}}{1-\\frac{1}{N}}+\\left(\\frac{M}{P}-\\left\\lfloor\\frac{M}{P}\\right\\rfloor\\right)\\frac{1}{N^{\\left\\lfloor \\frac{M}{P}\\right\\rfloor}}\\right)^{-1} \\label{laststep}\n\\end{align}\nwhere (\\ref{interstep}) follows from (\\ref{lastinduction}); and (\\ref{laststep}) follows from evaluating the sum in (\\ref{interstep}).\n\n\\section{Conclusions}\n\nIn this paper, we introduced the multi-message private information retrieval (MPIR) problem from an information-theoretic perspective. The problem generalizes the PIR problem in \\cite{JafarPIR} which retrieves a single message privately. We determined the exact sum capacity for this problem when the number of desired messages is at least half of the number of total stored messages to be $C_s^P=\\frac{1}{1+\\frac{M-P}{PN}}$. We showed that joint retrieval of the desired messages strictly outperforms repeating the single-message capacity achieving scheme for each message. Furthermore, we showed that if the total number of messages is an integer multiple of the number of desired messages, then the sum capacity is $C_s^P=\\frac{1-\\frac{1}{N}}{1-(\\frac{1}{N})^{M\/P}}$, which resembles the single-message PIR capacity expression when the number of messages is $\\frac{M}{P}$. For the remaining cases, we derived lower and upper bounds. We observed numerically that the gap between the lower and bounds decreases monotonically in $N$, and the worst case gap is $0.0082$ which occurs for the case $N=2$ when $M=5$, $P=2$.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nTo understand the fine structure of a \\cs-algebra, a good first step\nis to describe the primitive ideals in a systematic way.  Therefore\nproducing a prescription for a robust family of irreducible\nrepresentations is very important.  In the case of transformation\ngroup \\cs-algebras, it is well known that representations induced from\nirreducible representations of the stability groups are themselves\nirreducible.  (Furthermore, in many cases these representations\nexhaust the irreducible representations, or at least their kernels\nexhaust the collection of primitive ideals, and a fairly complete\ndescription of the primitive ideal space is possible.  For a more\nextensive discussion, see \\cite{wil:crossed}*{\\S \\S8.2--3}.)  In the\nseparable case, the irreducibility of representations induced from\nirreducible representations of the stability groups is due to Mackey\n\\cite{mac:pnasus49}*{\\S6} (see also Glimm's\n\\cite{gli:pjm62}*{pp.~900--901}).  The result for general\ntransformation group \\cs-algebras was proved in\n\\cite{wil:tams81}*{Proposition~4.2} (see also\n\\cite{wil:crossed}*{Proposition~8.27}).  The corresponding result for\ngroupoid \\cs-algebras has been proved in an \\emph{ad hoc} manner in a\nnumber of special cases (see Example~\\ref{ex-stab-grps} for specific\nreferences).  In this note, we want to prove the\nresult for general \\emph{separable} groupoids.  In so doing, we take\nthe opportunity to formalize the theory of inducing representations\nfrom a general closed subgroupoid.  Of course, induction is treated in\nRenault's thesis \\cite{ren:groupoid}*{Chap.~II \\S2}.  However, at the\ntime \\cite{ren:groupoid} was written, Renault did not yet have the\nfull power of his disintegration theorem\n(\\cite{ren:jot87}*{Proposition~4.2} or see\n\\cite{muhwil:nyjm08}*{Proposition~7.8}) available.  Nor was Rieffel's\ntheory of Morita equivalence fully developed.  So it seems appropriate\nto give a modern treatment here using a contemporary version of\nRieffel's theory, and the disintegration theorem in the form of the\nequivalence theorem from \\cite{mrw:jot87}*{Theorem~2.8}.\n\nIn Section~\\ref{sec:induc-repr}, we derive the general process for\ninducing groupoid representations of a closed subgroupoid $H$ of $G$\nto $G$\n--- actually, we induce representations \nfrom $\\cs(H)$ to $\\cs(G)$.  In\nSection~\\ref{sec:main-theorem}, we specialize to the case where $H$ is\nan isotropy group, $H=G(u)=G^{u}_{u}:=\\set{x\\in G:s(x)=r(x)}$, and\nprove the main result.\n\nThroughout, $G$ will be a second countable locally compact Hausdorff\ngroupoid.  Second countability, in the form of the separability of\n$\\cs(G)$, is necessary in Section~\\ref{sec:induc-repr} in order to\ninvoke the disintegration theorem.  Although separability might be\nunnecessary in the proof of the main theorem, we felt that, as much of\nthe deep theory of groupoid \\cs-algebras uses the disintegration\nresult in one form or another, there was little to be gained which\nwould justify the additional work of adjusting the proof to handle the\ngeneral case.  In addition, we always assume that $G$ and $H$ have\n(continuous) Haar systems.  We adopt the usual conventions that\nrepresentations of \\cs-algebras are nondegenerate and that\nhomomorphisms between \\cs-algebras are necessarily $*$-preserving.\n\n\\section{Inducing Representations}\n\\label{sec:induc-repr}\n\nWe assume that $G$ is a second countable locally compact groupoid with\nHaar system $\\set{\\lambda^{u}}_{u\\in\\go}$.  Let $H$ be a closed\nsubgroupoid of $G$ with Haar system $\\set{\\alpha^{u}}_{u\\in\\ho}$.\nSince $H$ is closed in $G$, we also have $\\ho$ closed in $\\go$.  Then\n$\\GsH:=s^{-1}(\\ho)$ is a locally compact free and proper right\n$H$-space and we can form the imprimitivity groupoid $H^{G}$ as\nfollows.  The space\n\\begin{equation*}\n  \\GsH*_{s}\\GsH:=\\set{(x,y)\\in \\GsH\\times \\GsH:s(x)=s(y)}\n\\end{equation*}\nis a free and proper right $H$-space for the diagonal action\n$(x,y)\\cdot h:=(x h,y h)$.  Consequently, the orbit space\n\\begin{equation*}\n  H^{G}:=(\\GsH*_{s}\\GsH)\/H\n\\end{equation*}\nis a locally compact Hausdorff space.  Following\n\\cite{mrw:jot87}*{\\S2}, $H^{G}$ is a groupoid in a natural way.  If\n$[x,y]$ denotes the orbit of $(x,y)$ in $H^{G}$, then the composable\npairs are given by\n\\begin{equation*}\n  (H^{G})^{(2)}:=\\set{\\bigl([x,y],[z,w]\\bigr):y\\cdot H=z\\cdot H},\n\\end{equation*}\nand the groupoid operations are given by\n\\begin{equation*}\n  [x,y][y h,z]:=[x,z\n  h^{-1}]\\quad\\text{and}\\quad[x,y]^{-1}:=[y,x]. \n\\end{equation*}\nWe can identify $(H^{G})^{(0)}$ with $\\GsH\/H$ and then\n\\begin{equation*}\n  r\\bigl([x,y]\\bigr)=x\\cdot H\\quad\\text{and}\\quad s\\bigl([x,y]\\bigr)\n  =y\\cdot H.\n\\end{equation*}\nIt is not hard to check that $H^{G}$ acts freely and properly on the\nleft of $\\GsH$:\n\\begin{equation*}\n  [x,y]\\cdot (y h)=x h,\n\\end{equation*}\nand that $\\GsH$ is then a $(H^{G},H)$-equivalence as in\n\\cite{mrw:jot87}*{Definition~2.1}. \n\nTo get a Haar system on $H^{G}$, we proceed as in\n\\cite{kmrw:ajm98}*{\\S5}.  Since $\\GsH$ is closed in $G$ and since\n$\\lambda$ is a Haar system on $G$, it is not hard to check that\n\\begin{equation*}\n  \\beta'(\\phi)(x):=\\int_{G} \\phi(y)\\,d\\lambda_{s(x)}(y)\\quad\\text{$\\phi \\in\n    C_{c}(\\GsH)$} \n\\end{equation*}\nis a full equivariant $s$-system for the map $s:\\GsH\\to \\ho$ (as\ndefined in \\cite{kmrw:ajm98}*{\\S5}).  Therefore\n\\cite{kmrw:ajm98}*{Proposition~5.2} implies that we get a Haar system\n$\\set{\\beta^{x\\cdot H}}_{x\\cdot H\\in (H^{G})^{(0)}}$ for $H^{G}$ via\n\\begin{equation*}\n  \\beta(F)(x\\cdot H)=\\int_{H^{G}}F\\bigl([x,y]\\bigr) \\, d\\beta^{x\\cdot\n    H}\\bigl([x,y]\\bigr) = \\int_{G}F\\bigl([x,y]\\bigr)\n  \\,d\\lambda_{s(x)}(y). \n\\end{equation*}\n\nSince both $H$ and $H^{G}$ have Haar systems,\n\\cite{mrw:jot87}*{Theorem~2.8} implies that $C_{c}(\\GsH)$ is a\npre-$C_{c}(H^{G},\\beta) \\sme C_{c}(H,\\alpha)$-\\ib\\ with actions and\ninner products given by\n\\begin{align}\n  \\label{eq:1}\n  F\\cdot \\phi(z)&=\\int_{G} F\\bigl([x,y]\\bigr) \\phi(y)\\,d\\lambda_{s(z)}\n  (y) \\\\\n\\phi\\cdot g(z)&=\\int_{H} \\phi(z h)g(h^{-1}) \\,d\\alpha^{s(z)} (h)\n\\label{eq:2}\\\\\n\\Rip<\\phi,\\psi>(h)&= \\int_{G} \\overline{\\phi(y)} \\psi(y h)\\,\nd\\lambda_{r(h)} (y) \\label{eq:3}\\\\\n\\Lip<\\phi,\\psi>\\bigl([x,y]\\bigr) &= \\int_{H} \\phi(y\nh)\\overline{\\psi(x h)} \\, d\\alpha^{s(x)}(h).\\label{eq:4}\n\\end{align}\nWe will write $\\X=\\X_{H}^{G}$ for the completion of $C_{c}(\\GsH)$ as\na $\\cs(H^{G})\\sme\\cs(H)$-\\ib.\n\nIf $L$ is a representation of $\\cs(H,\\alpha)$, then we write $\\xind L$\nfor the representation of $\\cs(H^{G},\\beta)$ induced via $\\X$ (see the\ndiscussion following \\cite{rw:morita}*{Proposition~2.66}).  Recall\nthat $\\xind L$ acts on the completion $\\Hind$ of $C_{c}(\\GsH)\\atensor\n\\H_{L}$ with respect to the pre-inner product given on elementary\ntensors by\n\\begin{equation*}\n  \\ip(\\phi\\tensor h|\\psi\\tensor\n  k)=\\bip(L\\bigl(\\Rip<\\psi,\\phi>\\bigr)h|k). \n\\end{equation*}\nIf $\\phi\\tensor_{H}h$ denotes the class of $\\phi\\tensor h$ in $\\Hind$,\nthen\n\\begin{equation*}\n  (\\xind L)(F)(\\phi\\tensor_{H}h)=F\\cdot \\phi\\tensor_{H}h.\n\\end{equation*}\n\nTo get an induced representation of $\\cs(G)$ out of this machinery\n(i.e., using \\cite{rw:morita}*{Proposition~2.66}), we need a nondegenerate\nhomomorphism of \\cs(G) into $\\mathcal{L}(\\X)$.  If $f\\in C_{c}(G)$ and\n$\\phi\\in C_{c}(\\GsH)$ we can define\n\\begin{equation}\\label{eq:10}\n  f\\cdot \\phi(z)=\\int_{G} f(y)\\phi(y^{-1}z)\\,d\\lambda^{r(z)}(y).\n\\end{equation}\n\n\\begin{remark}\n  \\label{rem-convolution}\n  Since $\\GsH$ is closed in $G$, each $\\phi\\in C_{c}(\\GsH)$ is the\n  restriction of an element $f_{\\phi}\\in C_{c}(G)$.  Thus we can write\n  \\begin{equation*}\n    \\Rip<\\phi,\\psi>=\\phi*\\psi\\quad\\text{and}\\quad f\\cdot \\phi=f*\\phi,\n  \\end{equation*}\nwhere, for example, $\\phi*\\psi$ should be interpreted as\n$f_{\\phi}*f_{\\psi}$ restricted to $\\GsH$ --- the point being that the\nrestriction is independent of our choice of $f_{\\phi}$ and\n$f_{\\psi}$.  Similarly, $f*\\phi$ is meant to be the restriction of\n$f*f_{\\phi}$ to $\\GsH$.\n\\end{remark}\n\nIf $\\rho$ is a state on $\\cs(H)$, then\n\\begin{equation*}\n  \\ip(\\cdot|\\cdot)_{\\rho}:=\\rho\\bigl(\\Rip<\\cdot,\\cdot>\\bigr)\n\\end{equation*}\nis a pre-inner product on $C_{c}(\\GsH)$ with Hilbert space completion\ndenoted by $\\H_{\\rho}$. If we let $V(f)\\phi:=f\\cdot\\phi$, then\nit follows from direct computation, or by invoking\nRemark~\\ref{rem-convolution} above, that\n\\begin{equation*}\n  \\Rip<V(f)\\phi,\\psi>=\\Rip<f\\cdot \\phi,\\psi> = \\Rip<\\phi,f^{*}\\cdot\n  \\psi> = \\Rip<\\phi,V(f^{*})\\psi>.\n\\end{equation*}\nThus $V$ induces a map of $C_{c}(G)$ into the\nlinear operators on the dense image of $C_{c}(\\GsH)$ in $\\H_{\\rho}$\nwhich clearly satisfies the\naxioms of Renault's disintegration theorem (see, e.g.,\n\\cite{muhwil:nyjm08}*{Theorem~7.8} or\n\\cite{ren:jot87}*{Proposition~4.2}).  In particular, we obtain a\nbona fide representation of $\\cs(G)$ on $\\Hind$,  and it follows that\n\\begin{equation*}\n \\rho\\bigl(\\Rip<f\\cdot \\phi,f\\cdot \\phi>\\bigr) \\le \\|f\\|_{\\cs(G)}^{2}\n  \\rho\\bigl(\\Rip<\\phi,\\phi>\\bigr). \n\\end{equation*}\nSince this holds for all $\\rho$,\n\\begin{equation*}\n  \\Rip<f\\cdot \\phi,f\\cdot \\phi>\\le \\|f\\|^{2}\\Rip<\\phi,\\phi>,\n\\end{equation*}\nand $V(f)$ is a bounded adjointable operator on $\\X$.  Therefore we\nobtain an induced representation $\\indgh L$ of $\\cs(G)$ on $\\Hind$\nsuch that\n\\begin{equation*}\n  (\\indgh L)(f)(\\phi\\tensor_{H}h)=f* \\phi\\tensor_{H}h.\n\\end{equation*}\n\n\\begin{remark}\n  \\label{rem-composition}\n  Since $M\\bigl(\\cs(H^{G})\\bigr)\\cong \\mathcal{L}(\\X)$\n  (\\cite{rw:morita}*{Corollary~2.54 and Proposition~3.8}), it is not\n  hard to see that $\\indgh L$ is the composition of $V$ with the\n  natural extension of $\\xind L$ to $\\mathcal{L}(\\X)$.\n\\end{remark}\n\n\\begin{example}\n  \\label{ex-stab-grps}\n  In the next section, we will be exclusively interested in the\n  special case of the above where $H$ is the stability group at a\n  $u\\in \\go$.  That is,\n  \\begin{equation}\n    \\label{eq:5}\n    H=G(u):=G^{u}_{u}=\\set{x\\in G:s(x)=u=r(x)}.\n  \\end{equation}\n(Thus, $H^{(0)}=\\set u$.)   In this case, we obtain the induced\nrepresentations used to establish special cases of\nTheorem~\\ref{thm-main} \nin  \n\\citelist{\\cite{muhwil:ms90}*{Lemma~2.4}\n\\cite{muhwil:ms92}*{Lemma~3.2}\n\\cite{mrw:tams96}*{Lemma~2.5}\n\\cite{cla:iumj07}*{Lemma~4.2}\\cite{facska:jot82}*{\\S5}}.\n\\end{example}\n\nOne advantage of having a formal theory of induction for\nrepresentations of groupoid \\cs-algebras is that we can apply the\nRieffel machinery.  An example is the following version of induction\nin stages.  The proof, modulo technicalities, is a\nstraightforward modification of Rieffel's original ``\\cs-version''\nfrom \\cite{rie:aim74}*{Theorem~5.9}.  For future reference, we've\nworked out the details of the proof in the last section.\n\n\\begin{thm}[Induction in Stages]\n  \\label{thm-stages}\n Suppose that $H$ and $K$ are closed subgroupoids of a second\n countable locally compact Hausdorff groupoid $G$ with $H\\subset K$.\n Assume that $H$, $K$ and $G$ have Haar systems. If $L$ is a\n representation of $\\cs(H)$, then \n \\begin{equation*}\n   \\Ind_{H}^{G}L\\quad\\text{and}\\quad \\Ind_{K}^{G}\\bigl(\\Ind_{H}^{K} L\\bigr)\n \\end{equation*}\nare equivalent representations of $\\cs(G)$.\n\\end{thm}\n\n\\section{The Main Theorem}\n\\label{sec:main-theorem}\n\n\\begin{thm}\n  \\label{thm-main}\n  Let $G$ be a second countable groupoid with Haar system\n  $\\set{\\lambda^{u}}_{u\\in\\go}$.  Suppose that $L$ is an irreducible\n  representation of the stability group $G(u)$ at $u\\in\\go$.  Then\n  $\\indgug L$ is an irreducible representation of $\\cs(G)$.\n\\end{thm}\n\nThe idea of the proof is straightforward.  Let $L$ be an irreducible\nrepresentation of $\\cs\\bigl(G(u)\\bigr)$.  Since $\\X$ is a\n$\\cs\\bigl(G(u)^{G}\\bigr)\\sme \\cs\\bigl(G(u)\\bigr)$-\\ib,\n\\cite{rw:morita}*{Corollary~3.32} implies that $\\xind L$ is an\nirreducible representation of $\\cs\\bigl(G(u)^{H}\\bigr)$.  \nWe will show that any $T$ in the commutant of $\\indgug L$ is a\nscalar multiple of the identity.  It will suffice to see\nthat any such $T$ commutes with $(\\xind L)(F)$ for all $F\\in\nC_{c}\\bigl(G(u)^{G}\\bigr)$.  Our proof will consist in\nproducing, given $F$, a net $\\set{f_{i}}$ in $C_{c}(G)$ such that\n\\begin{equation*}\n  (\\indgug L)(f_{i})\\to (\\xind L)(F)\n\\end{equation*}\nin the weak operator topology.  Since we will also arrange that this\nnet is uniformly bounded in the $\\|\\cdot\\|_{I}$-norm on $C_{c}(G)$ ---\nso that the net $\\set{(\\indgug L)(f_{i})}$ is uniformly bounded in\n$B\\bigl(\\Hind)\\bigr)$ --- we just have to arrange that\n\\begin{equation*}\n \n  \\bip((\\indgug L)(f_{i})(\\phi\\tensor_{G(u)}h) |\n  \\psi\\tensor_{G(u)}k) \\to \\bip((\\xind L)(F) (\\phi\\tensor_{G(u)}h) |\n  \\psi\\tensor_{G(u)}k)\n\\end{equation*}\nfor all $\\phi,\\psi\\in C_{c}(G_{u})$ and $h,k\\in\\H_{L}$.\n\nThe next lemma is the essential ingredient to our proof.\n\n\\begin{lemma}\n  \\label{lem-key}\n  Suppose that $F\\in C_{c}\\bigl(G(u)^{G}\\bigr)$.  Then there is\n  compact set $C_{F}$ in $G$ such that for each compact set $K\\subset\n  G_{u}$ there is a $f_{K}\\in C_{c}(G)$ such that\n  \\begin{enumerate}\n  \\item $f_{K}(zy^{-1})=F\\bigl([z,y]\\bigr)$ for all $(z,y)\\in K\\times\n    K$,\n  \\item $\\supp f_{K} \\subset C_{F}$ and\n  \\item $\\|f_{K}\\|_{I}\\le \\|F\\|_{I}+1$.\n  \\end{enumerate}\n\\end{lemma}\n\nThe proof of Lemma~\\ref{lem-key} is a bit technical, so we'll postpone\nthe proof for a bit, and show that the lemma allows us to prove\nTheorem~\\ref{thm-main}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm-main}]\n  For each $K\\subset G_{u}$, let $f_{K}$ be as in\n  Lemma~\\ref{lem-key}.  Then $\\set{f_{K}}$ and $\\set{(\\indgug\n    L)(f_{K})}$ are nets indexed by increasing $K$. Notice that\n  \\begin{multline}\\label{eq:9}\n    \\bip((\\indgug L)(f_{K})(\\phi\\tensor_{G(u)}h) |\n  \\psi\\tensor_{G(u)}k) -{}\\\\ \\bip((\\xind L)(F) (\\phi\\tensor_{G(u)}h) |\n  \\psi\\tensor_{G(u)}k) \\\\\n= \\bip( L\\bigl(\\Rip<\\psi,f_{K}*\\phi-F\\cdot \\phi>\\bigr)h|k)\n  \\end{multline}\nFurthermore, using the invariance of the Haar system on $G$, we can\ncompute as follows:\n\\begin{equation}\n  \\label{eq:7}\n  \\begin{split}\n    \\Rip<\\psi,f_{K}*\\phi>(s)&= \\int_{G}\\overline{\\psi(x)}\n    f_{K}*\\phi(xs) \\, d\\lambda_{u}(x) \\\\\n&= \\int_{G}\\int_{G} \\overline{\\psi(x)} f_{K}(xz^{-1})\\phi(zs)\n\\,d\\lambda_{u}(z) \\,d\\lambda_{u}(x),\n  \\end{split}\n\\end{equation}\nwhile on the other hand,\n\\begin{equation}\n  \\label{eq:8}\n  \\begin{split}\n    \\Rip<\\psi,F\\cdot \\phi>(s)&= \\int_{G}\\overline{\\psi(x)} F\\cdot\n    \\phi(xs) \\,d\\lambda_{u}(x) \\\\\n&=\\int_{G}\\int_{G} \\overline{\\psi(x)} F\\bigl([xs,z]\\bigr) \\phi(z)\n\\,d\\lambda_{u}(z) \\,d\\lambda_{u}(x) \\\\\n&= \\int_{G}\\int_{G} \\overline{\\psi(x)} F\\bigl([x,zs^{-1}]\\bigr) \\phi(z)\n\\,d\\lambda_{u}(z) \\,d\\lambda_{u}(x) \\\\\n&= \\int_{G}\\int_{G} \\overline{\\psi(x)} F\\bigl([x,z]\\bigr) \\phi(z)\n\\,d\\lambda_{u}(zs) \\,d\\lambda_{u}(x)\n  \\end{split}\n\\end{equation}\n\n\nNotice that $\\supp\\Rip<\\psi,\\phi>\\subset (\\supp \\psi)(\\supp\\phi)$.\nSince $\\supp f_{K}\\subset C_{F}$ for all $K$, we have\n\\begin{equation*}\n  \\supp f_{K}*\\phi\\subset (\\supp f_{K})(\\supp \\phi)\\subset\nC_{F}(\\supp\\phi).\n\\end{equation*}\n  Therefore if \\eqref{eq:7} does not vanish, then we\nmust have $s\\in (\\supp\\psi)C_{F}(\\supp \\phi)$.  Therefore there is a\ncompact set $K_{0}$ --- which does \\emph{not} depend on $K$ --- such\nthat both \\eqref{eq:7} and \\eqref{eq:8} vanish if $s\\notin K_{0}$.\nThus if $s\\in K_{0}$ and if $K \\supset (\\supp\\psi)\\cup (\\supp\n\\phi)K_{0}^{-1}$, then the integrand in \\eqref{eq:7} is either zero or\nwe must have $(x,z)\\in K\\times K$.  Therefore we can replace\n$f_{K}(xz^{-1})$ by $F\\bigl([x,z]\\bigr)$ and $f_{K}*\\phi-F\\cdot\\phi$\nis the zero function  whenever $K$ contains $(\\supp\\psi)\\cup (\\supp\n\\phi)K_{0}^{-1}$.  Therefore the left-hand side of\n\\eqref{eq:9} is eventually zero, and the\ntheorem follows.\n\\end{proof}\n\nWe still need to prove Lemma~\\ref{lem-key}, and to do that, we need some\npreliminaries.  In the sequel, if $S$ is a Borel subset of $G$, then\n\\begin{equation*}\n  \\int_{S}f(x)\\,d\\lambda^{u}(x):=\\int_{G}\\charfcn{S}(x)f(x)\\,d\\lambda^{u}(x)\n\\end{equation*}\nwhere $\\charfcn{S}$ is the characteristic function of $S$.\n\n\\begin{lemma}\n  \\label{lem-bdd}\n  Suppose that $f\\in C_{c}^{+}(G)$ and that $K\\subset G$ is a compact\n  set such that\n  \\begin{equation*}\n    \\int_{K}f(x)\\,d\\lambda^{u}(x)\\le M\\quad\\text{for all $u\\in\\go$.}\n  \\end{equation*}\nThere there is a neighborhood $V$ of $K$ such that\n\\begin{equation*}\n  \\int_{V}f(x)\\,d\\lambda^{u}(x)\\le M+1\\quad\\text{for all $u\\in\\go$.}\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\n  Let $K_{1}$ be a compact neighborhood\n  of $K$.  Since $G$ is second countable, we can find a countable\n  fundamental system $\\set{V_{n}}$ of neighborhoods of $K$ in $K_{1}$;\n  thus, given any neighborhood $V$ of $K$, there is a $n$ such that\n  $V_{n}\\subset V$ and \n$K=\\bigcap V_{n}$.  Certainly, we can assume that $V_{n+1}\\subset\nV_{n}$.\n\nIf no $V$ as prescribed in the lemma exists, then for each $n$\nwe can find $u_{n}\\in\\go$\nsuch that\n  \\begin{equation*}\n    \\int_{V_{n}}f(x)\\,d\\lambda^{u_{n}}(x)\\ge M+1.\n  \\end{equation*}\nSince we must have each $u_{n}\\in r(K_{1})$, we can pass to a\nsubsequence, relabel, and assume that $u_{n}\\to u_{0}$.  Since\n$\\charfcn{V_{n}}\\to \\charfcn{K}$ pointwise, the dominated convergence\ntheorem implies that\n\\begin{equation*}\n  \\int_{V_{n}}f(x)\\,d\\lambda^{u_{0}}(x)\\to\\int_{K}f(x)\\,d\\lambda^{u_{0}}(x).\n\\end{equation*}\nIn particular, there is a $n_{1}$ such that \n\\begin{equation*}\n  \\int_{V_{n_{1}}}f(x)\\,d\\lambda^{u_{0}}(x)\\le M+\\frac12\n\\end{equation*}\nLet $W$ be an open set such that $K\\subset W\\subset \\overline{W}\n\\subset V_{n_{1}}$, and let $f_{0}\\in C_{c}^{+}(G)$ be such that\n$f_{0}\\restr{\\overline{W}}=f$, $f_{0}\\le f$ and $\\supp f_{0}\\subset\nV_{n_{1}}$.  Then\n  \\begin{equation*}\n    \\int_{G} f_{0}(x) \\,d\\lambda^{u_{0}}(x)\\le M+\\frac 12.\n  \\end{equation*}\nHowever, since $\\set{\\lambda^{u}}$ is a Haar system,\n\\begin{equation*}\n  \\int_{G}f_{0}(x)\\,d\\lambda^{u_{n}}(x)\\to\n  \\int_{G}f_{0}(x)\\,d\\lambda^{u_{0}}(x)\\le M+\\frac12.\n\\end{equation*}\nBut for large $n$, we have $V_{n}\\subset W$ and therefore\n\\begin{equation*}\n  \\int_{G} f_{0}(x) \\,d\\lambda^{u_{n}}(x)\\ge \\int_{V_{n}}f_{0}(x)\n  \\,d\\lambda^{u_{n}}(x) = \\int_{V_{n}}f(x) \\,d\\lambda^{u_{n}}(x) \\ge M+1.\n\\end{equation*}\nThis leads to a contradiction and completes the proof of the lemma.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem-key}]\n  The map $(z,y)\\mapsto zy^{-1}$ is certainly continuous on\n  $G_{u}\\times G_{u}$ and factors through the orbit map\n  $\\pi:G_{u}\\times G_{u}\\to G(u)^{G}$.  In fact, if $zy^{-1}=xw^{-1}$,\n  then we must have $z=x(w^{-1}y)$ and $y=w(x^{-1}z)$.  But\n  $w^{-1}y=x^{-1}z$ and lies in $G(u)$.  Therefore, we have a\n  well-defined injection $\\Pi:G(u)^{G}\\to G$ sending $[z,y]$ to\n  $zy^{-1}$.  \nWe let $C_{F}$ be a compact neighborhood of $\\Pi(\\supp F)$.\n\nFix a compact set $K\\subset G_{u}$.\nThe restriction of $\\Pi$ to the compact set $\\pi(K\\times\n  K)$ is a homeomorphism so we can find a function $\\tfk\\in C_{c}(G)$\n  such that $\\supp\\tfk\\subset C_{F}$ and such that $\\tfk(zy^{-1}) =\n  F\\bigl([z,y]\\bigr)$ for all $(z,y)\\in K\\times K$.\n\nLet $K_{G}:=\\pi(K\\times K)$.  If\n\\begin{equation*}\n  \\int_{K_{G}}|\\tfk(y)|\\,d\\lambda^{w}(y)\\not=0,\n\\end{equation*}\nthen $K_{G}\\cap G^{w}\\not=\\emptyset$.  Thus there is a $z\\in K$ such\nthat $r(z)=w$ (and $s(z)=u$).  Then by left invariance\n\\begin{align*}\n  \\int_{K_{G}}|\\tfk(y)|\\,d\\lambda^{w}(y)&=\n  \\int_{G}\\charfcn{K_{G}}(zy)|\\tfk(zy)|\\, d\\lambda^{u}(y) \\\\\n&= \\int_{G}\\charfcn{K_{G}}(zy^{-1})|\\tfk(zy^{-1})|\\, d\\lambda_{u}(y)\\\\\n&= \\int_{G} \\charfcn{K_{G}}(zy^{-1}) |F\\bigl([z,y]\\bigr)| \\,d\\lambda_{u}(y) \\\\\n&\\le \\|F\\|_{I}.\n\\end{align*}\nSimilarly, if\n\\begin{equation*}\n  \\int_{K_{G}}|\\tfk(y^{-1})| \\,d\\lambda^{w}(y) \\not=0,\n\\end{equation*}\nthen as before there is a $z\\in K$ such that $r(z)=w$ and\n\\begin{align*}\n  \\int_{K_{G}}|\\tfk(y^{-1})|\\,d\\lambda^{w}(y) &=\n  \\int_{G}\\charfcn{K_{G}} (zy) |\\tfk(y^{-1}z^{-1})| \\,d \\lambda^{u}(y)\n  \\\\\n&= \\int_{G}\\charfcn{K_{G}}(zy^{-1})| \\tfk(yz^{-1})| \\,d\\lambda_{u}(y)\n\\\\\n\\intertext{which, since $K_{G}^{-1}=K_{G}$, is}\n&= \\int_{G}\\charfcn{K_{G}}(yz^{-1}) |\\tfk(yz^{-1})| \\,d\\lambda_{u}(y)\n\\\\\n&\\le \\int_{G}|F\\bigl([y,z]\\bigr)| \\,d\\lambda_{u}(y) \\\\\n&=\\int_{G(u)^{G}}|F\\bigl([z,y]^{-1}\\bigr) \\,d\\beta^{z\\cdot\n  G(u)}\\bigl([z,y]\\bigr) \\\\\n&\\le \\|F\\|_{I}.\n\\end{align*}\nUsing Lemma~\\ref{lem-bdd}, we can find a neighborhood $V$ of $K_{G}$\ncontained in $C_{F}$ such that both\n\\begin{equation*}\n  \\int_{V}|\\tfk(x)|\\,d\\lambda^{w}(x) \\quad\\text{and}\\quad\n\\int_{V} |\\tfk(x^{-1})|\\,d\\lambda^{w}(x)\n\\end{equation*}\nare bounded by $\\|F\\|_{I}+1$ for all $w\\in\\go$.  Since $K_{G}$ is\nsymmetric, we can assume that $V=V^{-1}$ as well.  We can now let\n$f_{K}$ be any element of $C_{c}(G)$ such that $f_{K}=\\tfk$ on\n$K_{G}$, $\\supp f_{K}\\subset V$ and $f_{K}\\le\\tfk$ everywhere.\nThen $\\|f\\|_{I}\\le\\|F\\|+1$, $\\supp f_{K}\\subset C_{F}$ and\n$f_{K}(yz^{-1})= F\\bigl([z,y]\\bigr)$ for all $(z,y)\\in K\\times K$.\nThis completes the proof of the lemma.\n\\end{proof}\n\n\\begin{example}[Holonomy Groupoid]\n  \\label{ex-holonomy}\n  Let $(V,\\mathcal{F})$ be a $C^{\\infty}$ compact foliated manifold,\n  and let $G$ be its holonomy groupoid\n  \\citelist{\\cite{win:agag83}\\cite{hae:a84}} \n  equipped with its usual locally compact topology as in\n  \\cite{con:pspm80}.  Naturally, we also assume that $G$ is Hausdorff\n  so that our results apply.  The stability groups $G(x)$ for $x\\in V$\n  are the holonomy groups for the foliation.  Using\n  Theorem~\\ref{thm-main}, each irreducible\n  representation $\\sigma_{x}$ of $G(x)$ provides an irreducible\n  representation $\\Ind_{G(x)}^{G}\\sigma_{x}$ of $\\cs(G)$.  This\n  representation is equivalent to the representations\n  $\\Ind_{x}\\sigma_{x}$ treated in \\cite{facska:jot82}*{\\S5}.\n  Thus we recover a part of \\cite{facska:jot82}*{Corollaire~5.7}.\n\\end{example}\n\n\\begin{example}\n  \\label{ex-deaconu}\n  If $\\sigma:X\\to X$ is a covering map for a compact Hausdorff space\n  $X$, then the associated \\emph{Deaconu-Renualt groupoid}\n  \\cite{dea:tams95} is\n  \\begin{equation*}\n    G:=\\set{(z,n-l,w)\\in X\\times \\Z\\times\n      X:\\sigma^{l}(z)=\\sigma^{n}(w)}. \n  \\end{equation*}\nMore concretely, we can let $X$ be the circle $\\T$ and\n$\\sigma(z):=z^{2}$.  Then the stability group $G(z)$ at $(z,0,z)$ is\ntrivial unless $z=1$ or $z$ is a primitive $2^{n}$-th root of unity.\nThen $G(1)=\\set{1,k,1):k\\in\\Z}$ and if $z$ is a primitive $2^{n}$-th\nroot of unity, then $G(z)=\\set{(z,nk,z):k\\in \\Z}$.  Now applying\nTheorem~\\ref{thm-main}, we see that if $z$ is not a primitive root,\n$\\Ind_{G(z)}^{G}\\delta(z,0,z)$ is an irreducible regular\nrepresentation.  If $z$ is a primitive $2^{n}$-th root of unity, then\nfor each $\\omega\\in \\T\\cong \\widehat\\Z$, we obtain an irreducible\nrepresentation $\\Ind_{G(z)}^{G}\\omega$. \n\\end{example}\n\n\\section{Proof of Theorem~\\ref{thm-stages}}\n\nWe let $\\lambda$, $\\beta$ and $\\alpha$ be Haar systems on $G$, $K$ and\n$H$, respectively. \nIt will be helpful to notice that the space $\\Hind$ of $\\indhg L$ is\nan internal tensor product $\\X_{H}^{G}\\tensor_{H}\\H_{L}$ for the\nappropriate actions of $\\cs(H)$.\\footnote{Internal tensor products of\n  Hilbert modules are discussed in\n  \\citelist{\\cite{lan:hilbert}\\cite{wil:crossed}*{App.~I}}.} Thus the\nspace of $\\Ind_{K}^{G}\\bigl(\\indgh L\\bigr)$ is\n$\\X_{K}^{G}\\tensor_{K}(\\X_{H}^{K}\\tensor_{H}\\H_{L})$.  Of course, the\nnatural map on the algebraic tensor products induces an isomorphism\n$U$ of $\\X_{K}^{G}\\tensor_{K}(\\X_{H}^{K}\\tensor_{H}\\H_{L})$ with\n$(\\X_{K}^{G}\\tensor_{K}\\X_{H}^{K)}\\tensor_{H}\\H_{L}$ (see\n\\cite{wil:crossed}*{Lemma~I.6}).  \nWe need to combine this with the following observation.\n\\begin{lemma}\n  \\label{lem-rieffel-tech}\n  The map sending $\\phi\\atensor\\psi\\in C_{c}(G_{K^{(0)}})\\atensor\n  C_{c}(K_{\\ho})$ to $\\theta(\\phi\\tensor\\psi)$ in $C_{c}(G_{\\ho})$,\n  given by\n  \\begin{equation*}\n    \\theta(\\phi\\tensor\\psi)(x):=\n\\int_{K}\\phi(xk)\\psi(k^{-1})\\,d\\beta^{s(x)}(k) ,\n  \\end{equation*}\ninduces an isomorphism, also called $\\theta$, of $\\X_{K}^{G}\\tensor\n\\X_{H}^{K}$ onto $\\X_{H}^{G}$.\n\\end{lemma}\n\\begin{proof}\n  The first step is to see that $\\theta$ is isometric.  Notice that we\n  have three sets of actions and inner products.  We have not tried to\n  invent notation to distinguish one from another.  Instead, we will\n  hope that it is ``clear from context'' which formula is being\n  employed.  In this spirit,\n  \\begin{align*}\n    \\Rip<\\phi_{1}&\\tensor \\psi_{1},\\phi_{2}\\tensor \\psi_{2}>(h) =\n    \\Rip<\\psi_{1}, {\\Rip<\\phi_{1},\\phi_{2}>\\cdot \\psi_{2}}>(h)\\\\\n\\intertext{which, using \\eqref{eq:3} on $C_{c}(K_{\\ho})$, is}\n&=\\int_{K}\\overline{\\psi_{1}(k)} \\Rip<\\phi_{1},\\phi_{2}>\\cdot \\psi_{2}(kh) \\,\nd\\beta_{r(h)}(k) \\\\\n\\intertext{which, using \\eqref{eq:10} for the $C_{c}(K)$-action on\n  $C_{c}(K_{\\ho})$, is}\n&=\n\\int_{K}\\int_{K} \\overline{\\psi_{1}(k)}\n\\Rip<\\phi_{1},\\phi_{2}>(kk_{1})\\psi_{2}\n(k_{1}^{-1}k)\\,d\\beta^{r(h)}(k_{1}) \\, d\\beta_{r(h) }(k) \\\\\n\\intertext{which, using \\eqref{eq:3}, is}\n&=\n\\int_{K}\\int_{K} \\int_{G}\\overline{\\psi_{1}(k)\\phi_{1}(x)} \\phi_{2}(xkk_{1})\n\\psi_{2}\n(k_{1}^{-1}k)\\, d\\lambda_{r(k)}(x) \n\\,d\\beta^{r(h)}(k_{1}) \\, d\\beta_{r(h) }(k)\\\\\n&=\\int_{K}\\int_{K}\\int_{G} \\overline{\\psi_{1}(k)\\phi_{1}(xk^{-1})} \\phi_{2}(xk_{1})\n\\psi_{2}\n(k_{1}^{-1}k)\\, d\\lambda_{r(h)}(x) \n\\,d\\beta^{r(h)}(k_{1}) \\, d\\beta_{r(h) }(k)\\\\\n\\intertext{which, after using Fubini and sending $k_{1}$ to $hk_{1}$,\n  is}\n& =\\int_{G}\\overline{\\theta(\\phi_{1}\\tensor\\psi_{1})(x)}\n\\theta(\\phi_{2}\\tensor \\psi_{2})(xh)\\,d\\lambda_{r(h)}(x) \\\\\n&= \\Rip<\\theta(\\phi_{1}\\tensor\\psi_{1}),\\theta(\\phi_{2}\\tensor \\psi_{2})>.\n \\end{align*}\nThus, $\\theta$ is isometric.  We just need to see that it has dense\nrange. \n\nHowever, notice that $\\theta(\\phi\\tensor\\psi)=\\phi\\cdot g_{\\psi}$ for\nthe right action on $C_{c}(K)$ on $C_{c}(G_{K^{(0)}})$ with $g_{\\psi}$\n\\emph{any} extension of $\\psi$ to $C_{c}(K)$ (see \\eqref{eq:2} and\nRemark~\\ref{rem-convolution}).  It follows from\n\\cite{mrw:jot87}*{Proposition~2.10} that there is an approximate\nidentity for $C_{c}(K)$ such that $\\phi\\cdot g_{i}\\to \\phi$ in the\ninductive limit topology for all $\\phi\\in C_{c}(_{K^{(0)}})$.  This\nimplies that the range of $\\theta$ is dense, and completes the proof\nof the lemma.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm-stages}]\n  Define a unitary\n  $V:\\X_{K}^{G}\\tensor_{K}(\\X_{H}^{K}\\tensor_{H}\\H_{L})\\to\n  \\X_{H}^{G}\\tensor_{H}\\H_{L}$ by $V=\\theta\\circ U$ (where $U$ is\n  defined prior to Lemma~\\ref{lem-rieffel-tech}).  Then on elementary\n  tensors,\n  $V\\bigl(\\phi\\tensor(\\psi\\tensor\n    h)\\bigr)=\\theta(\\phi\\tensor\\psi)\\tensor h$.\nThen on the one hand,\n\\begin{equation*}\n  V\\bigl(\\Ind_{K}^{G}\\bigl(\\Ind_{H}^{K}(L)(f)\\bigr)\\bigr)\n  \\bigl(\\phi\\tensor(\\psi\\tensor h)\\bigr) =\n  \\theta(f*\\phi\\tensor\\psi)\\tensor h.\n\\end{equation*}\nOn the other hand, $\\theta(f*\\phi\\tensor\\psi)=f*\\theta(\\phi\\tensor\n\\psi)$.  Therefore\n\\begin{equation*}\n  V(\\Ind_{K}^{G}\\bigl(\\Ind_{H}^{K}(L))=(\\indgh L)V.\n\\end{equation*}\nThis completes the proof.\n\\end{proof}\n\n\n\n\n\\def\\noopsort#1{}\\def$'$} \\def\\sp{^{$'$} \\def\\sp{^}\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{cla:iumj07}{article}{\n      author={Clark, Lisa~Orloff},\n       title={C{CR} and {GCR} groupoid {$C\\sp *$}-algebras},\n        date={2007},\n        ISSN={0022-2518},\n     journal={Indiana Univ. Math. J.},\n      volume={56},\n      number={5},\n       pages={2087\\ndash 2110},\n      review={\\MR{MR2359724}},\n}\n\n\\bib{con:pspm80}{incollection}{\n      author={Connes, Alain},\n       title={A survey of foliations and operator algebras},\n        date={1982},\n   booktitle={Operator algebras and applications, {P}art {I} ({K}ingston,\n  {O}nt., 1980)},\n      series={Proc. Sympos. Pure Math.},\n      volume={38},\n   publisher={Amer. Math. Soc.},\n     address={Providence, R.I.},\n       pages={521\\ndash 628},\n      review={\\MR{MR679730 (84m:58140)}},\n}\n\n\\bib{dea:tams95}{article}{\n      author={Deaconu, Valentin},\n       title={Groupoids associated with endomorphisms},\n        date={1995},\n        ISSN={0002-9947},\n     journal={Trans. Amer. Math. Soc.},\n      volume={347},\n       pages={1779\\ndash 1786},\n      review={\\MR{95h:46104}},\n}\n\n\\bib{facska:jot82}{article}{\n      author={Fack, T.},\n      author={Skandalis, G.},\n       title={Sur les repr\\'esentations et id\\'eaux de la {$C\\sp{\\ast}\n  $}-alg\\`ebre d'un feuilletage},\n        date={1982},\n        ISSN={0379-4024},\n     journal={J. Operator Theory},\n      volume={8},\n      number={1},\n       pages={95\\ndash 129},\n      review={\\MR{MR670180 (84d:46101)}},\n}\n\n\\bib{gli:pjm62}{article}{\n      author={Glimm, James},\n       title={Families of induced representations},\n        date={1962},\n        ISSN={0030-8730},\n     journal={Pacific J. Math.},\n      volume={12},\n       pages={885\\ndash 911},\n      review={\\MR{MR0146297 (26 \\#3819)}},\n}\n\n\\bib{hae:a84}{article}{\n      author={Haefliger, Andr{\\'e}},\n       title={Groupo\\\"\\i des d'holonomie et classifiants},\n        date={1984},\n        ISSN={0303-1179},\n     journal={Ast\\'erisque},\n      number={116},\n       pages={70\\ndash 97},\n        note={Transversal structure of foliations (Toulouse, 1982)},\n      review={\\MR{MR755163 (86c:57026a)}},\n}\n\n\\bib{kmrw:ajm98}{article}{\n      author={Kumjian, Alexander},\n      author={Muhly, Paul~S.},\n      author={Renault, Jean~N.},\n      author={Williams, Dana~P.},\n       title={The {B}rauer group of a locally compact groupoid},\n        date={1998},\n        ISSN={0002-9327},\n     journal={Amer. J. Math.},\n      volume={120},\n      number={5},\n       pages={901\\ndash 954},\n      review={\\MR{2000b:46122}},\n}\n\n\\bib{lan:hilbert}{book}{\n      author={Lance, E.~Christopher},\n       title={{H}ilbert {\\cs}-modules: A toolkit for operator algebraists},\n      series={London Math. Soc. Lecture Note Series},\n   publisher={Cambridge Univ. Press},\n     address={Cambridge},\n        date={1994},\n      volume={210},\n}\n\n\\bib{mac:pnasus49}{article}{\n      author={Mackey, George~W.},\n       title={Imprimitivity for representations of locally compact groups.\n  {I}},\n        date={1949},\n     journal={Proc. Nat. Acad. Sci. U. S. A.},\n      volume={35},\n       pages={537\\ndash 545},\n      review={\\MR{MR0031489 (11,158b)}},\n}\n\n\\bib{mrw:jot87}{article}{\n      author={Muhly, Paul~S.},\n      author={Renault, Jean~N.},\n      author={Williams, Dana~P.},\n       title={Equivalence and isomorphism for groupoid {$C^*$}-algebras},\n        date={1987},\n        ISSN={0379-4024},\n     journal={J. Operator Theory},\n      volume={17},\n      number={1},\n       pages={3\\ndash 22},\n      review={\\MR{88h:46123}},\n}\n\n\\bib{mrw:tams96}{article}{\n      author={Muhly, Paul~S.},\n      author={Renault, Jean~N.},\n      author={Williams, Dana~P.},\n       title={Continuous-trace groupoid {\\cs}-algebras, {III}},\n        date={1996},\n     journal={Trans. Amer. Math. Soc.},\n       pages={3621\\ndash 3641},\n}\n\n\\bib{muhwil:ms90}{article}{\n      author={Muhly, Paul~S.},\n      author={Williams, Dana~P.},\n       title={Continuous trace groupoid {\\cs}-algebras},\n        date={1990},\n     journal={Math. Scand.},\n      volume={66},\n       pages={231\\ndash 241},\n}\n\n\\bib{muhwil:ms92}{article}{\n      author={Muhly, Paul~S.},\n      author={Williams, Dana~P.},\n       title={Continuous trace groupoid {\\cs}-algebras. {II}},\n        date={1992},\n     journal={Math. Scand.},\n      volume={70},\n       pages={127\\ndash 145},\n}\n\n\\bib{muhwil:nyjm08}{book}{\n      author={Muhly, Paul~S.},\n      author={Williams, Dana~P.},\n       title={Renault's equivalence theorem for groupoid crossed products},\n      series={NYJM Monographs},\n   publisher={State University of New York University at Albany},\n     address={Albany, NY},\n        date={2008},\n      volume={3},\n        note={Available at http:\/\/nyjm.albany.edu:8000\/m\/2008\/3.htm},\n}\n\n\\bib{rw:morita}{book}{\n      author={Raeburn, Iain},\n      author={Williams, Dana~P.},\n       title={Morita equivalence and continuous-trace {$C^*$}-algebras},\n      series={Mathematical Surveys and Monographs},\n   publisher={American Mathematical Society},\n     address={Providence, RI},\n        date={1998},\n      volume={60},\n        ISBN={0-8218-0860-5},\n      review={\\MR{2000c:46108}},\n}\n\n\\bib{ren:groupoid}{book}{\n      author={Renault, Jean},\n       title={A groupoid approach to {\\cs}-algebras},\n      series={Lecture Notes in Mathematics},\n   publisher={Springer-Verlag},\n     address={New York},\n        date={1980},\n      volume={793},\n}\n\n\\bib{ren:jot87}{article}{\n      author={Renault, Jean},\n       title={Repr\\'esentations des produits crois\\'es d'alg\\`ebres de\n  groupo\\\"\\i des},\n        date={1987},\n     journal={J. Operator Theory},\n      volume={18},\n       pages={67\\ndash 97},\n}\n\n\\bib{rie:aim74}{article}{\n      author={Rieffel, Marc~A.},\n       title={Induced representations of {$C^*$}-algebras},\n        date={1974},\n     journal={Advances in Math.},\n      volume={13},\n       pages={176\\ndash 257},\n      review={\\MR{50 \\#5489}},\n}\n\n\\bib{wil:tams81}{article}{\n      author={Williams, Dana~P.},\n       title={The topology on the primitive ideal space of transformation group\n  {$C\\sp{\\ast} $}-algebras and {C}.{C}.{R}. transformation group {$C\\sp{\\ast}\n  $}-algebras},\n        date={1981},\n        ISSN={0002-9947},\n     journal={Trans. Amer. Math. Soc.},\n      volume={266},\n      number={2},\n       pages={335\\ndash 359},\n      review={\\MR{MR617538 (82h:46081)}},\n}\n\n\\bib{wil:crossed}{book}{\n      author={Williams, Dana~P.},\n       title={Crossed products of {$C{\\sp \\ast}$}-algebras},\n      series={Mathematical Surveys and Monographs},\n   publisher={American Mathematical Society},\n     address={Providence, RI},\n        date={2007},\n      volume={134},\n        ISBN={978-0-8218-4242-3; 0-8218-4242-0},\n      review={\\MR{MR2288954 (2007m:46003)}},\n}\n\n\\bib{win:agag83}{article}{\n      author={Winkelnkemper, H.~E.},\n       title={The graph of a foliation},\n        date={1983},\n        ISSN={0232-704X},\n     journal={Ann. Global Anal. Geom.},\n      volume={1},\n      number={3},\n       pages={51\\ndash 75},\n      review={\\MR{MR739904 (85j:57043)}},\n}\n\n\\end{biblist}\n\\end{bibdiv}\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{intro}\n\nThe relationship between AGN activity and star formation is a complex\none. In order to maintain the observed black hole mass\/bulge mass\nrelationship, black holes must grow as new stars form\n\\citep[e.g.][]{Magorrian+98} and black hole growth should result in\nAGN activity. The generally accepted picture is one in which mergers\ntrigger both AGN activity and star formation\n\\citep[e.g.][]{Granato+04,diMatteo+05} and in which the AGN activity,\nat some point, shuts down star formation by one of a range of\nprocesses generally referred to as `feedback'\n\\citep[e.g.][]{Croton+06}. The microphysics of this process presumably\ninvolves the driving of outflows either by luminous quasar activity\n\\citep[e.g.][]{Maiolino+12} and\/or radio jets\n\\citep[e.g.][]{Hardcastle+12}; an understanding of how feedback operates\nin populations of galaxies is crucial to models of galaxy and black\nhole evolution.\n\nRadio-loud active galaxies form a particularly interesting\nsub-population of AGN in the context of this question. Firstly, they\ntend to reside in massive elliptical galaxies, traditionally thought\nto be `red and dead' with little or no recent star formation;\nsecondly, the large amount of kinetic energy that they inject into\ntheir environment means that they must both influence and be\ninfluenced by the galactic environment in which they are embedded.\nThere is in fact long-standing observational evidence\n\\citep[e.g.][]{Heckman+86} that some powerful radio galaxies have\npeculiar optical morphologies, plausibly the results of mergers with\ngas-rich galaxies. In these systems, we might expect AGN activity and\nstar formation to go hand in hand, although the different timescales\nfor star formation and AGN triggering will mean that they will not\nalways be observed together; in radio-quiet systems, there may be\nseveral hundred Myr of delay between the starburst and the peak of AGN\naccretion \\citep{Wild+10}. In contrast to these objects, we know that\nother radio galaxies, often equally powerful when their kinetic powers\ncan be computed, reside in the centres of rich cluster or group\nenvironments where, on the one hand, gas-rich mergers must be very\nrare, and, on the other, the duty cycle of AGN activity must approach\n100 per cent to account for the nearly universal detection of radio\nsources in these systems \\citep[e.g.][]{Eilek+Owen06}. In these\nobjects, we would be surprised to see evidence for a direct link\nbetween AGN activity and star formation.\n\nIt may be possible, as originally suggested by \\cite{Heckman+86}, to\nunderstand these apparently contradictory results in the context of a\ntwo-population model of the AGN activity in radio galaxies. The two\npopulations in question probably correspond quite closely to classes A\nand B of \\cite{Hine+Longair79}, now known as high-excitation and\nlow-excitation radio galaxies \\citep[hereafter HERGs and LERGs:\n  e.g.][]{Laing+94, Jackson+Rawlings97}. In recent years it has become\nclear that the differences between these objects are not simply a\nmatter of emission-line strength but extend to optical\n\\citep{Chiaberge+02}, X-ray \\citep*{Hardcastle+06-2} and mid-IR\n\\citep*{Ogle+06,Hardcastle+09}. In the vast majority of\nLERGs\\footnote{The optical emission-line class does not correspond\n  completely reliably to other indicators of AGN activity; see\n  \\cite{Hardcastle+09} for a discussion of some anomalous or\n  intermediate objects and \\cite{RamosAlmeida+11a} for a particularly\n  well-documented `LERG' with a clear heavily absorbed, luminous\n  hidden AGN. Emission-line classification clearly does not have a\n  one-to-one relationship to radiative efficiency, but, for\n  simplicity, in this paper we will continue to refer to LERGs and\n  HERGs as though they represent the archetypes of their population.},\nthere is no evidence for any radiatively efficient AGN activity,\nsetting aside non-thermal emission associated with the nuclear jet\n(e.g. \\citealt{Hardcastle+09} and references therein); the AGN power\noutput is primarily kinetic and we observe it only through the\nradiation of the jet and lobes and through the work they do on the\nmedium in which they are embedded. On the other hand, the HERGs, which\ninclude the traditional classes of narrow-line radio galaxies with\nspectra like those of Seyfert 2s, the broad-line radio galaxies and\nthe radio-loud quasars, behave like textbook AGN with the addition of\njets and lobes. Although LERGs are more prevalent at low radio powers\nand HERGs at high powers, both classes are found across the vast\nmajority of the radio power range and, where they overlap, there is\noften no way of distinguishing between the radio structures that they\nproduce.\n\nThe reason for the fundamental differences between the AGN activity in\nthese two classes of radio source is not clear, but one proposal is\nthat the differences arise because of different fuelling mechanisms.\nIn this scenario \\citep{Hardcastle+07-2} the LERGs are fuelled\ndirectly from the hot gas halos of their host ellipticals and the\ngroups and clusters in which they lie, while the HERGs are fuelled,\noften at a higher rate, by cold gas, presumably brought into the host\nelliptical by mergers or interactions with gas-rich\nsystems\\footnote{It is not yet clear whether the difference in the AGN\n  results from the difference in the temperature of the accreted\n  material, as proposed by Hardcastle {et al}\\\/. , or simply from the lower\n  accretion rates as a fraction of Eddington expected for massive\n  black holes being fed at something approximating the Bondi rate in\n  the LERGs, as in the models of \\cite{Merloni+Heinz08} and as argued\n  by \\cite{Best+Heckman12}; Mingo {et al}\\\/.\\ (in prep) will discuss this\n  question in detail. However, the answer to this question makes very\n  little difference to the predictions of the model.}. Because the\nLERGs dominate the population at low power and low redshift, this\nallows a picture in which nearby radio-loud AGN are driven by\naccretion of the hot phase and are responsible for balancing its\nradiative cooling \\citep[e.g.][]{Best+06} while still allowing for\nmerger- and interaction-driven radio-loud AGN at higher radio\nluminosity and\/or redshift. This model makes a number of testable\npredictions. LERGs will tend to be associated with the most massive\nsystems, will therefore tend to inhabit rich environments, and will\nlargely have old stellar populations; as a population, they will\nevolve relatively slowly. HERGs can occur in lower-mass galaxies with\nlower-mass black holes, provided that there is a supply of (cold)\nfuel: we therefore expect them to be in less dense environments, to be\nassociated with merger and star-formation signatures, to be in less\nevolved, lower-mass galaxies and to evolve relatively fast with cosmic\ntime (since the merger rate was higher in the past). Many of these\npredictions have been tested. There is some evidence, particularly at\nlow redshifts, for a difference in the environments and the masses of\nthe host galaxies of LERGs and HERGs \\citep[][Ching {et al}\\\/., in prep.]{Hardcastle04,Tasse+08}, and there is strong evidence, also at low\nredshifts, for differences in the host galaxy colours in the sense\nexpected from the model described above\n\\citep{Smolcic09,Best+Heckman12,Janssen+12}. There is strong evidence\nfor an increased fraction of signatures of merger or interaction in\nthe galaxy morphologies of the HERGs with respect to the LERGs\n\\citep{RamosAlmeida+11b} and with respect to a background galaxy\npopulation \\citep{RamosAlmeida+12}. And, most importantly from the\npoint of view of the present paper, there is direct evidence for\ndifferent star-formation histories in the hosts of HERGs and LERGs, in\nthe sense predicted by the model, i.e. that HERGs show evidence for\nmore recent star formation both at low redshift\n\\citep{Baldi+Capetti08} and at $z \\sim 0.5$ \\citep{Herbert+10}.\n\nStudies of the star formation in the different classes of radio galaxy\nhave until recently been limited in size because of the techniques and\nsamples used (e.g. {\\it HST} imaging by Baldi \\& Capetti, analysis of\noptical spectroscopy by Herbert {et al}\\\/. ). Only recently have large\nsamples begun to be analysed \\citep{Best+Heckman12,Janssen+12} and so\nfar this work has been based only on optical colours at low redshift.\nMid-infrared observations with {\\it Spitzer} provide some evidence\nthat individual HERGs may have strong star formation (e.g. Cygnus A,\n\\citealt{Privon+12}) but systematic studies of large samples have\ngenerally shown that the luminosity in the mid-IR is dominated by\nemission from the AGN itself, by way of the dusty torus \\citep[e.g.][]{Hardcastle+09,Dicken+09};\ndetailed mid-IR spectroscopy in small samples \\citep{Dicken+12} has\nshown that there is not a one-to-one association between star\nformation signatures and AGN activity, but this type of work cannot\neasily be extended to very large samples. However, observations of\ncool dust in the far infrared (FIR) should, in principle, provide a\nvery clear way of studying star formation, which should be\nuncontaminated by AGN activity, since the emission from the dusty\n  torus of the AGN is found to peak in the rest-frame mid-IR\n  \\citep[e.g.][]{Haas+04}. FIR observations can be carried out simply for\nlarge samples, and the method can extend to relatively high redshifts, with\nthe only contaminant being emission from diffuse dust heated by the\nlocal interstellar radiation field rather than by young stars\n(at least until redshifts become so high that rest-frame mid-IR torus\n  emission starts to appear in the observer-frame FIR bands). Earlier\nwork on far-infrared\/sub-mm studies of star formation in samples of radio\ngalaxies necessarily concentrated on high-redshift objects, in which\nemission at long observed wavelengths (e.g. 850 $\\mu$m, 1.2 mm)\ncorresponds to rest-frame wavelengths around the expected peak\n  of thermal dust emission \\citep[e.g.][]{Archibald+01,Reuland+04} and thus\n  applied only to very radio-luminous AGN. Much larger and more\nlocal samples can be studied using the {\\it Herschel\n  Space Observatory} \\citep{Pilbratt+10} and in particular by\nwide-field surveys such as the {\\it Herschel} Astrophysical Terahertz\nLarge Area Survey \\citep[H-ATLAS;][]{Eales+10}.\n\nIn an earlier paper \\citep[][hereafter H10]{Hardcastle+10b}, we\nstudied the FIR properties of radio-loud objects in the\n14-square-degree field of the Science Demonstration Phase (SDP)\ndataset of H-ATLAS, and showed that, as a sample, their FIR properties\nwere very similar to those of normal radio-quiet galaxies of similar\nmagnitude; however, our sample size was small and we were not able to\nclassify our radio-loud objects spectroscopically. The full `Phase 1'\nATLAS dataset, consisting of three large equatorial regions, gives a\nfield almost twelve times larger (161 square degrees). Our work on\nradio galaxies in the Phase 1 dataset is divided between two papers.\nVirdee {et al}\\\/.\\ (2012; hereafter V12) use the same sample selection\nprocess as H10, but use the much larger sample available from the\nPhase 1 datasets to investigate the relationship between radio\ngalaxies and normal galaxies in more detail, dividing the radio-loud\nsample by properties such as host galaxy mass and radio source size.\nIn the present paper, we select our sample so as to be able to\nclassify our radio sources spectroscopically, using data derived from\nthe Galaxy and Mass Assembly project (\\citealt{Driver+09,Driver+11};\nhereafter GAMA) and search for differences in the far-infrared and\nstar-formation properties of HERGs and LERGs.\n\nThroughout the paper we use a concordance cosmology with $H_0 = 70$ km\ns$^{-1}$ Mpc$^{-1}$, $\\Omega_{\\rm m} = 0.3$ and $\\Omega_\\Lambda =\n0.7$. Spectral index $\\alpha$ is defined in the sense that $S \\propto\n\\nu^{-\\alpha}$.\n\n\\section{Sample selection and measurements}\n\n\\subsection{The GAMA sample}\nThe GAMA survey is a study of galaxy evolution using multiwavelength\ndata and optical spectrosopic data. In phase I of GAMA, target\ngalaxies are drawn from the SDSSDR6 photometric catalogue in three\nindividual $12^\\circ \\times 4^\\circ$ rectangles along the equatorial\nregions centred at around 9, 12 and 15 hours of right ascension. A\n$r$-band magnitude limit of 19.4 was used for the 9 and 15-h fields\nwhile the 12-h field had a deeper 19.8 mag limit \\citep{Driver+11}.\nThe H-ATLAS Phase I data is taken from regions corresponding closely\nto these three fields. Reliable spectroscopic redshifts from previous\nsurveys (e.g. SDSS, 6dF Galaxy Survey, etc.) were used for GAMA\nsources that had them. Those without reliable spectroscopic redshifts\nfrom previous surveys were spectroscopically observed on the\nAnglo-Australian Telescope (AAT).\n\nWe built a sample of candidate radio galaxies by cross-matching the\nFaint Images of the Radio Sky at Twenty-cm \\citep[FIRST,][]{Becker+95}\ncatalogue (16 July 2008) with optical sources ($i < 20.5$ mag,\nextinction corrected) from the Sloan Digital Sky Survey Data Release 6\n\\citep[SDSSDR6;][]{SDSSDR6} in all GAMA regions. The full details of\nthe cross-matching will be described by Ching {et al}\\\/.\\ (in prep.), but a\nshort summary is provided here. The cross-matching firstly involved\ngrouping FIRST components that were likely subcomponents of a single\noptical source (e.g. the core and lobes of a radio galaxy). The\noptical counterparts for the groups were matched automatically if they\nsatisfied certain criteria based on symmetries of the radio sources,\nand\/or manually when groups were more complex, by overlaying SDSS\nimages with FIRST and NRAO VLA Sky Survey \\citep[NVSS;][]{Condon+98}\ncontours. Groups that appeared to be separate individual radio sources\nwere split into appropriate subgroups matched to their individual\noptical counterpart. All FIRST components that were not identified as\na possible subcomponent were cross-matched to the nearest SDSS optical\ncounterpart with a maximum separation of 2.5 arcsec. This process gave\nus a sample of 3168 objects with radio\/optical identifications. Some\nof these objects, predominantly at low redshifts, had spectra from\nthe SDSS spectroscopic observations; GAMA does not re-observe such\nobjects. Others were part of the GAMA main sample. To increase the\nspectroscopic sample size, we identified galaxies that were not part\nof the GAMA main sample (internal data management unit TilingCatv16,\nSURVEY\\_CLASS$\\ne 1$), and observed some of them as spare-fibre\ntargets during the main GAMA observing programme (see Ching {et al}\\\/., in\nprep., for more details). The resulting sample, by construction,\ncontained only sources with usable spectra and spectroscopically\ndetermined redshifts ($nQ \\ge 3$, from GAMA data management unit\nSpecCatv08; \\citealt{Driver+11}), and is flux-limited in the radio,\nwith a lowest 1.4-GHz flux density around 0.5 mJy and most sources\nhaving flux density above 1.0 mJy, as a result of the use of FIRST in\nconstructing the sample. There were 2559 sources with spectroscopic\nredshifts in this parent sample.\n\n\\subsection{Spectral classification}\n\nSpectral classification of the objects with radio\/optical\nidentifications was carried out by inspection of their spectra. A\ndetailed description of the process will be given by Ching {et al}\\\/.\\ (in\nprep.); here we simply summarize the steps we followed. The emission\nline measurements used in this paper for GAMA spectra were made from\nthe Gas and Absorption Line Fitting (GANDALF; \\citealt{Sarzi+06}) code\nas part of the GAMA survey (see \\citealt{Hopkins+12} for a description\nof the GAMA spectroscopy and spectroscopic pipeline), while for SDSS\nspectra we used the measurements from the value-added MPA-JHU\nemission-line measurements derived from SDSS\nDR7\\footnote{http:\/\/www.mpa-garching.mpg.de\/SDSS\/DR7\/}. Both of these\nmeasurements fit the underlying stellar population before making\nemission line measurements, and hence take into account any stellar\nabsorption. Only high-quality GAMA spectra were used.\n\nWe firstly removed Galactic sources by imposing a lower redshift limit\nof $z>0.002$. Such objects are classified `Star' and play no further\npart in the analysis in this paper. Next, we visually selected objects\nwith broad emission lines; these are classified `AeB' in this paper,\nand are broad-line radio galaxies or radio-loud quasars.\n\nGalaxies that are within $z<0.3$ and have 1.4-GHz luminosity\n(hereafter $L_{1.4}$) below $10^{24}$ W Hz$^{-1}$ have a high\nprobability of having star-formation dominated radio emission (see\ne.g. \\citealt{Mauch+Sadler07}). For all such objects having\n[O{\\sc iii}], [N{\\sc ii}], H$\\alpha$ and H$\\beta$ emission lines detected\nwith a signal-to-noise ratio $>3$, we used a simple line diagnostic\n\\citep[BPT;][]{Baldwin+81} to classify `pure star-forming galaxies' as\nclassified by \\cite{Kauffmann+03}. These are classed as `SF' in the\nfollowing analysis. However, as pointed out by \\cite{Best+05}, line\ndiagnostics alone are not enough to ensure a clean sample of\nradio-loud AGN, since the emission lines may arise from a radio-quiet\nAGN, while the detected radio emission might arise from star formation\nin another region. In addition, the lines required for BPT analysis\nare not available at $z>0.3$. We therefore also classified as `SF' any\nobject whose H$\\alpha$ and 1.4-GHz radio emission placed it within\n$3\\sigma$ of the relation between these two quantities derived by\n\\cite{Hopkins+03} for star-forming objects. We emphasise that `SF'\nobjects are not discarded from the analysis at this stage -- therefore\nnothing in this classification prejudices the results of the H-ATLAS\nanalysis.\n\nFinally, we expected the remaining galaxies to be a reasonably robust\nsample of radio-loud AGN, possibly contaminated by $z>0.3$ and\/or\nextremely luminous ($L_{1.4} > 10^{24}$ W Hz$^{-1}$) star-forming objects.\nWe therefore classified them using a scheme intended to differentiate\nbetween HERG and LERG radio galaxies. Our preliminary classification\nwas visual, i.e. objects were classed as `Ae' (corresponding to HERGs)\nif they showed strong high-excitation lines such as [O{\\sc iii}],\n[N{\\sc ii}], [Mg{\\sc ii}], [C{\\sc iii}], [C{\\sc iv}] or Ly$\\alpha$,\nand as `Aa' (corresponding to LERG) otherwise, using a similar\nclassification scheme to that of \\cite{Mauch+Sadler07} -- see their\nSection 2.5 for more discussion of this approach and its reliability.\nHowever, we then found that the equivalent width of the [O{\\sc iii}]\nline gave a very similar division between objects with the advantage\nof removing the subjective element of the visual classification. In\nthe final analysis we classified all galaxies with SNR([O{\\sc\n    iii}])$>3$ and EW([O{\\sc iii}])$>5$\\AA\\ as `Ae' (HERG-like) and\nall objects not otherwise classified as `Aa' (LERG-like). The choice\nof 5\\AA\\ as the equivalent-width cut gives the best match to our\npreliminary visual analysis, but we verified that small variations in\nthis choice made little or no difference to the results presented in\nthe rest of the paper.\n\nA summary of the classification scheme and the number of objects in\nthe sample in each of the emission-line classes is given in Table\n\\ref{class}. Throughout the rest of the paper, we retain a distinction\nbetween the {\\it observational} classifications (SF, Aa, Ae, AeB) and\nthe {\\it physical} distinction between star-forming non-AGN sources,\nLERGs and HERGs; we discuss how well the observational emission-line\nclassifications map on to the physical distinctions in the course of\nthe paper, with a summary in Section \\ref{discussion}.\n\n\\begin{table*}\n\\caption{The classification scheme used in this paper and the\n  number of objects in each class in the H-ATLAS subsample. Also shown\nare the numbers of objects after the application of the `SF cut' based\non the radio\/FIR relation, as described in the text.}\n\\label{class}\n\\begin{tabular}{lllrr}\n\\hline\nName&Characteristics&RL AGN class&\\multicolumn{2}{c}{Number in\n  sample}\\\\\n&&&(Total)&(`SF cut' applied)\\\\\n\\hline\nAa&AGN spectra with EW([O{\\sc iii}])$\\le 5$\\AA&LERG&1247&1186\\\\\nAe&AGN spectra with EW([O{\\sc iii}])$>5$\\AA&HERG\/NLRG&199&156\\\\\nAeB&AGN spectra with strong broad high-excitation lines&HERG\/BLRG\/QSO&187&194\\\\\nSF&Star forming galaxy based on BPT or $H\\alpha$-radio correlation&--&191&8\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{{\\it Herschel} flux-density measurements}\n\nThe classification over the GAMA fields and the removal of stars gives\nus 2066 objects, all of which have positions, SDSS identifications,\nFIRST flux densities, spectroscopic redshifts from SDSS, GAMA proper or the\nspare-fibre programme, and spectroscopic classifications. Our next\nstep was to extract flux densities for these objects from the H-ATLAS `Phase\n1' images. `Phase 1' of H-ATLAS consists of observations of 161 square\ndegrees of the sky coincident with the GAMA fields, including the much\nsmaller SDP field discussed by H10; further information on the Phase 1\ndataset will be provided by Hoyos et al. and Valiante et al. (in\nprep). We discarded all GAMA objects which were outside the area\ncovered by H-ATLAS (i.e. where flux densities were not available):\nthis reduced the sample to 1836 objects, and it is this `H-ATLAS\nsubsample' that we discuss from now on.\n\nH-ATLAS maps the FIR sky with {\\it Herschel}'s Spectral and\nPhotometric Imaging Receiver \\citep[SPIRE;][]{Griffin+10} and the\nPhotodetector Array Camera and Spectrometer\n\\citep[PACS;][]{Poglitsch+10}. The process of deriving the images used\nin this paper is described by \\cite{Pascale+11} and \\cite{Ibar+10} for\nSPIRE and PACS respectively. For each of the objects in our H-ATLAS\nsubsample we derived the maximum-likelihood estimate of the flux\ndensity at the object position in the three SPIRE bands (250 $\\mu$m,\n350 $\\mu$m and 500 $\\mu$m) by measuring the flux density from the\nPSF-convolved H-ATLAS images as in H10, together with the error on the\nfluxes. We also extracted PACS flux densities and corresponding errors\nfrom the images at 100 and 160 $\\mu$m using circular apertures\nappropriate for the PACS beam (respectively 15.0 and 22.5 arcsec) and\nusing the appropriate aperture corrections, which take account of\nwhether any pixels have been masked. We add an estimated absolute flux\ncalibration uncertainty of 10 per cent (PACS) and 7 per cent (SPIRE)\nin quadrature to the errors measured from the maps for the purposes of\nfitting and stacking, as recommended in H-ATLAS documentation, but\nthis uncertainty is {\\it not} included when considering whether\nindividual sources are detected.\n\nOnly 368 of the H-ATLAS subsample (20 per cent) are detected in the\nconservative `$5\\sigma$' H-ATLAS source catalogue \\citep[created as\ndescribed by][]{Rigby+11}. This is a similar $5\\sigma$\ndetection fraction to that obtained by H10. We can relax this\ncriterion for detection slightly, as we know that there are objects\n(the host galaxies of the radio sources) at the positions of interest.\nA detection criterion of $2\\sigma$ implies that 2.3 per cent of\n`detected' sources will be spurious, which is acceptable for our\npurposes. However, care needs to be taken when applying such a\ncriterion to the H-ATLAS data. The images at 250, 350 and 500 $\\mu$m\nare badly affected by source confusion, and this means that the\nstatistics of the `noise' -- including confusing sources -- are not\nGaussian. We have therefore conservatively determined our $2\\sigma$\ncutoff by sampling a large number of random background-subtracted\nflux densities from the PSF-convolved maps, and determining the flux level\nbelow which 97.7 per cent of the random fluxes lie, to get a flux density\nlimit which takes account of confusion. This process returns twice the\nlocal r.m.s. noise if the noise is Gaussian, which turns out to be the\ncase for the PACS data, but gives substantially higher flux density limits of\n24.6, 26.5 and 25.6 mJy for the 250, 350 and 500-$\\mu$m SPIRE maps\nrespectively, corresponding to around 3.8 times the local noise\nestimates for 250 $\\mu$m. These limits are essentially independent of\nthe local noise estimates (from the noise maps), which is as expected\nsince the upper tail of the flux density distribution in the maps is dominated\nby the effects of confusing sources. In what follows, we say that a\nsource is `detected' in a given band if it lies above these confusion\nlimits (for the SPIRE data) or above the standard $2\\sigma$ value (for\nPACS). By these criteria, 486 sources (26 per cent) are detected at\n250 $\\mu$m, the most sensitive SPIRE band; the number falls to 244 (13\nper cent) at 500 $\\mu$m and 328 (18 per cent) at 100 $\\mu$m.\n\nWe compared this radio-galaxy sample to the sample of V12, which uses\nthe method described in H10 to select candidate radio galaxies,\nrequiring a cross-match between the NVSS and the UKIDSS-LAS\n\\citep{lawrence+07}, over the original 135-square-degree Phase 1\nfield. 786 of the current sample match objects in the sample of V12,\nand for those objects we find good agreement between the NVSS and\nFIRST flux densities, suggesting that there is little missing flux. The\nobjects that are in the H-ATLAS subsample but are not identified as\nradio galaxies in the sample of V12 are either not LAS sources or are\nfaint radio sources that fall below the NVSS flux density limit but are\ndetectable with FIRST, and so would not be expected to be in our NVSS\ncatalogue. We conclude that there is good consistency between the\nmethod used here and the method of H10, in the set of objects where\nthey overlap, and that there is no reason to suppose that the results\nare less robust for the population of faint radio sources that we\nstudy for the first time in this paper.\n\nAs noted above, our spectroscopically identified sample is not\ncomplete, in the sense that not all objects that would meet the\nselection criteria for spectroscopy have high-quality spectra, and\nthis should be borne in mind in what follows. No selection bias has\nbeen consciously imposed by our choice of objects for spectroscopic\nanalysis.\n\n\\subsection{Luminosity and dust mass calculations}\n\\label{luminosity}\n\nThe rest-frame 1.4-GHz radio luminosity of the sample sources is\ncalculated from the FIRST 1.4-GHz flux density and the spectroscopic\nredshift, assuming $\\alpha=0.8$ as in H10. (We comment on constraints\non the spectral index of objects in the sample in the next subsection.)\n\nH10 used integrated FIR luminosities, but these depend very strongly\non the assumptions made about the underlying spectrum, in particular\nthe $\\beta$ and temperature of the modified blackbody model which is\nassumed to describe the data. In this paper we instead use the\nmonochromatic luminosity at rest-frame 250 $\\mu$m, $L_{250}$. This has\nthe advantage that the assumptions we make about the spectrum only\naffect the $K$-correction, and so have negligible effect at low redshift. We\nstill have to make a choice of the spectrum to use for $K$-correction,\nsince we cannot fit models to the vast majority of our objects. H10\nused a modified blackbody with $T = 26$ K, $\\beta = 1.5$, but in this\npaper we use $T = 20$ K, $\\beta = 1.8$, for reasons that will be\njustified by temperature fits in Section \\ref{temperatures-ind}.\n\nThe disadvantage of this approach is that we lose the ability to\nestimate the star-formation rate directly from the integrated FIR\nluminosity, as we attempted to do in H10: however, the relationships\ncommonly used to do this \\citep[e.g., those given by][]{Kennicutt98}\nare calibrated using starburst galaxies and are not necessarily\napplicable in the temperature and luminosity range that most radio\ngalaxies occupy. Instead, we can consider the 250-$\\mu$m luminosity as\nrepresenting a dust {\\it mass} \\citep[as in][]{Dunne+11}; the\n`isothermal' dust mass, i.e. the mass derived on the assumption of a\nsingle temperature for the dust, is given by\n\\begin{equation}\nM_{\\rm iso} = \\frac{L_{250}}{4\\pi \\kappa_{250} B(\\nu_{250}, T)}\n\\label{dustmass}\n\\end{equation}\nwhere $\\kappa_{250}$ is the dust mass absorption coefficient, which\nDunne {et al}\\\/.\\ take to be 0.89 m$^2$ kg$^{-1}$, and $B(\\nu, T)$ is the\nPlanck function. It is clear for this mass estimation method, and also\nturns out to be the case for the more complex method discussed by\nDunne {et al}\\\/. , that for a roughly constant $T$ we have a linear\nrelationship between mass and luminosity, while we also expect a\nstrong correlation between $L_{\\rm 250}$ and $T$ for a fixed dust\nmass. Moreover, we expect high values of $T$ to be indicators of\nstrong star formation, independent of $M_{\\rm iso}$. In this paper we\nwill initially use $L_{250}$ to indicate possible differences in star\nformation, and use comparisons of fitted temperatures $T$ to confirm\nthem. Later we will show that $L_{250}$ can be calibrated to give a\nquantitative measure of star-formation rate, subject to some important\ncaveats.\n\nIt is important to note that the {\\it Herschel} SPIRE PSF has a FWHM\nof 18 arcsec at 250 $\\mu$m, which corresponds to linear sizes up to\n$\\sim 150$ kpc at the redshift of the most distant objects in our\nsample. As we noted in H10, the luminosities we measure, and any\ncorresponding dust masses or temperatures, apply not just to the host\ngalaxy of the radio source but also to its immediate environment. Star\nformation associated with a given AGN might actually be taking place\nin a merging system or a nearby companion galaxy.\n\n\\section{Results}\n\n\\subsection{Subsample properties}\n\\label{blrg}\n\nTable \\ref{class} gives the numbers of objects in the H-ATLAS\nsubsample that fall into the various emission-line classes defined\nabove. We see that absorption-line only or weak emission-line spectra\n(`Aa': unambiguously corresponding to the expected spectra of\n`low-excitation' radio galaxies or LERGs) dominate the population.\nThere are then roughly equal numbers of the `Ae' objects,\ncorresponding to the high-excitation narrow-line radio galaxies\n(HERGs, or NLRGs), broad-line objects (`AeB') and objects classed as\nstar-forming on the basis of their spectra (`SF').\n\nThe redshift distributions within the emission-line classes are\nsomewhat different. The objects in the Aa and Ae classes have very\nsimilar redshift distributions, with median redshifts around 0.4, as\nwe might expect for bright galaxies drawn from the parent (SDSS)\nsample, and maximum redshift $\\sim 1$. We cannot distinguish between\nthe redshift distributions of the Aa and Ae classes on a\nKolmogorov-Smirnov (KS) test at the $3\\sigma$ confidence level. The SF\ngalaxies have a clearly different distribution, with median $z \\sim\n0.08$ and maximum $z \\sim 0.3$, suggesting that these are mainly\nlocal, fainter galaxies \\citep[as expected from the known different\nluminosity functions of the AGN and SF populations; see][]{Mauch+Sadler07}. We retain the SF objects in the sample so as not to\nexclude the possibility, at this stage, that some are NLRG with strong\nstar formation. The broad-line objects have a much wider redshift\ndistribution, with median $z \\sim 1.3$ and maximum $z \\sim 3.7$. These\nobjects are clearly mostly quasars that are in the sample due to their\nbright AGN emission. Similarly, if we consider the radio flux density\ndistributions, we cannot distinguish between the Aa or Ae classes at\nhigh confidence with a KS test, but the SF objects have a significantly\ndifferent flux distribution from the Aa and Ae, tending to have\nfainter radio flux densities.\n\nThe AeB objects are systematically very much brighter in the radio,\nsuggesting that the combination of radio and optical selection for\nthese quasars is picking up strongly beamed objects, and this is true\neven if we consider only the $z<1$ subsample of the AeB objects. Among\nother things, this means that we need to be alert to the possibility\nof non-thermal contamination in the {\\it Herschel} bands. To check\nthis, we cross-matched the objects in our sample to the GMRT catalogue\nof Mauch {et al}\\\/.\\ (in prep.), who have imaged the majority of the Phase\n1 area at 325 MHz, using a simple positional matching algorithm with a\nmaximum offset of 5 arcsec. A total of 536\/1836 objects have\ncounterparts in the GMRT catalogue; the low matching fraction reflects\nthe incomplete sky coverage and variable sensitivity of the GMRT\nsurvey, as described in detail by Mauch et al. Nevertheless, we can\nlook for spectral index differences in the matching objects. The\nnumber of cross-matches, together with the mean spectral index and the\nvalues at the 10th and 90th percentile, are tabulated as\na function of emission-line class in Table \\ref{spix}.\n\n\\begin{table*}\n\\caption{Spectral indices between 325 MHz and 1.4 GHz for sample\n  sources with GMRT survey counterparts}\n\\label{spix}\n\\begin{center}\n\\begin{tabular}{lrrrr}\n\\hline\nSource type&Number of matches&Mean spectral index&10th percentile&90th\npercentile\\\\\n\\hline\nAll&536&0.70&0.27&1.09\\\\\nSF&45&0.88&0.58&1.34\\\\\nAa&356&0.77&0.37&1.09\\\\\nAe&55&0.68&0.34&0.99\\\\\nAeB&80&0.50&$-0.03$&1.12\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{figure}\n\\epsfxsize 8.5cm\n\\epsfbox{lopezplot.eps}\n\\caption{Radio sources with a 500-$\\mu$m detection plotted on the\n  diagnostic plot of \\cite{Lopez-Caniego+12}. Colours indicate\n  different emission-line classes. The dashed line indicates the\n  threshold in FIR\/radio radio used by \\cite{Lopez-Caniego+12}; below\n  this line, synchrotron emission might be bright enough to affect the\n  SPIRE bands.}\n\\label{lopezplot}\n\\end{figure}\n\nWhile the large number of non-detections in the GMRT survey means that\nwe cannot carry out a detailed analysis, we note first of all that the\nmean spectral index of detected sources is close enough to our\npreviously adopted value of 0.8 that our $K$-correction in the radio\nwill not be badly in error, and secondly that the mean spectral index\nof the AeB objects is very much flatter than any of the other\nemission-line classes, although there is still clearly a population of\nsteep-spectrum AeBs. Given that the detected objects are likely to be\nbiased, if at all, towards the steep-spectrum end of the intrinsic\ndistribution, it seems likely that the AeB objects contain a\nsignificant number of flat-spectrum quasars.\n\nWe investigated this issue further by considering the diagnostic\nmethods used by \\cite{Lopez-Caniego+12} in searching for blazars. They\nrelied on detections at 500 $\\mu$m, and, as noted above, only a small\nfraction of our sources have $2\\sigma$ detections at that band. We\nplotted the sources that do on the diagnostic radio\/FIR colour-colour\ndiagram used by \\cite{Lopez-Caniego+12}, which is intended to search\nfor non-thermal contamination in the SPIRE bands; the result is shown\nin Fig.\\ \\ref{lopezplot}. We see that of the 24 AeB objects with\n500-$\\mu$m detections, about half lie in the region occupied by the\nL\\'opez-Caniego blazar candidates in which synchrotron emission might\naffect the SPIRE bands, a much higher fraction than for any\nother emission-line class. While red 500\/350-$\\mu$m colours may just\nbe an indication of low dust temperatures, and the AeB sources have higher\nredshifts than the comparison objects, this is a further sign that the\nAeBs cannot safely be merged with the Ae objects in what follows.\n\n\\subsection{{\\it Herschel} and radio luminosity}\n\n\\begin{figure*}\n\\epsfxsize 14cm\n\\epsfbox{ll-j-bins.eps}\n\\caption{250-$\\mu$m luminosity against radio luminosity for\n  all the objects in the sample. Stars indicate {\\it Herschel}\n  detections at $2\\sigma$ or better as defined in the text, points\n  show $2\\sigma$ upper limits in IR luminosity derived from the\n  confusion limit. Colours correspond to emission-line classes as\n  follows: Aa, blue; Ae, green; AeB, red; SF, black. The solid magenta\n  lines indicate the expected radio-FIR correlation for star-forming\n  objects, $q_{\\rm 250} = 1.78$, and the approximate scatter about\n  this relation, $1.4 < q_{250} < 2.1$ \\citep[from][]{Jarvis+10}.\n  The solid orange line shows our adopted `SF cut' at $q_{250}=1.3$, and the\n    dot-dashed vertical line shows the nominal FRI\/FRII break luminosity.}\n\\label{llplot}\n\\end{figure*}\n\nFig.\\ \\ref{llplot} shows the IR luminosity, $L_{250}$, against the\nradio luminosity for all the objects in the sample. This plot shows\nseveral important features of the sample. First, we note that the vast\nmajority of the broad-line objects (in red) lie at the very\nhigh-luminosity end of the plot, presumably due to their high\nredshifts. As we noted above that some of these objects may well have\nFIR fluxes contaminated by non-thermal emission, and as their high\nredshift makes it difficult to compare them with radio galaxies in any\ncase, we exclude them from further analysis.\n\n\\begin{figure*}\n\\epsfxsize 8.5cm\n\\epsfbox{zdist.eps}\n\\epsfxsize 8.5cm\n\\epsfbox{ldist.eps}\n\\caption{The distribution of (left) redshift and (right) radio\n  luminosity in the Aa (blue) and Ae (green) objects after the SF cut.\n  The redshift and luminosity distributions of the samples are very\n  similar.}\n\\label{lzdist}\n\\end{figure*}\n\nFig. \\ref{llplot} also shows the expected linear radio-FIR correlation\nfor star-forming objects (magenta lines), together with the dispersion\nseen in that relationship, based on the parameter $q_{250}$, which is\ndefined as $\\log_{10}(L_{250}\/L_{1.4})$ \\citep{Jarvis+10}. We see that\nobjects classed as SF on the basis of their emission-line properties\nor their radio\/H$\\alpha$ relation (black points) almost all lie in\nthis region of the plot and close to the best-fitting line; there is\nsome positive deviation above the line at low luminosities\/redshifts,\nbut this was also seen by Jarvis {et al}\\\/. . However, at higher\nluminosities, a number of objects of other emission-line classes also\nfall in the star-forming region, meaning that their radio emission is\nnot bright enough to definitively classify them as radio galaxies.\nConservatively, every object that lies in the star-forming region of\nthis plot should be excluded from a discussion of the FIR properties\nof radio galaxies; following V12, we adopt a cut at $q_{250}\\ge 1.3$.\nThe numbers of sources remaining, if these objects are excluded, are\ngiven in Table \\ref{class}. The vast majority of the SF objects are\nremoved by the cut (hereafter the `SF cut'), and although some of the 8 remaining sources may\nbe radio galaxies which have erroneously been classified as SF, we\nconservatively exclude them from subsequent analysis (given the small\nnumbers involved, including them as though they were Ae objects would\nhave little effect on our results).\n\nConsidering only the remaining objects, which we expect to be\nradio-loud AGN, we see that these span a very large range in radio\nluminosity, from $10^{22}$ to $10^{27}$ W Hz$^{-1}$ if we ignore the\nbroad-line objects. The vast majority of these lie below the nominal\nFRI-FRII luminosity divide \\citep{Fanaroff+Riley74} of $1.2\\times\n10^{25}$ W Hz$^{-1}$ (plotted on Fig.\\ \\ref{llplot} for reference) and\nso would normally be classed as low-luminosity radio galaxies, though\nwe emphasise that the FRI\/FRII division is a morphological one and we\nhave made no attempt to classify these objects morphologically. In\nterms of our observational emission-line classifications, we see that\nAa objects dominate numerically by a large factor, but that there are\nAe objects at all powers. Assuming that Aas trace LERGs and Aes HERGs,\nthis is consistent both with what is seen in brighter radio-selected\nsamples at low redshift \\citep[see, e.g.,][]{Hardcastle+09} and with\nthe work of \\cite{Best+Heckman12} over a comparable luminosity range.\nThe Aa and Ae objects left after the SF cut has been made have\nredshift and radio luminosity distributions that are indistinguishable\non a KS test (Fig.\\ \\ref{lzdist}), but this is not surprising, since\nthe differences in the slope of the luminosity function for the two\npopulations, leading to the dominance of HERGs at high luminosities,\nstart to become significant only at $L_{1.4} > 10^{25}$ W Hz$^{-1}$\n\\citep{Best+Heckman12}, where we have relatively few sources.\n\n\\subsection{LERG\/HERG comparisons and stacking}\n\\label{stacking}\n\nSome differences between the FIR properties of the Aa and Ae objects\nafter the SF cut are immediately obvious on inspection of the data.\nFor example, 53\/156 (34 per cent) of the Ae objects are detected at\nthe $2\\sigma$ level or better at 250 $\\mu$m, while only 93\/1186 (8\nper cent) of the Aa objects are detected at this level.\n\n\\begin{figure*}\n\\epsfxsize 14cm\n\\epsfbox{ll-j-bins2.eps}\n\\epsfxsize 14cm\n\\epsfbox{lz-j-bins.eps}\n\\caption{Comparisons between the IR luminosities of the Aa and Ae\n  objects, after excluding objects near the star-formation line, as a\n  function of (top) radio luminosity and (bottom) redshift.\n  Symbols and colours as for Fig.\\ \\ref{llplot}. The\n  large crosses indicate the results of stacking the IR luminosities\n  of all Ae (green) and Aa (blue) objects in the corresponding radio\n  luminosity or redshift range. Other types of object are plotted but\n  not stacked. The orange bins indicate stacking of comparison\n  galaxies, as described in the text.}\n\\label{compar}\n\\end{figure*}\n\nOverall, both individual sub-samples still being considered (i.e. Aa\nand Ae after SF cut) are significantly detected with respect to the\nbackground at all three {\\it Herschel}-SPIRE bands. We follow H10 in\ntesting this with a KS test on the distribution of flux densities compared to\nrandom flux densities from the field; the highest null hypothesis probability\nis $6.6 \\times 10^{-8}$ for the Aa sources at 500 $\\mu$m,\ncorresponding to a {\\it minimum} significance of $5.5\\sigma$ for both\nclasses and all SPIRE bands, and the significance is much higher at\n250 $\\mu$m. For the two PACS bands, the Aa sub-sample\nis detected at around 98 per cent confidence (i.e. a marginal\ndetection, $2.3\\sigma$) but the Ae objects are significantly detected\nwith a null hypothesis probability around $2\\times 10^{-7}$\n($5.2\\sigma$).\n\nWe are therefore able to adopt the approach of H10 and divide our\nsources into luminosity and redshift bins for a stacking analysis.\nSince we have relatively few Ae sources, we use only three bins in\nboth, ensuring that the highest-luminosity bin includes all sources\nabove the nominal FRI\/II luminosity boundary at\n$L_{1.4} = 1.2\\times 10^{25}$ W Hz$^{-1}$, which, as noted above,\ncan roughly be taken to separate `low-power' and `high-lower' radio\ngalaxies. We then used KS tests to see whether these subsamples were\ndetected (distinguished from the background flux density distribution) at each\nof the five H-ATLAS wavelengths. The results are shown in Tables\n\\ref{zstest} and \\ref{lstest}.\n\n\\begin{table*}\n\\caption{Mean bin flux densities and K-S probabilities that the {\\it\n    Herschel} fluxes of objects in redshift bins (after the SF cut) are\n  drawn from the background distribution, as a function of\n  emission-line class and wavelength. Low probabilities (below 1 per\n  cent) imply significant differences between the bin being considered\n  and the distribution of flux densities measured from randomly\n  selected positions in the sky, as described in the text. Note that\n  the bins do not include quite all the objects in the sample.}\n\\label{zstest}\n\\begin{tabular}{llrrrrrrrrrrr}\n\\hline\nClass&$z$ range&Objects&\\multicolumn{5}{c}{Mean bin flux density\n  (mJy)}&\\multicolumn{5}{c}{K-S probability (\\%)}\\\\\n&&in bin&\\multicolumn{3}{c}{SPIRE bands}&\\multicolumn{2}{c}{PACS bands}&\\multicolumn{3}{c}{SPIRE bands}&\\multicolumn{2}{c}{PACS bands}\\\\\n&&&250 $\\mu$m&350 $\\mu$m&500 $\\mu$m&100 $\\mu$m&160 $\\mu$m&250 $\\mu$m&350 $\\mu$m&500 $\\mu$m&100 $\\mu$m&160 $\\mu$m\\\\\n\\hline\nAa  & 0.00 -- 0.30 & 399  & $5.5 \\pm 0.3$  & $1.0 \\pm 0.4$  & $2.5 \\pm 0.4$  & $2.7 \\pm 1.5$  & $8.7 \\pm 1.9$ & $<10^{-3}$ & $<10^{-3}$ & 0.3 & 2.2 & 3.6 \\\\\n & 0.30 -- 0.50 & 475  & $4.7 \\pm 0.3$  & $0.7 \\pm 0.3$  & $2.1 \\pm 0.4$  & $3.1 \\pm 1.4$  & $7.0 \\pm 1.7$ & $<10^{-3}$ & 0.004 & 0.09 & 46.1 & 9.9 \\\\\n & 0.50 -- 0.90 & 310  & $5.3 \\pm 0.4$  & $1.4 \\pm 0.4$  & $2.6 \\pm 0.5$  & $0.2 \\pm 1.7$  & $5.9 \\pm 2.1$ & $<10^{-3}$ & $<10^{-3}$ & 0.01 & 41.1 & 35.4 \\\\\nAe  & 0.00 -- 0.30 &  62  & $27.9 \\pm 0.9$  & $10.2 \\pm 0.9$  & $6.4 \\pm 1.1$  & $68.2 \\pm 3.8$  & $50.3 \\pm 4.6$ & $<10^{-3}$ & $<10^{-3}$ & 0.05 & $<10^{-3}$ & 0.06 \\\\\n & 0.30 -- 0.50 &  57  & $22.8 \\pm 0.8$  & $8.1 \\pm 0.9$  & $3.8 \\pm 1.1$  & $25.6 \\pm 3.8$  & $30.7 \\pm 4.7$ & $<10^{-3}$ & 0.005 & 2.1 & 0.3 & 1.5 \\\\\n & 0.50 -- 0.90 &  37  & $20.4 \\pm 1.0$  & $9.8 \\pm 1.2$  & $7.6 \\pm 1.4$  & $23.5 \\pm 5.0$  & $14.2 \\pm 6.1$ & $<10^{-3}$ & 0.002 & 0.10 & 11.9 & 20.8 \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\caption{Mean bin flux densities and K-S probabilities that the {\\it\n    Herschel} fluxes of objects in luminosity bins (after the SF cut)\n  are drawn from the background distribution, as a function of\n  wavelength. Notes as for Table \\ref{zstest}.}\n\\label{lstest}\n\\label{ks-l}\n\\begin{tabular}{llrrrrrrrrrrr}\n\\hline\nClass&Range in&Objects&\\multicolumn{5}{c}{Mean bin flux density (mJy)}&\\multicolumn{5}{c}{K-S probability (\\%)}\\\\\n&$L_{1.4}$&in bin&\\multicolumn{3}{c}{SPIRE bands}&\\multicolumn{2}{c}{PACS bands}&\\multicolumn{3}{c}{SPIRE bands}&\\multicolumn{2}{c}{PACS bands}\\\\\n&&&250 $\\mu$m&350 $\\mu$m&500 $\\mu$m&100 $\\mu$m&160 $\\mu$m&250 $\\mu$m&350 $\\mu$m&500 $\\mu$m&100 $\\mu$m&160 $\\mu$m\\\\\n\\hline\nAa  & 22.0 -- 24.0  & 457  & $4.7 \\pm 0.3$  & $0.5 \\pm 0.3$  & $1.9 \\pm 0.4$  & $3.2 \\pm 1.4$  & $6.9 \\pm 1.8$ & $<10^{-3}$ & 0.002 & 0.2 & 2.1 & 14.5 \\\\\n & 24.0 -- 25.0  & 589  & $5.7 \\pm 0.3$  & $1.5 \\pm 0.3$  & $2.6 \\pm 0.3$  & $2.4 \\pm 1.2$  & $7.5 \\pm 1.5$ & $<10^{-3}$ & $<10^{-3}$ & $<10^{-3}$ & 49.6 & 8.1 \\\\\n & 25.0 -- 28.0  & 140  & $3.6 \\pm 0.5$  & $0.7 \\pm 0.6$  & $2.8 \\pm 0.7$  & $-1.7 \\pm 2.5$  & $7.4 \\pm 3.1$ & 0.006 & 0.4 & 1.8 & 34.0 & 19.8 \\\\\nAe  & 22.0 -- 24.0  &  71  & $19.0 \\pm 0.8$  & $6.3 \\pm 0.9$  & $3.7 \\pm 1.0$  & $27.2 \\pm 3.4$  & $30.7 \\pm 4.2$ & $<10^{-3}$ & $<10^{-3}$ & 1.7 & 0.006 & 2.0 \\\\\n & 24.0 -- 25.0  &  57  & $35.3 \\pm 0.8$  & $14.2 \\pm 0.9$  & $8.0 \\pm 1.1$  & $69.2 \\pm 4.0$  & $52.5 \\pm 4.8$ & $<10^{-3}$ & $<10^{-3}$ & 0.01 & 0.002 & 0.09 \\\\\n & 25.0 -- 28.0  &  28  & $15.2 \\pm 1.2$  & $7.2 \\pm 1.3$  & $6.4 \\pm 1.6$  & $24.3 \\pm 5.6$  & $7.8 \\pm 7.0$ & $<10^{-3}$ & 0.004 & 0.4 & 4.0 & 94.5 \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\nAs found by H10, the detection of all our subsamples is best at 250\n$\\mu$m, although with this larger sample most bins are significantly\ndetected at 350 $\\mu$m as well (see Tables \\ref{zstest} and\n\\ref{lstest}). 500-$\\mu$m detections are less robust, and only the\nlow-redshift Ae subsamples are significantly detected in the PACS\nbands. We therefore rely on the 250 $\\mu$m flux densities for our first\nestimate of luminosities. With the division of the samples into the\ntwo emission-line classes, we can see that the mean 250-$\\mu$m flux\ndensity for the Ae objects is much higher than for the Aa objects in\nevery bin.\n\nOur stacking analysis follows the method of H10; we determine the\nluminosity for each source from the background-subtracted flux\ndensity, even if negative, on the grounds that this is the\nmaximum-likelihood estimator of the true luminosity, and take the\nweighted mean within a bin to estimate stacked bin luminosities.\nUnlike H10, we determine errors on the bins by bootstrapping, having\nverified that this method gives very similar results to the much more\ntime-consuming and complex method used in the earlier paper. As\nFig.\\ \\ref{compar} shows, we find a clear and significant difference\nbetween the FIR luminosities of the Ae and Aa objects in every bin in\neither radio luminosity and redshift.\n\n\\subsection{Comparison with normal galaxies}\n\\label{normal}\n\n\\begin{figure*}\n\\epsfxsize 17cm\n\\epsfbox{rz.eps}\n\\caption{The $M_R$-redshift distribution ($K$-corrected as described\n  in the text) of sample objects in the Aa (blue) and Ae (green)\n  classes, after the `SF cut', plotted on top of the same information\n  for the comparison galaxies discussed in the text (greyscale shows a\n  density plot with square-root transfer function). Red lines show the\n  range of absolute magnitudes of the radio-selected sample, used to\n  generate the comparison galaxy sample discussed in the text. The\n  peak in redshift around $z\\approx 0.8$ is probably an artefact of\n  the photometric redshifting, but does not affect our comparison\n  sample.}\n\\label{rz}\n\\end{figure*}\n\nAs a result of our procedure for generating our radio galaxy catalogue\nin H10, we automatically had a comparison galaxy population. There is\nno equivalent in the present work, in the sense that there is no\ngalaxy population selected in the same way as the radio-loud objects.\nSpectroscopic redshifts for the GAMA sample run out at around $z \\sim\n0.6$ because of the magnitude limit used by GAMA \\citep{Driver+11}\nwhile our spectroscopic sample extends to fainter galaxies and higher redshifts.\nHowever, a rough comparison with radio-quiet galaxies is useful to put\nour results in context. We therefore constructed a comparison galaxy\nsample as follows:\n\\begin{enumerate}\n\\item We based the sample on the galaxy catalogue over the Phase 1\n  fields provided as part of the H-ATLAS data release, constructed in\n  the manner described by \\cite{SmithD+11}, and selected galaxies that\n  had either a spectroscopic redshift (from GAMA or SDSS) or a\n  photometric redshift with nominal error $<0.1$, had measured SDSS\n  $r$ and $i$ magnitudes, and were not point-like in $r$ or $i$.\n\\item From this sample we took all objects which lay on the observed H-ATLAS\n  fields and measured their background-subtracted 250-$\\mu$m flux densities as\n  for the radio galaxies (giving 318,244 objects in total, the vast\n  majority with only photometric redshifts). We excluded at this point\n  all objects that formed part of the radio-galaxy sample.\n\\item We $K$-corrected the $r$-band absolute magnitudes of the radio\n  galaxies and the comparison sample to $z=0$ using {\\sc kcorrect} v.\\ 4.2\n  \\citep{Blanton+07}.\n\\item Comparing the range of $r$-band absolute magnitude in the\n  radio-galaxy sample with that in the comparison galaxies\n  (Fig.\\ \\ref{rz}) we saw that the radio-selected objects tend to be\n  bright galaxies at all redshifts. At lower $z$ there is a tendency\n  for the Ae galaxies to be fainter than the Aa (as seen by, for\n  example, \\citealt{Tasse+08} and \\citealt{Best+Heckman12}; note that our\n  sample is not complete, which reduces the extent to which we can\n  draw conclusions from this observation), but they occupy similarly\n  bright galaxies at high $z$ (Fig.\\ \\ref{amhist}).\n\\item Clearly for even a\n  rough comparison we should compare the radio galaxies with optical\n  objects of comparable magnitudes. In each of 14 bins of width\n  $\\Delta z =0.05$ between $z=0.1$ and $z=0.8$, we selected only the\n  comparison galaxies that lay in the absolute $r$ magnitude range\n  spanned by the Aa and Ae objects (post-SF cut) in that redshift\n  range.\n\\end{enumerate}\n\n\\begin{figure*}\n\\epsfxsize 8.5cm\n\\epsfbox{amhist.eps}\n\\epsfxsize 8.5cm\n\\epsfbox{amhist-cg.eps}\n\\caption{Histogram of (left) the absolute $r$ magnitudes of Aa (blue) and Ae (green)\n  galaxies in the sample, after application of the SF cut, and (right)\n  the absolute magnitudes of the comparison objects after the\n  magnitude range selection. Aa objects\n  are generally more massive galaxies than Aes, but the distributions have\n  substantial overlap; the distribution of comparison galaxy absolute\n  magnitudes has an intermediate peak.}\n\\label{amhist}\n\\end{figure*}\n\nWe then stacked the {\\it Herschel} FIR luminosities of the galaxies in\nthose 14 bins, deriving them from the 250-$\\mu$m flux densities on the\nassumption $T = 20$ K, $\\beta = 1.8$ in the same way as for the radio\ngalaxies. These stacks are plotted as a function of $z$ on\nFig.\\ \\ref{compar}. We emphasise that this is intentionally a crude\ncomparison: for example, we could also have performed a colour\nselection on the comparison galaxies, but this would have involved an\ninvestigation of the optical colours of the Aa and Ae objects,\naccounting for possible AGN contamination, which we wish to defer to a\nlater paper (Ching {et al}\\\/., in prep.); similarly, we are not attempting\nto separate spirals and ellipticals in the comparison sample, and we\nhave made no attempt to match the actual distributions of magnitudes\n(and thus stellar masses) of the comparison galaxies within the\nabsolute magnitude ranges used (Fig. \\ref{amhist}). Our optical\nselection, which is required to allow matching to the radio galaxies,\nalso potentially biases us against the most strongly star-forming\nradio-quiet galaxies, which will tend to be more dust-obscured.\nHowever, the result of the comparison is clear. The luminosities for\nthe comparison galaxies tend to lie in between those for the Aa and\nthe Ae radio-loud objects; thus, in the redshift range where we have\ndata, Ae galaxies are on average {\\it more} luminous in the FIR than\nthe average galaxy of comparable optical magnitude at a given\nredshift, and Aa galaxies are on average {\\it less} luminous, at\n  least in the low-redshift bins where a comparison is possible. We note\nthat the comparison sample, though differently selected, is behaving\nin a very similar manner to that of H10 in terms of its FIR luminosity\nas a function of redshift; but now that we have a larger sample and\ncan separate radio galaxies by emission-line class, we are able to see\ndifferences between the radio-loud and radio-quiet populations.\n\n\\subsection{Individual dust temperatures and masses}\n\\label{temperatures-ind}\n\nOur large sample and the availability of the PACS data allow us to\ninvestigate the temperatures and isothermal dust masses (Section\n\\ref{luminosity}) of radio-selected objects for the first time.\n\n\\begin{figure*}\n\\epsfxsize 8.5cm\n\\epsfbox{t-l.eps}\n\\epsfxsize 8.5cm\n\\epsfbox{pdf.eps}\n\\caption{Left: a temperature-luminosity plot for the SF,\n  Aa and Ae sources with individually determined temperatures.\n  Right: stacked, normalized posterior distribution functions (PDFs) for\n  temperature fits to the SF, Ae and Aa emission-line classes.}\n\\label{temps-plot}\n\\end{figure*}\n\nWe began by fitting single-temperature modified black-body models to\nall the sources for which this was possible. We selected all objects\nwhich had a $2\\sigma$ detection, as defined above, in at least two\n{\\it Herschel} bands, in order to ensure that there was at least in\nprinciple sufficient signal-to-noise to constrain parameters, and then\nused standard Levenberg-Marquardt $\\chi^2$ minimization to find the\nbest-fitting values of temperature and normalization for the modified\nblackbody model to the flux densities measured at all five H-ATLAS\nbands.\n\nOne decision that has to be taken here concerns the emissivity\nparameter $\\beta$. Earlier work, including H10 and \\cite{Dunne+11},\ntakes this to be 1.5, but work on local galaxies\n\\citep[e.g.][]{Davies+12} has obtained good fits (using much better\ndata than available to us) with $\\beta=2$. Fitting for $\\beta$ across\nthe whole sample (marginalizing over temperature and normalization for\neach object) we find that the best fits are found with $\\beta = 1.8$\n(the validity of this approach will be discussed by Smith {et al}\\\/., in prep).\nWe fixed $\\beta$ to this value and then fitted for temperature and\nnormalization, determining errors by mapping the $\\Delta \\chi^2 = 2.3$\nerror ellipse (corresponding to $1\\sigma$ for two parameters of\ninterest). For these final fits, only individual fits with an\nacceptable $\\chi^2$ value (defined as a reduced $\\chi^2 <2$) and a\nwell-constrained temperature ($\\Delta T\/T < 2$) are considered in what\nfollows. This process gives us 385 measured temperatures and\nnormalizations, including 128 Aa objects (10 per cent of the total),\n70 Aes (35 per cent) and 170 SF objects (88 per cent); the `SF cut'\nwas not applied to the parent sample. Integrated FIR luminosity\n($L_{\\rm IR}$, integrated between 8 and 1000 microns) and isothermal\ndust masses were then calculated from the fitted temperature and\nnormalization.\n\nClearly the quantities we measure in this way are expected to be\nbiased towards the brightest and hottest objects, but it is still\ninstructive to see how they relate to our emission-line\nclassifications. Fig.\\ \\ref{temps-plot} shows the\ntemperature-luminosity and temperature-dust mass plots for these\nobjects broken down by emission-line class. We exclude the broad-line\nobjects, because of concerns noted above about contamination of the\nFIR bands by non-thermal emission. We see what appears to be a bimodal\ndistribution of temperatures, with one set of objects, here seen to be\nmostly Aas, having temperatures in the range $10<T<20$ K, while the\nother, comprising most of the SF and Ae objects together with a\nsignificant minority of Aas, has $25 < T < 40$ K. The typical error\nbar on fitted temperature (not plotted for clarity) is of order 10 per\ncent. These temperatures generally seem realistic: the isothermal dust\ntemperatures measured by \\cite{Dunne+11} span the range 10--50 K, and,\nunsurprisingly, our temperatures also lie in this range, while\ntemperatures measured for early-type galaxies in the {\\it Herschel}\nReference Survey \\citep{SmithM+12} are $\\sim 24$ K, which is\ncomparable to what we see for the Ae sample. What is more surprising\nis the cold temperatures found for a number of the Aa objects.\nIntegrated temperatures of the order of 10 K are perhaps just\nplausible for dust in thermal equilibrium with the old stellar\npopulation of elliptical galaxies, if the distribution of dust is\nchosen carefully; \\cite{Goudfrooij+deJong95} show that temperatures of\ntens of K are expected for dust in the centres of elliptical galaxies,\nbut the effective temperature should be an emission-weighted average\nover the dust distribution throughout the galaxy, and the energy\ndensity in photons falls off very rapidly with distance from the\ncentre of the galaxy, so the integrated temperature depends on dust\ndistribution, but could be substantially lower than the peak value.\nHowever, the lowest temperatures found by the fits would imply dust\nmasses of up to $10^{10} M_\\odot$, which is probably not realistic.\nInspection of the images for the sources with very low temperatures\nsuggests that several of them are the result of flux confusion: given\nour $2\\sigma$ flux cuts, up to 30 of the Aa sources that we have\nfitted could be spurious detections, and while we do not expect the\nnumbers to be this high in practice, confusion seems likely to account\nfor a number of the sources with the lowest fitted temperatures and\nhighest dust masses. In addition, synchrotron contamination of the\nSPIRE bands, which is observed in nearby LERGs like M87\n\\citep{Baes+10} could be affecting a few Aas -- several\nfall below the dividing line in the plot of Fig.\\ \\ref{lopezplot},\nalthough not all of these will have the flat integrated radio spectrum\nrequired for non-thermal emission to appear at 500 $\\mu$m.\n\nAnother way of investigating the temperature differences between the\nsamples, which does not put so much weight on individual objects, is\nto consider the stacked posterior probability distribution over $T$,\nmarginalizing over normalization, for the fitted objects, and this is\nshown in the right-hand panel of Fig.\\ \\ref{temps-plot}. Here a prior\nof $5<T<55$ K is used. We see that this plot reproduces the broad\ntrends seen in the left-hand panel: the SF objects have a fairly\nwell-defined peak in temperature at around 26 K, the Aes have a\nbroader peak at around 30 K, and the Aas span a range between around\n10 and 30 K. Even taking into account possible contamination by\nconfused sources and\/or synchrotron emission in a few cases, it does\nnot seem likely that the Aa and Ae sources have the same intrinsic\ntemperature distribution. In addition, it is hard to see how this\ndifference in the PDFs could be explained by, for example, different\n$\\beta$ values for different populations.\n\n\\begin{figure*}\n\\epsfxsize 8.5cm\n\\epsfbox{l-l.eps}\n\\epsfxsize 8.5cm\n\\epsfbox{m-m.eps}\n\\caption{The relationships between (left) the integrated IR luminosity\nand (right) the dust mass for objects with fits of both a\nsingle-temperature and a two-temperature model.}\n\\label{twot-plots}\n\\end{figure*}\n\nIt is natural to interpret the wide range of temperatures in the\ntemperature-luminosity plots in terms of two populations of dust: (1),\na cold dust component which is always present, which is essentially in\nthermal equilibrium with the old stellar population ($T \\sim 15$ K might\nbe reasonable for this as an average over the dust properties of an\nelliptical galaxy, as noted above) and whose mass scales with the\ntotal galaxy mass, perhaps with some redshift dependence; and (2), a\nwarmer dust component with $T \\sim 30$ K which traces current star\nformation and whose mass and luminosity are primarily an indicator of\nthe star-formation rate \\citep[cf.][]{Dunne+11,SmithD+12}. In fact,\nsuch a two-temperature model might help to explain the lack of objects\nwith $T \\sim 20$ K in Fig.\\ \\ref{temps-plot}: although there should be\nobjects where the two components contribute roughly equally to the\n{\\it Herschel} SED, these would tend to be poorly fitted with a\nsingle-temperature model and would be rejected by our fitting\nprocedure.\n\nThe type of broad-band fitting to the FIR through ultraviolet SEDs\ncarried out by \\cite{SmithD+12} is beyond the scope of this paper, but\nwe investigated a two-temperature model by fitting the same dataset\nfor the normalizations of two modified-blackbody models with $\\beta =\n1.8$ and fixed temperatures of 15 K and 30 K (normalizations here are\nexplicitly constrained to be positive to avoid trading off negative\nflux in one component against positive flux in another). These models\nin general fit less well and therefore give fewer sources with\nacceptable fits, presumably because there are objects with\nwell-determined temperatures for the warm component that are\nsignificantly different from 30 K, but we do note that they provide\ngood fits to a population of objects that are rejected by the $\\chi^2$\ncriterion for the single-temperature fits and that generally have\nnon-zero contributions to the dust luminosity from both hot and cold\ncomponents. Moreover, and importantly for what follows, we find a good\ncorrelation between the total luminosities for objects where these can\nbe obtained using both methods (Fig.\\ \\ref{twot-plots}), while the\nestimated dust masses show some systematic differences. This suggests\nthat, at least for this sample, the total IR luminosity can be used\nwithout worrying too much about more complex models, while the dust\nmass must be interpreted with a little more care. If we interpret the\nmass of {\\it warm} dust (or, equivalently for this model, its\nluminosity) in these fits as tracing star-formation rate, then, using\nthe results of \\citep{SmithD+12}, the typical SF object in our sample\nhas a star-formation rate of $\\sim 10 M_\\odot$ yr$^{-1}$, while the\nmost luminous Ae objects might have star-formation rates more than ten\ntimes higher. More detailed methods for estimating star-formation\nrates are discussed in the following subection.\n\nFinally, we note that the weighted mean of the best-fitting single\ntemperatures of the Aas and Aes, for $\\beta = 1.8$, is 20.3 K. This\njustifies the assumptions we used for $K$-corrections in the luminosity\nstacking of Section \\ref{stacking}.\n\n\\subsection{$L_{250}$ as a star-formation rate indicator; comparing\n  emission-line classes}\n\\label{SFR}\n\nAs we noted in the previous subsection, contributions\nto $L_{250}$ are made both by cold dust (driven by the old stellar\npopulation) and warm dust (driven by star formation). It follows that neither\n$L_{250}$ nor the integrated $L_{\\rm IR}$ are reliable indicators of\nstar formation rates (SFR) in general. However, they should\nboth be usable to estimate SFR for an object whose FIR emission is\n{\\it dominated} by emission from warm dust: these will be the objects\nwhose best-fitting temperatures are $\\sim 25$ K or more. For objects\nwith a contribution from cold dust, the SFR estimated from $L_{\\rm\n  250}$ or $L_{\\rm IR}$ will be an upper limit.\n\nTo use the quantity that we have used for stacking, $L_{\\rm 250}$, in\nthis way we need to calibrate the relationship between it and SFR. We\nchoose to do this by considering the objects classed optically as\n`SF', as, where temperature information is available, all of these\nhave FIR temperatures consistent with being dominated by star\nformation (Section \\ref{temperatures-ind}). For SF objects with SDSS\nspectra estimated star formation rates, derived using the methods of\n\\cite{Brinchmann+04} (i.e. by model-fitting to the optical emission\nlines and stellar continuum), are available in the MPA-JHU database.\nCross-matching our SF objects against this database gives a sample of\n158 objects with both SFR and $L_{\\rm 250}$ estimates; we use the\nmedian likelihood estimates given in the MPA-JHU database, as being\nthe most robust, and take half the difference between the 16th and\n84th percentiles as an estimate of the error on the SFR. As these\nobjects are all at low redshifts ($z<0.24$, and median $z = 0.08$) we\nneed not be concerned about the $K$-correction used to derive $L_{\\rm\n  250}$. When we plot $L_{\\rm 250}$ against SFR derived in this way,\nwe see a good correlation (Fig.\\ \\ref{l-sfr}) with a slope that is, by\neye, close to unity. A Markov-Chain Monte Carlo regression, taking the\nerrors on both SFR and $L_{\\rm 250}$ into account and incorporating an\nintrinsic dispersion in the manner described by \\cite{Hardcastle+09},\ngives Bayesian estimates of the slope and intercept of the correlation:\n\\[\n\\log_{10}(L_{250}\/{\\rm W}\\ {\\rm Hz}^{-1}) = 23.64 + 0.96\n\\log_{10}({\\rm SFR}\/M_\\odot\\ {\\rm yr}^{-1})\n\\]\nAlthough a slope\nof unity is not ruled out, we will use this slightly non-linear\nrelationship in what follows. We emphasise again that it is only valid for\nobjects whose FIR emission is known to be dominated by warm dust\nheated by star formation.\n\nAs a sanity check on this approach, we can also estimate the\nrelationship between SFR and integrated $L_{\\rm IR}$ by using our\ntemperature fits from Section \\ref{temperatures-ind}. The vast majority (143)\nof SF objects with SFR estimates also have estimates of $T_{\\rm IR}$\nand thus $L_{\\rm IR}$, and this quantity also correlates well with\nSFR (Fig.\\ \\ref{l-sfr}). Regression gives a linear relation\n\\[\n\\log_{10}(L_{\\rm IR}\/L_\\odot) = 9.90 + 1.00 \\log_{10}({\\rm\n  SFR}\/M_\\odot\\ {\\rm yr}^{-1})\n\\]\nwhose normalization is only a factor 1.4 away from the standard\nrelation given by \\cite{Kennicutt98}, derived for starbursts. Thus we\ncan use our IR-derived SFR with reasonable confidence.\n\nApplying the $L_{250}$\/SFR relation to Fig.\\ \\ref{compar}, we can see\nthat the most luminous detected radio galaxies, at around $9 \\times\n10^{25}$ W Hz$^{-1}$, should correspond to star-formation rates around\n250 $M_\\odot$ yr$^{-1}$, which does not seem unreasonable -- these\nwould be radio galaxies associated with starbursts -- although it\nshould be noted that there is a non-negligible uncertainty associated\nwith the $L_{250}$ values of these luminous, high-$z$ sources because\nof the poorly known $K$-correction. Individual powerful, high-$z$ radio galaxies have\n  been associated with star formation at levels even higher than this\n  \\citep{Barthel+12,Seymour+12}. The mean $L_{250}$ of the most\nradio-luminous Aes, with $L_{1.4}>10^{25}$ W Hz$^{-1}$, corresponds to\n$15M_\\odot$ yr$^{-1}$, which is well above the SFRs expected for\nnormal ellipticals in the local universe. The factor $\\sim 4$ between\nthe stacked $L_{250}$ values for Aes and Aas means that the mean star\nformation rate in the latter is {\\it at least} a factor 4 below that\nin the Aes; `at least' because the temperature measurements suggest\nthat the emission from some, and perhaps most, Aas is dominated by\ncold dust.\n\n\\begin{figure*}\n\\hbox{\\hskip-1.3cm\n\\input{sfr-l250}\n\\hskip-1.2cm\n\\input{sfr-lir}}\n\\caption{The relationship between star-formation rate (SFR),\n  determined for the SF class from the MPA-JHU emission-line database\n  using the methods of \\cite{Brinchmann+04}, and the\n  two types of FIR luminosity discussed in this paper, $L_{250}$\n  (left) and $L_{\\rm IR}$ (right). The green lines\nshow the results of a MCMC regression, as described in the text. Error\nbars are not plotted for clarity -- the uncertainty on SFR can be $\\pm\n0.5$ dex.}\n\\label{l-sfr}\n\\end{figure*}\n\n\\subsection{Stacked dust temperatures, masses and SFR}\n\\label{temperatures-stacked}\n\n\\begin{table*}\n\\caption{Results of temperature fitting, dust mass and star formation\n  rate estimation in\n  redshift bins.}\n\\label{tfit-z-results}\n\\begin{tabular}{lrrrrrrr}\n\\hline\nClass&Range in&Objects&Best-fit $T$&Reduced&$\\log_{10}(L_{250}\/{\\rm\n  W}\\ {\\rm Hz}^{-1})$&$\\log_{10}(M_{\\rm iso}\/M_\\odot)$&$\\log_{10}({\\rm SFR}\/M_\\odot\\ {\\rm yr}^{-1})$\\\\\n&$z$&in bin&(K)&$\\chi^2$\\\\\n\\hline\nAa& 0.00 -- 0.30 & 398 & $17.0^{+0.9}_{-0.8}$ & 1.65 & $23.9 \\pm 0.1$ & $7.6 \\pm 0.1$ & $0.2 \\pm 0.1$ \\\\\n& 0.30 -- 0.50 & 472 & $17.4^{+0.6}_{-0.6}$ & 1.52 & $24.3 \\pm 0.1$ & $8.0 \\pm 0.1$ & $0.7 \\pm 0.1$ \\\\\n& 0.50 -- 0.90 & 309 & $27.8^{+2.1}_{-3.4}$ & 1.44 & $24.6 \\pm 0.1$ & $7.7 \\pm 0.1$ & $1.0 \\pm 0.1$ \\\\\nAe& 0.00 -- 0.30 & 62 & $32.8^{+0.7}_{-0.8}$ & 2.11 & $24.3 \\pm 0.1$ & $7.3 \\pm 0.1$ & $0.7 \\pm 0.1$ \\\\\n& 0.30 -- 0.50 & 55 & $30.9^{+0.7}_{-0.8}$ & 1.45 & $24.8 \\pm 0.1$ & $7.8 \\pm 0.1$ & $1.2 \\pm 0.1$ \\\\\n& 0.50 -- 0.90 & 36 & $34.2^{+1.4}_{-1.5}$ & 1.99 & $25.1 \\pm 0.1$ & $8.0 \\pm 0.1$ & $1.5 \\pm 0.1$ \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\caption{Results of temperature fitting, dust mass and star formation\n  rate estimation in\n  luminosity bins.}\n\\label{tfit-l-results}\n\\begin{tabular}{lrrrrrrr}\n\\hline\nClass&Range in&Objects&Best-fit $T$&Reduced&$\\log_{10}(L_{250}\/{\\rm\n  W}\\ {\\rm Hz}^{-1})$&$\\log_{10}(M_{\\rm\n  iso}\/M_\\odot)$&$\\log_{10}({\\rm SFR}\/M_\\odot\\ {\\rm yr}^{-1})$\\\\\n&$\\log_{10}(L_{1.4})$&in bin&(K)&$\\chi^2$\\\\\n\\hline\nAa& 22.0 -- 24.0 & 456 & $15.1^{+0.6}_{-0.5}$ & 1.50 & $24.0 \\pm 0.1$ & $7.9 \\pm 0.1$ & $0.4 \\pm 0.1$ \\\\\n& 24.0 -- 25.0 & 589 & $23.8^{+0.8}_{-0.7}$ & 1.65 & $24.5 \\pm 0.1$ & $7.7 \\pm 0.1$ & $0.9 \\pm 0.1$ \\\\\n& 25.0 -- 28.0 & 140 & $15.3^{+0.7}_{-0.6}$ & 1.48 & $24.7 \\pm 0.1$ & $8.6 \\pm 0.1$ & $1.1 \\pm 0.1$ \\\\\nAe& 22.0 -- 24.0 & 71 & $26.3^{+0.6}_{-0.7}$ & 1.70 & $24.3 \\pm 0.1$ & $7.5 \\pm 0.1$ & $0.7 \\pm 0.1$ \\\\\n& 24.0 -- 25.0 & 57 & $30.9^{+0.5}_{-0.6}$ & 2.08 & $25.0 \\pm 0.1$ & $8.0 \\pm 0.1$ & $1.5 \\pm 0.1$ \\\\\n& 25.0 -- 28.0 & 28 & $27.0^{+1.7}_{-1.8}$ & 1.59 & $24.9 \\pm 0.2$ & $8.0 \\pm 0.2$ & $1.3 \\pm 0.2$ \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\nAs noted above, direct estimation of dust temperatures can only be\ncarried out for the brightest (and possibly hottest) objects, and so\nmight give misleading results if used in the interpretation of our\nstacking analysis of the whole sample. As an alternative, we can\nestimate mean temperatures for objects in the sample as follows. We\nbin our objects in redshift or radio luminosity as in the previous two\nsections. For each redshift\/luminosity bin, we determine the {\\it\n  single} dust temperature that gives the best $\\chi^2$ fit to the\nobserved flux densities of every galaxy in the bin, allowing each galaxy to\nhave a free normalization (which may be negative) and taking a fixed\n$\\beta = 1.8$. Errors in this fitted temperature are estimated by\nfinding the range that gives $\\Delta \\chi^2 = 1$. We can then use the\nbest-fitting temperature and normalizations for all the sources to\nestimate the 250-$\\mu$m luminosity of the bins, determining error bars\nby bootstrap as before. The results of this process are tabulated in\nTables \\ref{tfit-z-results} and \\ref{tfit-l-results}. The $\\chi^2$\nfitting gives acceptable, though not particularly good results, as we\nwould expect since, from the analysis of Section\n\\ref{temperatures-ind}, we know that there is a wide range of\ntemperatures in each bin. Nevertheless, we can attempt to interpret\nthe results.\n\nThree points are of interest. Firstly, we note that the luminosities\nwe estimate are broadly consistent, within the errors, with the\nluminosities estimated from the stacking analysis of Section\n\\ref{stacking}; this gives us confidence that the luminosities from\nthe earlier analysis are reasonable and that the assumption of a\nsingle temperature for the $K$-correction does not have a big effect\non the inferred monochromatic luminosities. The luminosity difference\nbetween the Aa and Ae spectral classes clearly persists in this\nanalysis. Secondly, we see that the temperatures are systematically\ndifferent for the two emission-line classes: Aes have systematically\nhigher dust temperatures. Thirdly, we can compute isothermal dust\nmasses from eq.\\ \\ref{dustmass} using the best-fitting temperature and\nmean luminosity -- these are of course a complicated weighted mean of\nthe dust masses of all the objects in the bin, but still gives us some\ninformation on the properties of the galaxies. These mean isothermal\ndust masses are tabulated for each bin in Tables \\ref{tfit-z-results}\nand \\ref{tfit-l-results}. No very strong difference between the dust\nmasses for the emission-line classes is seen in these mean masses. It\ntherefore seems plausible that the clear observed difference in\nmonochromatic FIR luminosity at 250 $\\mu$m between the populations is\ndriven by a difference in dust temperature rather than by dust mass.\n\nFinally, we can attempt to convert the $L_{\\rm 250}$ values from this\nfitting into star formation rates, using the results of Section\n\\ref{SFR}. As already noted, this gives us upper limits if we have\nreason to suppose that some of the FIR emission comes from cold dust\nunrelated to star formation. The results of this conversion, given in\nthe final column of Tables \\ref{tfit-z-results}\nand \\ref{tfit-l-results}, must be treated with caution, therefore.\nSince the mean fitted temperatures of the Aes are $>30$ K, their SFR\nestimates may be a reasonable estimate of the true mean SFR in these\nsystems; the same is not true of the Aas, and so, again, the safest\ninterpretation is to say that the mean SFR in the Aes is of order a\nfew tens of solar masses per year, and is {\\it at least}\n$\\sim 0.5$ dex higher than that in the Aas.\n\n\\subsection{Radio source sizes}\n\\label{length}\n\n\\begin{figure*}\n\\epsfxsize 8.43cm\n\\epsfbox{pd.eps}\n\\epsfxsize 8.57cm\n\\epsfbox{lenl-j-bins.eps}\n\\caption{The physical sizes of radio sources in our sample and their\n  relationship to radio and FIR luminosity. Left: the\n  power\/linear-size plot for the sample, broken down by emission-line\n  class, and without the SF cut. Colours are as in Fig.\\ \\ref{llplot};\n  for comparison, 3CRR sources are plotted in magenta. Right: the FIR\n  luminosity\/linear size plot, with stacking in bins at $<40$ and\n  $>40$ kpc, after the SF cut. Colours and symbols as in\n  Fig.\\ \\ref{llplot}; upper limits in length are denoted by arrows.}\n\\label{length-plot}\n\\end{figure*}\n\nV12 noted a strong relationship between the FIR\nluminosity or temperature of the objects they studied and the radio\nsource size, in the sense that larger objects had systematically lower\n$L_{250}$ or $T$. They also showed that this discrepancy was largely\ndriven by the most massive objects in their sample.\n\nIt is clearly interesting to ask how this result relates to the\nobserved differences between emission-line classes. One of us (JSV)\ntherefore determined the largest angular size of every object in the\npresent sample, taking the sizes used by V12 for\nobjects in common between the two samples and otherwise making\nmeasurements directly from the FIRST images. Where objects were\nunresolved in FIRST, an upper limit was assigned, as described by\nV12. Scaling by the angular size distance, this gives the distribution\nof source physical sizes for the current sample.\n\nAn important caveat in this analysis is that the current sample is not\ncomplete. This is illustrated by the left panel of Fig.\n\\ref{length-plot}, which shows the power\/linear-size plot (the `$PD$\ndiagram') for the current sample together with the equivalent plot for\nthe complete and well-studied 3CRR sample. For clarity, upper limits\nare not marked on this plot, but it should be noted that all sources\nwith a physical size $\\ge 40$ kpc are actually resolved. We see that\nall emission-line classes are heavily biased towards smaller physical\nsizes with respect to 3CRR: of course, this is not a completely fair\ncomparison, since the 3CRR objects are the most luminous objects in\nthe radio sky at any given redshift, and we might expect\nlower-luminosity sources to be systematically smaller than the most\nluminous ones. However, there are more subtle signs of bias, such as\nthe fact that there are more large AeB sources than there are Aes:\nthis arises principally because a broad-line object is more likely to\nhave a bright radio core and so to be identified with a galaxy or\nquasar in our original selection (though there is an additional effect\ndue to the different redshift distribution of AeBs and Aes). The Aa\nsources, which should have the full range of angles to the line of\nsight, have a length distribution intermediate between the Aes and\nAeBs, as expected. This bias towards compact sources, or sources with\ncompact cores, is particularly problematic for our sample because of\nthe selection from FIRST radio images, which resolve out large-scale\nemission: the sample of V12 will be closer to being complete.\n\nHaving said this, it is still possible to investigate the FIR\nproperties as a function of length. To do this we apply the SF cut and\nthen stack the FIR luminosities as in Section \\ref{stacking}, binning\nby length: we use only two length bins and the division is set at 40\nkpc to ensure that all upper limits are in the correct bin. The\nresults are shown in Fig.\\ \\ref{length-plot} (right panel). We see\nfirst of all that the Ae\/Aa difference persists in this analysis: Ae\nsources have higher $L_{250}$ than Aas irrespective of length.\nSecondly, we see no evidence for any length dependence of the\n$L_{250}$ of the Ae population, although the error bars are large\nbecause the sample is small. Thirdly, we note a marginally significant\ndifference between the $L_{250}$ values for the small and large Aa\nsources, in the sense noted by V12: the null hypothesis that these two\nare equal can be rejected at the 95 per cent confidence level. If,\ninstead of stacking $L_{250}$, we fit temperatures to all the sources\nin each bin as described in Section \\ref{temperatures-stacked}, we\nfind {\\it no} significant difference in luminosities as a function of\nlength for either emission-line class, but there are significant\ndifferences in best-fitting temperature ($29.4 \\pm 0.5$ vs $22 \\pm 2$\nK for Ae; $21.8 \\pm 0.5$ vs $12.2 \\pm 0.4$ K for Aa), in both cases in\nthe sense that the larger objects have lower best-fitting\ntemperatures. This is again broadly consistent with the results of\nV12.\n\n\\section{Discussion}\n\\label{discussion}\n\nThe results of the previous section show very clearly that there is a\ndifference between the average far-infrared properties of radio\ngalaxies whose spectra show strong emission lines and those of radio\ngalaxies that do not. How can we interpret this?\n\nIt is first of all important to consider the issue of possible AGN\ncontamination, discussed in Section \\ref{intro}, in more detail. There\nare two possible sources for this: (1) emission from the warm dusty\ntorus, which is expected to be seen predominantly in HERGs, and (2)\nsynchrotron emission from the jets and lobes, which may appear in all\nobjects. The first of these is particularly important, as\ntorus-related emission in the FIR bands might give rise to a HERG\/LERG\ndifference, but we are confident that it is not a significant effect\nin our sample, for several reasons. Firstly, when the required mid-IR\ndata are available, which is not the case for our sample at present,\ndecompositions of the SEDs of radio-loud and radio-quiet AGN tend to\nshow that observer-frame {\\it Herschel} SPIRE bands are dominated by\ncool dust rather than by the torus component, even for powerful AGN\nwith luminous tori \\citep[e.g][]{Barthel+12,DelMoro+12}. Secondly, we\nknow, as pointed out by H10, that the mid-IR torus luminosities even\nfor the most powerful HERGs in our sample, if they follow the\ncorrelation between radio power and mid-IR luminosity established by\n\\cite{Hardcastle+09}, should be about two orders of magnitude less\nthan the total FIR luminosities estimated e.g. in\nFig.\\ \\ref{temps-plot}, implying that even if the torus SEDs were\nstrikingly different from those of known radio-loud AGN, it would\nstill be energetically impossible for them to affect the observed FIR\nemission significantly. The second possible source of contamination,\nsynchrotron emission, we believe to affect mostly the broad-line\nobjects, as discussed in Section \\ref{blrg}, together with at most a\nvery few of the Aas; there is certainly no reason to expect that it\nwould give rise to the observed Aa\/Ae difference, since the radio\nfluxes and spectra of these two classes are very similar. We therefore\nconsider it safe to discuss our observations in terms of a difference\n  in the properties of cool dust in the two populations.\n\nOur result cannot be significantly affected by contamination by pure\nstar-forming objects whose radio emission is bright enough to cause\nthem to be misidentified as radio galaxies. While our spectroscopic\nclassification alone does not identify all objects whose radio\nemission is dominated by star formation (Fig.\\ \\ref{llplot}) the\ncombination of optical spectroscopy and the `SF cut' that we impose on\nthe radio-FIR luminosity plane, where a clear star-forming sequence is\nvisible, should remove all such objects. Moreover, the highest radio\nluminosities in our sample are well above even the $\\sim 10^{25}$ W\nHz$^{-1}$ expected from a starburst of a few thousand $M_\\odot$\nyr$^{-1}$, and we see a clear difference between the different\nemission-line classes in this luminosity range.\n\nAlong similar lines, we do not believe that the relationship between\nemission-line class and FIR emission can be a result of optical\nemission-line activity due to the star formation process itself. Our\nemission-line classification uses [O{\\sc iii}], which, at least at\nhigh luminosities, is widely used as an AGN indicator, although it can\nbe produced by hot, young stars. We do not have a direct estimate of\nthe [O{\\sc iii}] luminosities of our sample objects in the version of\nthe GANDALF-derived database we use, but we have derived a rough\nindicator of luminosity from the measured equivalent widths and the\n$K$-corrected absolute magnitude in the SDSS $g$ band. Calibrating\nthis indicator using the MPA-JHU emission-line measurements, for which\nboth equivalent width and [O{\\sc iii}] flux are tabulated, we see that\nalmost all the Ae objects would be expected to have $L([\\rm{O\\hskip\n    0.02cm\\scriptscriptstyle{III}}])>10^{40}$ erg s$^{-1}$, and are\nthus in the range classified by e.g. \\cite{Kauffmann+03} as `strong\nAGN'. Further work in this area will require measurements of the\n[O{\\sc iii}] fluxes, and ideally those of other lines, for a large\nradio-galaxy sample, but we are confident that we are assessing\ngenuine AGN activity in the vast majority of cases.\n\nWe can therefore move on to interpreting the relationship as being one\nbetween FIR properties of the host galaxy and the AGN-related\nemission-line properties of radio galaxies that we discussed in\nSection \\ref{intro}, with the Aa population corresponding to LERGs and\nthe Ae population to HERGs. It then appears that HERGs, on average,\nhave significantly higher $L_{250}$ than LERGs (Section\n\\ref{stacking}); moreover, HERGs appear to have higher $L_{250}$ than\nnormal galaxies of comparable absolute magnitude at all redshifts\n(Section \\ref{normal}). In a simple isothermal model, higher\nluminosity can arise either because of higher masses of dust or higher\ntemperatures; what we see from the analysis of Sections\n\\ref{temperatures-ind} and \\ref{temperatures-stacked} is that it is\nplausible that the dust masses of the different systems are similar,\nbut that the mean isothermal temperatures of the HERGs are higher. In\nresolved local galaxies, it has been shown that low-temperature dust\nemission ($T \\sim 15$ K) is driven by the old stellar population,\nwhile significantly hotter temperatures are seen from star-forming\nregions \\citep[e.g.][]{Bendo+10,Boquien+11}. By far the most obvious\ninterpretation of our result is therefore that the star-formation\nrates are significantly higher in the HERG subsamples than in the\nLERGs, giving rise to a significant component of emission from hot\ndust which raises the isothermal temperature as seen in the analysis\nof \\cite{Dunne+11}. If so, this is strong confirmation, using a much\nlarger sample, of the picture that emerges from the earlier work\ndiscussed in Section \\ref{intro}\n\\citep[e.g.][]{Baldi+Capetti08,Herbert+10,RamosAlmeida+11b,RamosAlmeida+12}.\nBy calibrating $L_{\\rm 250}$ as a star formation indicator using SFRs\nderived from local radio-loud star-forming galaxies (Section\n\\ref{SFR}) we have been able to quantify this, showing that the mean\nSFR in the most luminous\/high-$z$ Aes is probably at the level of\naround $30 M_\\odot$ yr$^{-1}$, and is {\\it at least} $\\sim 0.5$ dex\nhigher than that in the Aas at all redshifts and radio luminosities.\n\nWhat does this tell us about the association between star formation\nand AGN activity in radio galaxies? The first point to note is that\nthe association between a HERG classification and increased FIR\nluminosity (and therefore star-formation rate) is statistical only. It\ncan be seen from Fig.\\ \\ref{compar} that there are individual LERGs\nwith high FIR luminosities, while at the same radio luminosity we see\nHERGs with FIR luminosities 0.5-1 dex lower. Similarly, the\ntemperature analysis of Section \\ref{temperatures-ind} shows that\nthere are LERGs with best-fitting temperatures comparable (within the\nlarge errors) to those of the warm dust in known star-forming\ngalaxies. Nothing appears to require HERGs to be associated with high\nstar-formation rates or LERGs to be associated with completely\nquiescent galaxies, consistent with the conclusions of\n\\cite{Tadhunter+11}. This suggests that the mechanism of the\nassociation is not the simplest possible one, in which some single\nevent, such as a merger, always triggers both HERG activity and star\nformation. If this were the case, we would not see individual LERGs\nwith high star-formation rates (setting aside the possibility, which\nwe regard as remote, that these sources are all misidentified HERGs).\n\nAnother piece of evidence supporting this picture comes from the lobe\nlength analysis of Section \\ref{length}. If HERGs were associated with\nAGN triggering following a merger, we might expect them to show a very\nstrong relationship between FIR properties and lobe physical size, since star\nformation would be expected to peak on average at early times in the radio\nsource's lifetime. This was a possible interpretation of the results\nof V12, who showed that larger sources in general have lower FIR\ntemperatures and luminosities. However, our analysis shows that this\nresult is {\\it not} driven by the HERG (Ae) population, and in fact for\nour sample is more obvious for the LERGs (which, however, have\nconsiderably better statistics).\n\nWe would therefore argue that we are not seeing a simple triggering\nrelationship, but rather that the difference between HERGs and LERGs\nis that HERGs tend to inhabit environments in which star formation is\nfavoured relative to the general galaxy population, while by contrast\nstar formation is disfavoured in the environments of LERGs. The FIR\ndifferences as a function of source length would then be explained by\nsome other process, such as jet-induced star formation when the bow\nshock of the source is within the host galaxy, which can in principle\ntake place in both emission-line classes (although we note there is\nnot yet any direct evidence for this process affecting emission seen\nin the FIR band). Such a model is consistent both with all the\nobservations to date\n\\citep[e.g.][]{Baldi+Capetti08,Herbert+10,Best+Heckman12,Janssen+12}\nand with the explanation of the HERG\/LERG dichotomy in terms of\naccretion mode discussed in Section \\ref{intro}. It will be of great\ninterest to see whether this result is confirmed by the larger samples\nthat will be made available by the full H-ATLAS dataset, whether it\ncan be extended to higher redshifts using deeper spectroscopic or\nphotometric surveys, and whether the same results are obtained when\nthe HERG\/LERG classification is made using data at other wavebands\n(e.g. X-ray or mid-IR).\n\n\\section{Summary and conclusions}\n\nThe key results from the analysis and discussion above can be\nsummarized as follows:\n\\begin{itemize}\n\\item We have used individual measurements and stacking analyses to\n  determine the FIR properties (mean luminosities and temperatures) of\n  a large sample of radio-selected sources with spectroscopic redshifts\n  and HERG\/LERG classifications from optical spectroscopy. Sources near\n  the known FIR-radio correlation are excluded from our analysis; the\n  vast majority of the objects we study should be bona fide radio galaxies.\n\\item We find a clear difference between the FIR properties of the two\n  populations in the sense that the rest-frame 250-$\\mu$m luminosities\n  are systematically higher in the HERGs than in the LERGs; the host\n  galaxies of LERGs in fact occupy galaxies with lower FIR luminosities\n  than normal galaxies matched in absolute magnitude, while HERGs tend\n  to have higher FIR luminosities. This difference is apparent at all\n  redshifts and all radio luminosities sampled by our targets.\n\\item A comparison of the temperatures and dust masses of HERGs and\n  LERGs, stacked in coarse bins, suggests that the dust masses are\n  reasonably comparable for the two samples but that the temperatures\n  in the HERGs are systematically higher. This provides strong evidence\n  that the higher FIR luminosities we are seeing imply, on average,\n  higher star formation rates (which are required to raise the mean\n  temperature of the dust) rather than just higher dust masses. The\n  low mean temperatures seen for LERGs are consistent with what would\n  be expected for quiescent dust which is in thermal equilibrium with\n  the photon field of the old stellar population of the host galaxy,\n  although the fact that these objects are detected at all implies\n  that large masses of dust are present.\n\\item Quantifying the SFR by calibrating $L_{250}$ as a star-formation\n  indicator in the `SF' sources known to be dominated by hot dust, we\n  find that the mean SFR in the radio-luminous Aes is $\\sim 30M_\\odot$\n  yr$^{-1}$, and is {\\it at least} $\\sim 0.5$ dex higher than that in\n  the Aas at all luminosities and redshifts.\n\\item Consistent with the results of V12, we find that both\n  emission-line classes in our sample show some evidence for a\n  dependence of FIR properties on radio source size.\n\\item We argue that there is certainly not a simple triggering\n  relation, and not even a one-to-one association, between enhanced\n  star formation and a particular AGN type \\citep[a conclusion\n    consistent with detailed studies of starburst radio galaxies such\n    as that of][]{Tadhunter+11}. However, the statistical trend for\n  HERGs to have higher star formation rates is consistent both with\n  what is known from other wavebands\n  \\citep[e.g.][]{Best+Heckman12,Janssen+12} and with the general class\n  of models \\citep{Hardcastle+07-2} in which HERG activity takes place\n  in lower-mass galaxies where the black hole is able to accrete\n  significant quantities of cold gas.\n\\end{itemize}\n\nAs more H-ATLAS data and supporting optical imaging and spectroscopy\nbecome available we expect to extend this work to much larger samples,\nallowing more detailed binning and temperature analysis, to\ninvestigate different methods of carrying out the LERG\/HERG\nclassification, and to consider radio galaxies at higher redshifts in\norder to search for evidence of cosmological evolution of the\nstar-formation properties of the radio-loud AGN population.\n\n\\section*{Acknowledgements}\n\nThe Herschel-ATLAS is a project with Herschel, which is an ESA space\nobservatory with science instruments provided by European-led\nPrincipal Investigator consortia and with important participation from\nNASA. The H-ATLAS website is http:\/\/www.h-atlas.org\/.\n\nGAMA is a joint\nEuropean-Australasian project, based around a spectroscopic campaign\nusing the AAOmega instrument, and is funded by the STFC, the ARC, and\nthe AAO. The GAMA input catalogue is based on data taken from the\nSloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey.\nComplementary imaging of the GAMA regions is being obtained by a\nnumber of independent survey programs including GALEX MIS, VST KIDS,\nVISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to\nradio coverage. GAMA is funded by the STFC (UK), the ARC (Australia),\nthe AAO, and the participating institutions. The GAMA website is\nhttp:\/\/www.gama-survey.org\/. Funding for the SDSS and SDSS-II has been\nprovided by the Alfred P. Sloan Foundation, the Participating\nInstitutions, the National Science Foundation, the U.S. Department of\nEnergy, the National Aeronautics and Space Administration, the\nJapanese Monbukagakusho, the Max Planck Society, and the Higher\nEducation Funding Council for England. The SDSS website is\nhttp:\/\/www.sdss.org\/.\n\nThe National Radio Astronomy Observatory (NRAO)\nis a facility of the National Science Foundation operated under\ncooperative agreement by Associated Universities, Inc.\n\nWe thank an anonymous referee for helpful comments on the paper.\n\n\\bibliographystyle{mn2e}\n\\renewcommand{\\refname}{REFERENCES}\n\\setlength{\\bibhang}{2.0em}\n\\setlength\\labelwidth{0.0em}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}