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+{"text":"\\section{Introduction}\n Pour nous, une $C$-relation est une relation ternaire satisfaisant les axiomes suivants:\n\\begin{enumerate}\n\\item $\\forall x,y,z \\;\\:  { C}(x,y,z)\\rightarrow { C}(x,z,y) $,\n\\item $\\forall x,y,z \\;\\: { C}(x,y,z) \\rightarrow \\neg { C}(y,x,z)$,\n\\item $\\forall x,y,z,w \\;\\:  { C}(x,y,z)\\rightarrow ({ C}(w,y,z)\\vee { C}(x,w,z)),$\n\n\\item $\\forall x,y \\;\\: x\\neq y \\rightarrow { C}(x,y,y)$.\n\\end{enumerate}\n\n\n\nUne $C$-relation interpr\u00e8te toujours un arbre (cf. \\cite{Adeleke1998}): pour des cha\u00eenes maximales $x,y$ et $z$,  ${ C}(x,y,z)$, exprime que $y$ et $z$  branchent au-dessus de l\u00e0 o\u00f9 $x$ et $y$ (ou $z$) branchent.\n\nOn rappelle qu'une\ndistance ultram\u00e9trique, est un triplet $(M,\\Gamma, d)$,  o\\`u $M$ est un ensemble, $\\Gamma$ est un ordre linaire ayant un minimum $0$, et $d:M^2\\to \\Gamma$, une application surjective, telle que, pour tout $x,y,z \\in M$, \n\\begin{enumerate}\n\t\\item $d(x,y)=0$ si et seulement si $x=y$,\n\t\\item $d(x,z)\\leq \\min\\{d(x,y),d(y,z)\\}$.\n\\end{enumerate}  \n\nUne distance ultram\\'etrique induit la $C$-relation \n$${ C}(x,y,z) \\Leftrightarrow d(x,y)=d(x,z)>d(y,z).$$\nDonc un groupe ab\u00e9lien ou un corps valu\u00e9 porte en particulier la $C$-relation canonique en inversant l'in\\'egalit\\'e ci-dessus:\n$${ C}(x,y,z) \\Leftrightarrow v(x-y)=v(x-z)<v(y-z).$$ \nUne valuation triviale, c'est-\u00e0-dire  prenant au plus deux valeurs, induit la\n$C$-relation triviale: ${ C}(x,y,z) \\Leftrightarrow x\\neq y=z$.\n\n\n\n\\begin{defn}On dit qu'une expansion $\\mathcal{U}=(U,{ C}, \\dots)$ d'une $C$-relation est $C$-minimale si, pour tout $\\mathcal{U}'=(U',{ C}, \\dots)$, \u00e9l\u00e9mentairement \u00e9quivalente \u00e0 $\\mathcal{U}$, tout sous-ensemble d\u00e9finissable (avec param\u00e8tres) de $U'$, est d\u00e9finissable sans quantificateur en n'utilisant que la relation ternaire $C$.\n\\end{defn}\n \n\\begin{remarque} En pr\u00e9sence d'une distance ultram\u00e9trique, la $C$-minimalit\u00e9 se traduit par le fait que tout sous-ensemble d\u00e9finissable  (de la structure elle-m\u00eame) est une combinaison bool\u00e9enne de boules \u00abouvertes\u00bb ou \u00abferm\u00e9es\u00bb.\n\\end{remarque}\n\n\n\n\n\n \n  \n   La $C$-minimalit\u00e9 appara\u00eet comme une g\u00e9n\u00e9ralisation commune de la minimalit\u00e9 forte et de l'$o$-minimalit\u00e9. Par exemple, une structure munie de la $C$-relation triviale est $C$-minimale si et seulement si elle est \nfortement minimale. Plus g\u00e9n\u00e9ralement ces trois notions sont \u00e9troitement li\u00e9es. Notons \u00e9galement que les th\u00e9ories $C$-minimales sont d\u00e9pendantes, i.e, elles ont la propri\u00e9t\u00e9  NIP, comme les th\u00e9ories fortement minimales, $o$-minimales et $P$-minimales (on pourra consulter \\cite{Haskell1997} pour les d\u00e9finitions des th\u00e9ories $o$-minimales, $C$-minimales et $P$-minimales, dont  le th\u00e9or\u00e8me 4.2.6 et la discussion qui le pr\u00e9c\u00e8de pour le r\u00e9sultat concernant NIP). Noter que si une structure est  $o$-minimale, alors il va de m\u00eame pour sa th\u00e9orie compl\u00e8te, i.e. toute structure qui lui est \u00e9l\u00e9mentairement \u00e9quivalente est  $o$-minimale. Cette derni\u00e8re propri\u00e9t\u00e9 est fausse pour les structures fortement minimales et par  voie de cons\u00e9quence pour les structures $C$-minimales.\n\nLe groupe additif de $\\mathbb{Q}_{ p}$, muni de sa valuation  $p$-adique,  est $C$-minimal mais le corps $\\mathbb{Q}_{p}$ ne l'est pas, par exemple\nl'ensemble des carr\u00e9s n'y est pas d\u00e9finissable sans quantificateur dans le langage des corps valu\u00e9s, donc  dans le langage des corps \u00e9quip\u00e9s\nde la $C$-relation non plus. De fait, un corps valu\u00e9 est $C$-minimal si et seulement s'il est alg\u00e9briquement clos (cf. \\cite{Haskell1994}).\n\nDans cette \\'etude, on reprend les notations et les d\\'efinitions (cf. Section 2 pour un rappel) figurant dans nos articles \\cite{Onay2017} et \\cite{Onay2018a}. Soit $K$ un corps et $\\vfi$ un endomorphisme de $K$. On pose $R:=K[t;\\vfi]$, $K$-alg\\`ebre engendr\\'e par $t$ selon la loi de commutativit\\'e \n$$at=ta^{\\vfi}$$ pour tout $a\\in K$. \nDans cette article, on  caract\u00e9rise les $R$-modules valu\u00e9s  $C$-minimaux.\n\n\n\n\nNos r\u00e9sultats principaux  se r\u00e9sument dans les trois th\u00e9or\u00e8mes \nci-dessous. \n\n\\begin{thm}\\label{thmfrtmin}\nUn $R$-module trivialement valu\u00e9 infini $M$ est $C$-minimal (i.e. fortement minimal dans ce cas, justifiant la notation $M$, \u00e0 la place de $(M,v)$) si et seulement si l'une des conditions ci-dessous est v\u00e9rifi\u00e9e:\n\\begin{enumerate}[$1.$]\n\\item $M$ est divisible et, pour tout $r \\in R\\setminus\\{0\\}$, $\\text{ann}_M(r)$ est fini\n\\item l'id\u00e9al annulateur de $M$, $\\{q \\in R \\;\\: | \\:\\: x.q=0, \\;\\: \\forall x \\in M\\}$, est engendr\u00e9 par un polyn\u00f4me irr\\'eductible $r$, $R\/rR$ est un corps et \n$M$ est un $(R\/rR)$-espace vectoriel. \n\\end{enumerate}\n\\end{thm}\n\n\n\\begin{thm}\\label{thmCmintrivi}\nSoit $(M,v)$ un module $K$-trivialement valu\u00e9 infini et non-trivialement valu\u00e9.  Alors\n$(M,v)$ est $C$-minimal si et seulement si $v(M)$ est $o$-minimal dans le langage $L'_V$, pour tout $r \\in R\\setminus\\{0\\}$, $\\text{ann}_M(r)$ est fini, $(M,v)$ est henselien et si l'une des conditions ci-dessous est v\u00e9rifi\u00e9e;\n\\begin{enumerate}[$1.$]\n\\item $(M,v)$ est divisible,\n\\item  $M_{>\\theta}.t$ contient une boule d'indice fini et $M=M_{tor}\\oplus M_{>\\theta}$, o\u00f9 $M_{tor}$ est divisible et fortement minimal s'il n'est pas fini.  \n\\end{enumerate}\n\\end{thm}\n\n\\begin{thm}\\label{thmKnontrivi}\nSoit $(M,v)$ un module  valu\u00e9  non $K$-trivialement. Alors\nil est $C$-minimal si et seulement s'il est affinement maximal, r\u00e9siduellement divisible et tel que $v(M)$ soit une $L_V$-structure $o$-minimale. \n\\end{thm}\n\n\n\nCet article est organiz\\'ee comme suit. La section 2 rappelle les d\\'efinitions introduites dans \\cite{Onay2017} et \\cite{Onay2018a}. La section 3,5,6 tra\\^itent respectivement les cas d'une valuation triviale, d'une valuation non-triviale et $K$-triviale et d'une valuation non $K$-triviale. La section 4\ns'agit d'une interlude des r\\'esultats g\\'en\\'eraux sur la $C$-minimialit\\'e utilis\\'es dans les sections 5 et 6.\n\\section{Pr\\'eliminaires}\n\nSoit $K$ un corps infini, $\\vfi$ un endomorphisme\nde $K$, et $R:=K[t,\\vfi]$. On renvoi le lecteur \\`a la section 2 de \\cite{Onay2017} pour les g\\'en\\'eralit\\'es sur l'anneau $R$ et les $R$-modules.\n\nRappelons les faits suivants.\n\nOn appelle polyn\\^ome un \\'el\\'ement de  $R$ et donc un mon\\^ome  un term de la forme $t^na$, o\\`u $a\\in K$ et  $n\\in \\mathbb{N}$.  Cet anneau est euclidien \\`a droite, en cons\u00e9quence tout $r \\in R$ s'\\'ecrit comme \n$$r=t^ns_1\\ldots s_1,$$ o\\`u les $s_i$ sont irr\\'educibles et s\\'eparables (i.e., non divible par $t$). Il s'en suit l'existence du $\\text{ppcm}(r,q)$ pour $r,q \\in R$. Il vient aussi qu'un $R$-module est divisible si et seulement \ns'il l'est par $t$ et par tout $s$ s\\'eparable. \nPar un $R$-module, on entendra un $R$-module \\`a droite.\nPar $ann_M(r)$ $(r\\in R)$ on d\\'esigne l'ensemble des \\'el\\'ements annul\\'es par $r$ dans un $R$-module $M$, i.e., l'ensemble \n$$\\{x\\in M \\;\\: | \\:\\: x.r=0\\}.$$\n La cl\\^oture divisible d'un sous module $A$ dans un $R$-module divisible $M$, est un sous-module $A' \\supseteq A$ tel que pour tout $a \\in A'$ il existe un $r\\in R$, non nul avec $a.r \\in A$. Les cl\\^otures divisibles de $A$ sont isomorphes au-dessus de $A$ en tant que $R$-modules (cf. Lemma 2.6 (4) \\cite{Onay2017} et la discussion suivant la preuve).\n\n  Une $K$-cha\\^ine est un ordre lin\\'eaire $\\Delta$ avec un maximum $\\infty$, muni d'une action de $K$ satisfaisant, pour tout $a,b \\in K^{\\times}$ et $\\delta, \\gamma \\in \\Delta\\setminus\\{\\infty\\}$, les axiomes suivantes:\n\n  \\begin{enumerate}\n  \t\\item $\\gamma > \\delta \\to \\gamma \\cdot a> \\delta \\cdot a$,\n  \t\\item $\\gamma \\cdot ab = (\\gamma \\cdot a) \\cdot b$,\n  \t\\item $\\gamma \\cdot a > \\gamma \\to \\delta \n  \t\\cdot a > \\delta$,\n  \t\\item $\\gamma \\cdot (a \\pm b)\\geq \n  \t\\min\\{\\gamma\\cdot a, \\gamma \\cdot b\\}$,\n  \t\\item $\\gamma \\cdot 0= \\infty \\; ;\n  \t\\gamma \\cdot 1 =\\gamma\\; ; \\infty\\cdot b=\\infty$.\n  \\end{enumerate}\n  \n Cet action induit une valuation $v_K$ sur $K$ (cf. \\cite{Onay2018a} Proposition 2.4). \n \\begin{defn}\n \tOn dit que cette action est $K$-triviale si $v_K$ est la valuation triviale, sinon on dit qu'elle est $K$-non-triviale. \n \\end{defn}\n \n \\begin{defn}[$R$-chains]\\label{rchains}\n \tUne $R$-cha\\^ine  $(\\Delta, <, \\infty, \\cdot r_{r \\in R})$ est une  $K$-cha\\^ine telle que\n \t\\begin{enumerate}\n \t\t\\item $\\cdot t$ is strictement croissant $\\Delta \\setminus \\{\\infty\\}$, et $\\infty \\cdot t=\\infty$,\n \t\t\\item $\\gamma\\cdot ta=(\\gamma \\cdot t)\\cdot a$ et $\\gamma\\cdot t^na = (\\gamma \\cdot t^{n-1})\\cdot ta$ pour tout $\\gamma \\in \\Delta$, et $a\\in K$,\n \t\t\\item $\\gamma \\cdot r = \\min_{i} \\{\\gamma \\cdot \\mathbf{m}_i\\}$, pour tout $r \\in R$ non nul, o\\`u\n \t\tles $\\mathbf{m}_i$ sont les mon\\^omes de $r$ et pour tout $\\gamma \\in \\Delta$,\n \t\t\\item $\\gamma \\cdot \\mathbf{m}_1 \\leq \\gamma \\cdot \\mathbf{m}_2 \\to \\left( \n \t\t\\forall \\delta \\;\\: (\\delta <\\gamma \\to \\delta \\cdot \n \t\t\\mathbf{m}_1 < \\delta \\cdot \\mathbf{m}_2)\\right)$ pout tout mon\\^omes \n \t\t$\\mathbf{m}_1, \\mathbf{m}_2 \\in R$ tels que $0\\leq\\deg(\\mathbf{m}_2)<\\deg(\\mathbf{m}_1)$,\n \t\tet pour tout $\\gamma\n \t\t\\neq \\infty$.\n \t\\end{enumerate}\n \\end{defn}\n \n\n \\begin{defn} Un $R$-module valu\\'e est un $R$-module, qui est de plus un groupe ab\\'elien valu\\'e \n \t$(M,\\Delta,v)$ (en particulier $v:M\\to \\Delta$ est surjective),  o\\`u $\\Delta$ est une $R$-cha\\^ine\n \ttelle que pour tout $x\\in M$, l'on ait\n \t\\begin{enumerate}\n \t\t\\item $v(x.a)=v(x)\\cdot a$, for all $a \\in K$,\n \t\t\\item  $v(x.t)=v(x)\\cdot t.$\n \t\\end{enumerate}\nUn $R$-module valu\\'e est  dit $K$-trivialement  valu\\'e (respectivement $K$-non-trivialement valu\\'e) si l'action de $K$ sur $\\Delta$ est $K$-triviale (respectivement $K$-non-triviale), i.e., $v(x.a)=v(x)$ pour tout $x\\in M$ et $a\\in K$. Enfin, $(M,\\Delta,v)$ est dit trivialement valu\\'e si $\\vert v(M) \\vert \\leq 2$.  \n \\end{defn}\n \nIl est clair que la d\\'efinition ci-dessus d'un $R$-module $K$-trivialement valu\\'e co\\\"incide bien avec la celle donn\\'ee dans \\cite{Onay2017} (cf. la d\\'efinition 3.4). En particulier l'application $x \\mapsto x.t$ est injective dans tout $R$-module valu\\'e.\n \nSoit $(M,\\Delta,v)$ un $R$-module valu\\'e. \n\\begin{nota}\n\tOn rappelle que l'on note  $\\theta$ la coupure de $\\Delta$, d\\'efinie par l'axiome (4) en prenant $\\mathbf{m}_1=t$ et $\\mathbf{m}_2=1$. Par cons\\'equent, on note, dans un $R$-module valu\\'e $(M,\\ldots)$,  ind\\'ependemment du fait que cette coupure soit realis\\'ee dans $\\Delta$;\n\t\\begin{itemize}\n\t\t\\item $M_{>\\theta}:=\\{x\\in M \\;\\: | \\:\\: v(x.t)>v(x)\\}$\n\t\t\\item $M_{\\geq \\theta}:=\\{x\\in M \\;\\: | \\:\\: v(x.t)\\geq v(x)\\}$.\n\t\\end{itemize}\n\\end{nota} \n\nPar ailleurs, tout $\\gamma \\in v(M)\\setminus\\{\\theta, \\infty\\}$, on note $M_{\\geq \\gamma}$ (respectivement $M_{>\\gamma}$) la boule ferm\\'ee (respectiement la boule ouverte) centr\\'ee  $0$ et de rayon $\\gamma$.\nNotez  que si la sous-module de torsion $M_{tor}$  est non triviale alors $v(M_{tor})=\\{\\theta, \\infty\\}$ est un sous $R$-module trivialement valu\\'e (cf. Lemma 3.10 (3) \\cite{Onay2017}). \n\n\\begin{nota}\n\tComme dans une $R$-cha\\^ine munie d'une action $K$-triviale,\n\tl'application $\\gamma \\mapsto  \\gamma \\cdot r$  est \\'egale \\`a $\\gamma \\mapsto \\gamma \\cdot t^k$, pour un certain $k \\in \\mathbb{N}$, suivant \\cite{Onay2017}, on notera parfois $\\tau$,  l'application $\\gamma \\mapsto   \\gamma \\cdot t$.  \n\\end{nota}\n \n \n\\begin{defn}[cf. Definition 3.23 \\cite{Onay2017}]\n\tSoit $(M,\\Delta,v)$ un $R$-module $K$-trivialement valu\\'e. On dit qu'il est hens\\'elien si pour tout $s$ s\\'eparable, $x\\mapsto x.s$ est une bijection de $M_{>\\theta}$ (autrement dit, le sous-module $M_{>\\theta}$ est divisible par les polyn\\^omes s\\'eparables).\n\\end{defn} \n\nOn peut prendre l'\\'enonc\\'e ci-dessous comme la d\\'efition d'un $R$-module {\\it r\\'esiduellement divisible et affinement maximal}. \n \n\\begin{fait}[reformulation du  Th\\'eor\\`eme 3.35, \\cite{Onay2018a}] $(M,\\Delta,v)$\n\test r\\'esiduelement divisible et affinement maximal si et seulement si pour tout non z\\'ero $r\\in R$ pour tout $z\\in M$, il y a un $y \\in M$ tel que \n\t\\begin{itemize}\n\t\t\\item $y.r=z$,\n\t\t\\item $v(y.r)=v(y)\\cdot r$.\n\t\\end{itemize}\t\n\\end{fait}\n\n\\begin{defn}\n\tUn \\'el\\'ement $y \\in M$, tel que $v(y.r)=v(y)\\cdot r$, est appell\\'e {\\it r\\'egulier} pour $r$.\n\\end{defn}\n Rappelons que $L_V=\\{<,(\\cdot r)_{r \\in R},\n \\infty\\}$, dit, le langage des $R$-cha\\^ines et $L_V'=L_V\\cup\\{(R_n)_{n \\in \\mathbb{N}} \\}$, o\\`u  les $R_n$ sont des pr\\'edicats unaires telles que,  $M \\models R_n(\\gamma)$ si et seulement si, $\\vert M_{\\geq \\gamma}\/M_{>\\gamma} \\vert\\geq n$.\nPar la suite, on notera $(M,v)$ pour d\\'esigner un $R$-module valu\\'e.\n\\section{Cas fortement minimal}\n\nOn consid\\`ere les $R$-modules trivialement valu\\'es. Notez qu'un $R$-module trivialement valu\\'e est $C$-minimale si et seulement s'il est fortement minimal.\n\\begin{lem}\\label{frtmin} \nSi $M$ est un $R$-module fortement minimal infini alors, pour tout $r \\in R \\setminus \\{0\\}$,\n$${ann}_M(r) \\;\\: \\text{fini} \\;\\: \\Leftrightarrow (M.r \\neq 0) \\Leftrightarrow M.r=M.$$\n\\begin{proof}\nPour tout $r \\in R\\setminus\\{0\\}$, ${ann}_M(r)$ et $M.r$, \u00e9tant d\u00e9finissables, sont finis ou cofinis; de plus, \\'{e}tant des sous-groupes, ils ne peuvent \u00eatre cofinis propres. Si tous les deux sont finis, en consid\u00e9rant l'application\n$M \\to M, x \\mapsto x.r$, qui a comme noyau $ann_M(r)$, on devrait avoir $M$ fini. Donc la seule possibilit\u00e9 restante est celle qui est \u00e9nonc\u00e9e.\n\\end{proof}\n\\end{lem}\n\n\n\\begin{lem}\\label{generateurI}\nSoit $I$ un id\u00e9al bilat\u00e8re non nul de $R$. Si $\\vfi^n=1$ pour un certain $n \\in \\mathbb{N}$, alors il existe $q\\in R$ \u00e0 coefficients dans $Fix(\\vfi)$  tel que $I=t^nqR$. Si $\\vfi$ n'est pas\nd'ordre fini, alors $I$ est de la forme $t^kR$ pour un certain $k \\in \\mathbb{N}$.\n \n\\begin{proof}\nVoir la proposition 2.2.8 dans \\cite{cohn}.\n\\end{proof}\n\\end{lem}\n\n\n\n\n\n\\begin{corr}\nSoit $(M,v)$ un $R$-module   trivialement valu\u00e9, infini et fortement minimal. Soit $I=\\{r \\in R \\;\\: | \\:\\: M.r=0\\}$. Alors, ou bien  $I=\\{0\\}$ et dans ce cas $M$ est divisible et les annulateurs \n$\\text{ann}_M(r)$ sont finis, ou bien $I\\neq \\{0\\}$ et dans ce cas, $I=rR$ pour un $r\\in R$ irr\\'eductible et $M$ est un $R\/rR$-espace vectoriel.\n\\begin{proof}\nLe cas o\u00f9 $I=\\{0\\}$ d\u00e9coule du lemme \\ref{frtmin}. \nSupposons donc $I\\neq \\{0\\}$. Remarquons que $I$ est un id\u00e9al bilat\u00e8re et que,\npar le lemme \\ref{generateurI},  il existe\n$r \\in K_0[t;\\vfi]$, o\\`u $K_0=Fix(\\vfi)$), tel que $I=rR$. Soit $d$ le degr\u00e9 de $r$. Alors pour tout $q$ de degr\u00e9 $<d$ v\u00e9rifiant $r=qs$ on a $M.q \\neq 0$ d'o\u00f9 par \\ref{frtmin}, $M.q=M$. Par cons\u00e9quent $M.r=M.qs=M.s=0$, d'o\u00f9 $s \\in I$. Donc\n$q \\in K$. C'est-\u00e0-dire que $r$ est irr\\'eductible. Par l'existence du ppcm, et par le th\u00e9or\u00e8me de B\u00e9zout, le quotient $R\/rR$ est un corps.\n\\end{proof}\n\\end{corr}\n\\begin{lem}\\label{excorr4.6}\n Si $M$ est un $R$-module fortement minimal, si l'action de $t$ est injective et si $\\vfi$ n'est pas d'ordre fini, alors l'id\u00e9al $I$ ci-dessus est nul.\n\\begin{proof}\nSi $\\vfi$ n'est pas d'ordre fini et si $I\\neq \\{0\\}$, par le lemme \\ref{generateurI}, $I$ est engendr\u00e9 par $t^k$, pour un certain $k \\in \\mathbb{N}_{>0}$. Puisque la multiplication par $t$ est injective ceci n'est pas possible. \n\\end{proof}\n\\end{lem}\n\n\n\n\n{\\bf Preuve du th\u00e9or\u00e8me \\ref{thmfrtmin}}.\nPar ce qui pr\u00e9c\u00e8de il suffit de montrer qu'un $R$-module satisfaisant $1$ ou \n$2$ est fortement minimal. Si $M$  est un $R$-module satisfaisant $1$,  alors,  par la proposition 2.15 \\cite{Onay2017}, toute partie d\u00e9finissable de $M$ est finie ou cofinie. Si $M$ satisfait $2$, alors \nla structure de $R$-module  sur $M$ co\u00efncide avec sa structure de $R\/rR$-espace vectoriel. Or tout espace vectoriel est fortement minimal.\n\n\n\n\\section{Interlude: cons\u00e9quences de la $C$-minimalit\u00e9}\n\nRappelons que dans un groupe (ou module )valu\\'e $G$, on note $G_{\\geq \\gamma}$ (respectivement $G_{>\\gamma}$ la boule ferm\\'ee de centre $0$ et de rayon $\\gamma$ (respectivement la boule ouverte de centre $0$ et de rayon $\\gamma$). On utilisera tr\u00e8s fr\u00e9quemment la proposition ci-dessous.\n\\begin{prop}\\label{c-min_d'indice_fini}\nSi $G$ est un groupe valu\u00e9  ${C}$-minimal alors, pour tout sous-groupe  d\u00e9finissable infini propre $F$ de G, il existe $\\gamma \\in v(G)$, tel que $F$ contient $G_{\\geq \\gamma}$ (ou $G_{> \\gamma}$), et $G_{\\geq \\gamma}$ (ou $G_{> \\gamma}$) est d'indice fini dans $F$.\n\\begin{proof}\nVoir  \\cite{Macpherson1996} ou \\cite{Simonetta2001}  1.6(ii).\n\\end{proof}\n\\end{prop}\n\n\\begin{prop}\nSoit $\\mathbb{G}=(G,v,\\dots)$  un groupe valu\u00e9, porteur de structure additionnelle, et $C$-minimal. Alors \\begin{enumerate}[$1.$]\n\\item pour tout $\\gamma \\in v(G)$, le groupe $G_{\\geq \\gamma}\/G_{>\\gamma}$ \nmuni de la structure induite par $\\mathbb{G}$ est fortement minimal,\n\\item la cha\u00eene $v(G)$ munie de la structure induite par $\\mathbb{G}$ est\n$o$-minimale.\n\\end{enumerate} \n\\begin{proof}\nVoir  \\cite{Macpherson1996}.\n\\end{proof}\n\\end{prop}\n\n\n\\begin{corr} Soit $(M,v)$ un $R$-module  valu\u00e9  $C$-minimal.\n\tNotons $K\/v_K$ le corps r\\'esiduel associ\\'e \\`a la valuation $v_K$ induit par l'action $K$ sur $v(M)$. Alors,\n\\begin{enumerate}\n\\item pour tout  $\\gamma \\in v(M)$, $M_{\\geq \\gamma}\/M_{>\\gamma}$ est un $(K\/v_K)$-espace vectoriel fortement minimal et,\n\\item la $R$-cha\u00eene $v(M)$ est $o$-minimale.\n\\end{enumerate}\n\\end{corr}\n\n\n\\begin{remarque}\nDans le point $1$ ci-dessus, si $(M,v)$ est $K$-trivialement valu\u00e9, i.e. si $v_K$ est triviale, alors $K\/v_K=K$ et donc tout   $M_{\\geq \\gamma}\/M_{>\\gamma}$ est un $K$ espace-vectoriel. \\end{remarque}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Cas d'une valuation non triviale et $K$-triviale}\nUne des implications du th\u00e9or\u00e8me \\ref{thmCmintrivi}  est contenue  dans la proposition suivante.\n\n\\begin{prop}\\label{divhens->c-min}\nSi $(M,v)$  est un module $K$-trivialement valu\u00e9 divisible henselien, si pour tout $q \\in R\\setminus\\{0\\}$, $\\text{ann}_M(q)$ est fini  et si\n$v(M)$ est ${o}$-minimal dans le langage $L'_V$, alors $(M,v)$ est ${ C}$-minimal.\n\\begin{proof}\nNoter qu'un tel $(M,v)$ est r\\'esiduellement divisible et affinement maximal par le corollaire 3.29, \\cite{Onay2017}.  Nous renvoyons donc \u00e0 la preuve de la proposition \\ref{cmin}  \ndont la pr\u00e9sente proposition n'est qu'un cas particulier. \n\\end{proof}\n\\end{prop}\n\n\n\n\n\n\\begin{lem}\\label{puiseux-stayla-c_min}\nSoit $(M,v)$ un module $K$-trivialement valu\u00e9, tel que \n\\begin{enumerate}\n\\item $M_{tor}$ est fini et $M=M_{tor} \\oplus M_{>\\theta}$, et\n\\item $M_{>\\theta}$ est divisible et $v(M)$ est $o$-minimal.\n\\end{enumerate}\nAlors $(M,v)$ est $C$-minimal.\n\\begin{proof} Si $M_{tor}=0$ alors, $M=M_{>\\theta}$ est  divisible henselien donc $C$-minimal par la proposition pr\u00e9c\u00e9dante. Sinon, soit $N'$ une cl\u00f4ture divisible de $M_{tor}$. Posons $N:=N'\\oplus M_{>\\theta}$ et on \u00e9tend $v$ \u00e0 $N$ en posant $v(x_{tor} + x_{>\\theta})=\\theta$ si $x_{tor}$ est non nul. D'o\u00f9 $(N,v)$ est divisible henselien et $v(N)=v(M)$, donc $(N,v)$ est $C$-minimal par la proposition pr\u00e9c\u00e9dente. Alors $M=\\bigcup_{a \\in M_{tor}} M_{>\\theta} + a$, est une union finie de boules de $N$, par cons\u00e9quent il est $C$-minimal.\n\\end{proof}\n\\end{lem}\n\n\nUne cons\u00e9quence imm\u00e9diate des deux r\u00e9sultats pr\u00e9c\u00e9dents est:\n\\begin{corr}\\label{puiseuxCmin}\nL'anneau de valuation d'un corps de caract\u00e9ristique $p>0$, parfait, muni d'une valuation henselienne,  de corps r\u00e9siduel fini ou p-clos, et de groupe de valuation divisible, est {C}-minimal en tant que $\\mathbb{F}_p[t;x\\to x^p]$-module valu\u00e9.\n\\begin{proof}\nUn groupe ab\u00e9lien ordonn\u00e9 divisible est $o$-minimal, donc a fortiori sa restriction \u00e0 la structure de  $\\mathbb{F}_p[t;x\\to x^p]$-cha\u00eene. \nLe corollaire d\u00e9coule donc du \n lemme 3.24 \\cite{Onay2017}, avec la proposition \\ref{divhens->c-min} si le corps r\u00e9siduel est $p$-clos et avec le lemme  \\ref{puiseux-stayla-c_min}\nsi le corps r\u00e9siduel est fini.\n\\end{proof}\n\\end{corr}\n\n\\begin{exemple}\nSoit $R=\\mathbb{F}_p[t;x\\mapsto x^p]$.\nL'anneau des s\u00e9ries de Puiseux sur $\\mathbb{F}_p$ ou sur un corps $p$-clos est $C$-minimal en tant que $R$-module valu\u00e9, mais le corps des s\u00e9ries de Puiseux ne l'est pas (en tant que $R$-module). \n\\end{exemple}\n\n\nLe lemme ci-dessous, avec le lemme pr\u00e9c\u00e9dent, montreront que la condition $2.$ cit\u00e9 dans le th\u00e9or\u00e8me \\ref{thmCmintrivi}, implique la $C$-minimalit\u00e9. \n\n\n\\begin{lem}\nSoit $(M,v)$  un module $K$-trivialement valu\u00e9 henselien et tel que\n\\begin{enumerate}\n\\item $M_{tor}$ est fini  et $M=M_{tor} \\oplus M_{>\\theta}$,\n\\item $M_{>\\theta}.t$ contient une boule  de la forme $M_{\\geq \\gamma}$ ou $M_{>\\gamma}$,  qui est d'indice fini dans $M_{>\\theta}.t$   et $v(M_{>\\theta})$ est $o$-minimal.\n\\end{enumerate}\nAlors $(M,v)$ est $C$-minimal.\n\\begin{proof}\nSi $M_{>\\theta}$ est divisible, alors l'assertion suit de lemme ci-dessus. \n\nMontrons d'abord que la valuation sur $M_{>\\theta}$  d\u00e9termine une et une seule valuation sur chacune de ses cl\u00f4tures divisibles (toutes sont d\\'ej\\`a isomorphes en tant que $R$-modules). \nSoit $N$ une cl\u00f4ture divisible   de $M_{>\\theta}$. Alors, puisque $M_{>\\theta}$ est divisible par tous les polyn\u00f4mes s\u00e9parables, \n $x \\in N$ si et seulement si  $x.t^n \\in M_{>\\theta}$ pour un certain $n \\in N$.  Soit alors $x \\in N\\setminus M_{>\\theta}$ et $n$ minimal tel que $x.t^n \\in M_{>\\theta}$. Soit  $\\gamma=v(x.t^n)$. On d\u00e9finit $\\bar{v}(x)$ comme le couple $(\\gamma, n)$. Si $y \\in N\\setminus M_{>\\theta}$ est tel que $\\bar{v}(y)=(\\delta, k)$, on d\u00e9cr\u00e8te que \n$\\bar{v}(x) < \\bar{v}(y)$ si et seulement si $\\tau^k(\\gamma) < \\tau^n(\\delta)$. Il est imm\u00e9diat de v\u00e9rifier que $(N,\\bar{v})$ est un module $K$-trivialement valu\u00e9, et que $\\bar{v}$ prolonge $v$. \n\nMontrons maintenant que $M_{>\\theta}$ s'identifie \u00e0 une union finie de boules de $(N,\\bar{v})$. Soit $B$ la boule de $M$,  d'indice fini dans $M_{>\\theta}.t$, dont l'existence est donn\u00e9e par l'hypoth\u00e8se du corollaire. Montrons que  $B$ est en fait une boule de $N$.  Soit  $\\gamma$ le rayon de $B$, i.e. $B=M_{>\\gamma}$ ou $B=M_{\\geq \\gamma}$. Soit $x \\in N \\setminus M_{>\\theta}$ , avec $v(x) > \\gamma$, et soit $k\\geq 1$ minimal tel que $\\overline{x}.t^k \\in  M_{>\\theta}$. On a, $v(x.t^k) > \n\\tau^{k}(\\gamma)> \\gamma$. Or $M_{>\\theta}.t \\supset B$ implique $x.t^{k-1} \\in M_{>\\theta}$. Contradiction. D'o\u00f9 $B=N_{\\geq \\gamma}$ ou\n$B=N_{>\\gamma}$. En \u00e9crivant $M_{>\\theta}.t= \\bigcup_i B +a_i$, on a $M_{>\\theta}=B.t^{-1} + a_.t^{-1}$, o\u00f9 $B.t^{-1}$ est n\u00e9cessairement la boule ouverte ou ferm\u00e9e de rayon $\\tau^{-1}(\\gamma)$ et $a_i.t^{-1}$ est l'unique $b_i \\in N$ tel que $b_i.t=a_i$. Par cons\u00e9quent, $M_{>\\theta}$   est une union finie de boules de $N$ donc $M$ est une union finie de boules de $N\\oplus M_{tor}$ et il suffit de montrer que $N\\oplus M_{tor}$ est $C$-minimal.  \nPuisque $(N, \\overline{v})$ est divisible henselien sans-torsion, le lemme \\ref{puiseux-stayla-c_min} dit  qu'il suffit de montrer que \n$\\overline{v}(N)$ est $o$-minimal.\n \nMontrons  d'abord que  $\\overline{v}(N)$\nest la cl\u00f4ture de $v(B)$ par $\\tau^{-1}$.\n Soit  $x \\in M_{>\\theta}$, tel que $x.t \\notin B$. On a $v(x.t-a_i)>v(x.t)=v(a_i)$ pour un certain $i \\in \\{1,\\dots, n\\}$. Dans ce cas , $v(x.t^2)>v(x.t)$ et donc pour tout $i$, $v(x.t^2-a_i)=v(a_i)$. Alors $x.t^2 \\in B$ n\u00e9cessairement. Par ailleurs, si $x.t \\in B$ de m\u00eame pour  $x.t^2$. Donc $N$ est aussi la cl\u00f4ture divisible de $B$. \nD'o\u00f9 $\\overline{v}(N)$\nest la cl\u00f4ture de $v(B)$ par $\\tau^{-1}$,  $\\overline{v}(N)=\\bigcup_{i \\in \\mathbb{N} } \\tau^{-i}v(B)$, et chaque   $\\tau^{-i}v(B)$ est isomorphe \u00e0 $v(B)$ donc ils sont  tous $o$-minimaux. \n\nPuisque $v(M)$ est $o$-minimal $v(M)\\setminus\\{\\theta,\\infty\\}$ est dense ou discret: Soit  $DE$ l'ensemble des points  ayant un voisinage dense et $DI$ l'ensemble des points ayant un pr\u00e9d\u00e9cesseur et un successeur; par $o$-minimalit\u00e9, $v(M)$ se partionne comme $DE\\cup DI\\cup F$, avec $F$ fini, plus pr\u00e9cisement $DE$ et $DI$ sont des unions finis d'intervalles s\u00e9par\u00e9s par au moins un point de $F$; par ce que $\\tau$ est strictement croissant, il pr\u00e9serve chacun des trois ensembles, $DE$, $DI$ et $F$, et parce qu'il n'admet aucun point fixe sur $v(M)\\setminus\\{\\theta,\\infty\\}$,  $F\\setminus\\{\\theta,\\infty\\}=\\emptyset$, par cons\u00e9quent $v(M)\\setminus\\{\\theta,\\infty\\}=DE$ ou $v(M)\\setminus\\{\\theta,\\infty\\}=DI$.\n\nMaintenant, par \\cite{Pillay1987}, si $v(M)$ est discret alors $\\tau$ est une translation sur $v(M)$ et donc $\\overline{v}(N)$ est discret et $\\tau$ reste une translation. Sinon si $v(M)$ est dense, par construction $\\overline{v}(N)$ est dense. Dans le \ndeux cas, la $\\overline{v}(N)$ est $o$-minimal par le corollaire 1.16 de \\cite{Maalouf2010a}.\n\\end{proof}\n\\end{lem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nLes lemmes qui suivent   \u00e9tablissent des r\u00e9ciproques  aux r\u00e9sultat ci-dessus et ainsi compl\u00e8tent la preuve du th\u00e9or\u00e8me \\ref{thmCmintrivi}.\n\\begin{lem}\\label{c-min2div}\n Soit $(M,v)$  un module $K$-trivialement valu\u00e9, ${ C}$-minimal, infini.\nS'il existe  $a \\in M$ tel que $v(a)< \\theta$   alors M est divisible.\n\\begin{proof}\nSoit $q\\in R$, non nul. S'il existe un $a \\in M$ comme ci-dessus, alors $v(M.q)$\nn'a pas de premier \u00e9l\u00e9ment. Si $M.q \\neq M$ alors par la proposition \\ref{c-min_d'indice_fini} il contient un sous-groupe $H$ de la forme\n$M_{\\geq \\gamma}$ ou de la forme $M_{>\\gamma}$ qui est d'indice fini dans $M.q$. Alors $M.q$ est la r\u00e9union disjointe finie des $H + a_i$ et donc $v(M.q)$ a un  premier \u00e9l\u00e9ment. Contradiction.  \n\\end{proof}\n\\end{lem}\n\n\\begin{lem}\\label{anninfini_implique_Cmin}\nSoit $(M,v)$ un module non trivialement valu\u00e9 qui est $K$-trivialement valu\u00e9 et $C$-minimal. Alors, pour tout $r \\in R\\setminus\\{0\\}$, $\\text{ann}_M(r)$   est fini.\n\\begin{proof}\nSi $\\text{ann}_M(r)$,  est une partie propre infini de $M$, alors par la proposition \n\\ref{c-min_d'indice_fini} il existe $\\gamma \\in \\Delta$ tel que $\\text{ann}_M(r)$  contient une boule de la forme $M_{\\geq \\gamma}$ ou $M_{>\\gamma}$ qui est d'indice fini dans $\\text{ann}_M(r)$. Comme $v(\\text{ann}_M(r)) = \\{\\theta\\}$, on obtient  $\\gamma=\\theta$; dans ce cas l\u00e0, on a n\u00e9cessairement $M_{>\\theta}={0}$ et donc $\\text{ann}_M(r)=M_{\\geq\\theta}=M_{tor}$. Cela implique en particulier $\\infty$ est l'unique \u00e9l\u00e9ment qui est strictement$>\\theta$. Puisque $(M,v)$ est non trivialement valu\u00e9, il existe donc des \u00e9l\u00e9ments de valuation $<\\theta$. Par le lemme ci-dessus $M$ est alors divisible; d'o\u00f9 $M_{tor}$ est divisible.  Or ceci contredit le fait que $M_{tor}$ est annul\u00e9 par un seul \u00e9l\u00e9ment.\n\\end{proof}\n\\end{lem}\n\n\\begin{corr}\nSoit $(M,v)$ non trivialement valu\u00e9, $K$-trivialement valu\u00e9 et $C$-minimal. Si $M_{tor}$ est infini alors $M_{tor}$ est divisible et fortement minimal.\n\\end{corr}\n\\begin{proof} $M_{tor}$ se plonge dans  le $R$-module $M_{\\geq \\theta}\/M_{>\\theta}$ qui est fortement minimal, on a la divisibilit\u00e9 de $M_{\\geq \\theta}\/M_{>\\theta}$ par le lemme ci-dessus et par le lemme \\ref{frtmin}, d'o\u00f9 la divisibilit\u00e9 de $M_{tor}$. Alors par la proposition 2.15 dans \\cite{Onay2017}, tout sous-ensemble d\u00e9finissable de $M_{tor}$ est une combinaison bool\u00e9enne  d'ensembles du type $\\text{ann}_M(r)$, avec $r \\in R\\setminus\\{0\\}$. D'o\u00f9, par le th\u00e9or\u00e8me \\ref{frtmin} et par le fait\n\tque chaque annulateur est fini, la forte minimalit\u00e9 de $M_{tor}.$ \n\\end{proof}\n\n\\begin{lem}\nSoit $(M,v)$  un module $K$-trivialement valu\u00e9 ${C}$-minimal. Alors $(M,v)$ est henselien.\n\\begin{proof}\nOn va consid\u00e9rer $2$ cas:\\\\\n{\\bf a.} {\\it $v(M_{>\\theta})$ n'a pas de premier \u00e9l\u00e9ment.} \nPar la proposition\n\\ref{c-min_d'indice_fini}, si  $M_{>\\theta}.r$ ($r \\in R\\setminus 0$) n'est pas \u00e9gal \u00e0 $M_{>\\theta}$, alors il contiendrait \nune boule ouverte ou ferm\u00e9e d'indice fini. \nEn particulier, ou bien $v(M_{>\\theta }.r)$ aurait un plus petit \u00e9l\u00e9ment ou bien pour un certain $\\gamma>\\theta$ on aurait \n$M_{>\\theta }.r=M_{>\\gamma}$. Ces deux cas sont impossibles car si $r$ est s\u00e9parable et $x \\in M_{>\\theta}$, on a  $v(x.r)=v(x)$.\n\n\\noindent\n{\\bf b.} {\\it $v(M_{>\\theta})$ a un plus petit \u00e9l\u00e9ment.} \nSoit $\\gamma_{0}$ ce plus petit \u00e9l\u00e9ment $> \\theta$. Soit $s \\in R_{sep}$. \n\u00c9crivons $s$ comme $tq+ a$ avec $a \\in K$. Encore par le lemme \n\\ref{c-min_d'indice_fini}, si  $M_{>\\theta}.s \\neq M_{>\\theta}$ alors il contient\nun certain $M_{>\\gamma}$ ou $M_{\\geq \\gamma}$ qui est d'indice fini dans $M_{>\\theta}.s$. Supposons que c'est $M_{> \\gamma}$, l'autre cas se traite de m\u00eame. Puisque $M_{>\\gamma}$ est d'indice fini dans $M_{>\\theta}.s$, il existe  $\\gamma_1, \\dots \\gamma_k$ tel que $\\gamma_0 < \\gamma_1 < \\dots < \\gamma_k= \\gamma$ avec $\\gamma_{i+1}$  le successeur de $\\gamma_{i}$ dans $v(M)$. Soit $x$ de valuation $\\gamma_0$, alors \n$x_0:=(x - x.a^{-1}(tq + a))=(x - x.a^{-1}s)$ est de valuation $ \\geq \\gamma_1 >\\gamma_{0}$. Ainsi\non d\u00e9finit $x_i:=(x_{i-1} - x_{i-1}.a^{-1}s)$, avec $v(x_i) > \\gamma_i$, pour $1\\leq i \\leq k$. Donc $v(x_{k})>\\gamma$. Par cons\u00e9quent $x_k$ est divisible\npar $s$, et donc $x$ aussi. On a montr\u00e9 que $M_{>\\theta}$ est divisible\npar les polyn\u00f4mes s\u00e9parables, i.e. $(M,v)$ est henselien.\n\\end{proof}\n\\end{lem}\n\n\n\n\n\n\n\n\n\n\nPar ce qui pr\u00e9c\u00e8de, on peut supposer que $M=M_{\\geq \\theta}$ et que $M_{tor}$ est fini. Il nous  reste alors \u00e0 montrer le r\u00e9sultat suivant, avec lequel s'ach\u00e8ve la preuve du th\u00e9or\u00e8me \\ref{thmCmintrivi}.\n\n\\begin{lem}\nSoit $(M,v)$ un module $K$-trivialement valu\u00e9 $C$-minimal non divisible et tel que \n$M_{\\geq \\theta}=M$ et $M_{tor}$ est fini. Alors $M=M_{tor} \\oplus M_{>\\theta}$.\n\\begin{proof}\nPuisque $M_{tor}$ est fini la multiplication par $t$ induit une bijection\nde $M_{tor}$. Donc $M_{tor}=\\text{ann}_M(t^k-1)$ pour un certain $k \\in \\mathbb{N}\\setminus \\{0\\}$. Par cons\u00e9quent, $M_{tor}$ n'est pas divisible donc $M$ non plus. Donc $M.(t^k-1)$ contient une boule (ouverte ou ferm\u00e9e) qui y est d'indice fini, d'apr\u00e8s le lemme \\ref{c-min_d'indice_fini}. En particulier le quotient \n$M.(t^k-1)\/(M_{>\\theta} \\cap M.(t^k - \n1))$ est fini; de plus il est sans torsion. Or, puisque $R$ est infini, le \nseul module sans torsion fini est le module $0$. Cela implique  que \n$M.(t^k-1) \\subset M_{>\\theta}$. \n\nMontrons  l'inclusion r\u00e9ciproque $M_{>\\theta}\\subset M.(t^k-1)$: $(M,v)$ est henselien par le lemme ci-dessus; en particulier $M_{>\\theta}$ est divisible par le polyn\u00f4me $t^k-1$; cela implique  que la suite  ci-dessous est exacte (et clairement scind\u00e9e):\n$$0 \\longrightarrow M_{tor} \\longrightarrow M \\xrightarrow{.(t^k-1)} (M_{>\\theta})  \\longrightarrow 0.$$\nD'o\u00f9 $M_{>\\theta}$ est facteur direct dans $M$.\n\\end{proof}\n\\end{lem}\n\n\n\n\\section{Cas d'une valuation non $K$-triviale}\nLes propositions ci-dessous \u00e9tablissent la preuve du th\u00e9or\u00e8me \\ref{thmKnontrivi}. Rappelons que $(M,v)$ est dit r\\'esiduellement divisible, si pour tout non zero $r\\in R$, non zero $z\\in M$ il existe $y\\in M$, tel que \n$$v(z-y.r)>v(y.r)=v(y)\\cdot r.$$\n\nNotons que l'\\'egalit\\'e dans l'expression ci-dessus  peut ne peut \\^etre toujours v\\'erifi\\'ee (on peut penser au cas o\\`u $y$ est non zero et annul\\'e par $r$.)\n\n\n\\begin{prop}\\label{cmin}\nUn module  valu\u00e9 $(M,v)$ affinement maximal, r\u00e9siduellement divisible, ayant sa cha\u00eene $v(M)$ $o$-minimale et tel que, pour chaque $r \\in R\\setminus\\{0\\}$, $\\text{ann}_M(r)$ est fini, est $C$-minimal.\n\\begin{proof}\nPar \\cite{Onay2018a}, Th\\'eor\\`eme 4.6, on sait que toute formule \u00e0 une seule variable libre $x$, avec param\u00e8tres, est \u00e9quivalente \u00e0 une formule de la forme $$\\phi(x) \\wedge \\psi(v(t_1(x)),\\dots,v(t_k(x)), \\overline{\\gamma}),$$ o\u00f9 \n$\\phi$ est une formule sans quantificateur avec param\u00e8tres dans le langage des $R$-modules, $\\psi$ est\nune formule sans param\u00e8tres de $L_V'$,  les $t_i$ sont des termes avec param\u00e8tres dans le langage des $R$-modules  et $\\overline{\\gamma}$ est un $n$-uplet d'\u00e9l\u00e9ments de $v(M)$.\nEn particulier, $\\phi$ est une combinaison bool\u00e9enne de formules de la forme $x.r=b$. Puisque $\\text{ann}_M(r)$ est fini, elle d\u00e9finit un ensemble fini ou cofini. On se ram\u00e8ne donc \u00e0 montrer que $\\psi(v(t_i(x))_{i})$ d\u00e9crit une combinaison bool\u00e9enne de boules ouvertes ou ferm\u00e9es.\nOn se contentera de montrer que l'on peut remplacer les termes $v(t_i(x))$ par des termes\nde la forme $\\tau^k(v(x-c_i))$ o\u00f9 $k \\in \\mathbb{N}$ et $c_i \\in M$. Ainsi $\\psi$ sera de la forme $\\psi'(v(x-c_1),\\dots,v(x-c_n))$. \nLa $C$-minimalit\u00e9 de $(M,v)$ suit alors de l'$o$-minimalit\u00e9 de la $R$-cha\u00eene $v(M)$   par des consid\u00e9rations ultram\u00e9trique g\u00e9n\u00e9rales (comme il a \u00e9t\u00e9 explicit\u00e9 dans la preuve de la proposition 36 de \\cite{Maalouf2010a}). \n\n\\noindent\nLes termes $t_i(x)$ sont de la forme $x.r_i - a_i$ ($r_i \\in R$, $a_i \\in M$). \nPar divisiblit\u00e9 de $M$, il existe $b_i \\in M$ tel que \n$b_i.r_i= a_i$. Ainsi chaque terme $t_i(x)$ est \u00e9gal \u00e0 un terme de la forme $(x-b_i).r_i$. La preuve sera donc achev\u00e9e d\u00e8s que nous aurons prouv\u00e9 l'assertion suivante:\n\n\\noindent\nEtant donn\u00e9 $r \\in R\\setminus\\{0\\}$, il existe un recouvrement fini de $M$ par des ensembles dont chacun, disons $E$, est une combinaison bool\u00e9enne de boules de $M$ et tel que pour un certain param\u00e8tre $d$,  \n$(M,v) \\models \\forall x \\in E, \n           v(x  \\cdot r ) = v(x-d) \\cdot r$. \n\n\\noindent\nEn effet, d'une part par le r\u00e9sultat 3.36 dans \\cite{Onay2018a}, on a \n$$\\forall x \\bigvee_{c \\in ann_M(r)} v(x.r)=v((x-c).r)=v(x-c)\\cdot r.$$\nEt d'autre  part, pour tout $c_0 \\in \\text{ann}_M(r)$, l'ensemble \n$\\{x \\in M \\;\\: | \\:\\: v((x-c_0).r)=v(x-c_0)\\cdot r\\}$ est exactement l'ensemble des \u00e9l\u00e9ments $x$ tel que $x-c_0$ est r\u00e9gulier pour $r$,  et est \u00e9gal, par le m\\^eme r\\'esultat, l'ensemble $\\{x \\;\\: | \\:\\: v(x-c_0)=\\max\\{v(x-c_0-c); c\\in \\text{ann}_M(r) \\}\\}$. Ce dernier ensemble est clairement une combinaison bool\u00e9enne de boules.\n\n\\end{proof}\n\\end{prop}\n\nPour la d\\'emonstration suivante on rappelle que l'ensemble $Saut_M(r)$ (not\\'e $\\jumpm{M}{r}$ dans \\cite{Onay2018a}) est l'ensemble des valeurs $\\gamma$ tel qu'il existe un $x\\in M$ de valuation $\\gamma$ satisfaisant \n$$v(x.r)>v(x)\\cdot r.$$\n\n\\begin{prop}\nSoit $(M,v)$ un module valu\u00e9 non $K$-trivialement  et $C$-minimal. Alors\n$(M,v)$ est affinement maximal et r\u00e9siduellement divisible, et pour tout \n$r\\in R\\setminus \\{0\\}$, \n$\\text{ann}_M(r)$ est fini.\n\\begin{proof}\nPuisque $v_K$ est non triviale sur $K$,  $v(M)$ n'est pas minor\u00e9. Or \n$Saut_M(r)$ est fini pour tout $r \\in R \\setminus \\{0\\}$ donc $\\text{ann}_M(r)$\nne peut pas contenir de boule propre d'indice fini et doit \u00eatre fini. \nDe plus, pour les m\u00eames raisons, $v(M.r)$ n'est pas minor\u00e9, donc $M.r$ \nne peut contenir de boule propre d'indice fini et $M.r=M$. D'o\u00f9 $M$ est divisible. \n\nSupposons que $(M,v)$ n'est pas affinement maximal. Comme $M$ est d\\'ej\\`a divisible, il est r\\'esiduellement divisible. Dans ce cas, par le th\u00e9or\u00e8me 3.35 \\cite{Onay2018a},\n  il existe $r \\in R \\setminus\\{0\\}$ et $y \\in M\\setminus\\{0\\}$ tels que pour tout $x$ v\u00e9rifiant $x.r=y$, $x$ est irr\u00e9gulier pour $r$. Puisque \n$M$ est divisible, l'ensemble $A:=\\{x \\in M \\;\\: | \\:\\: x.r=y\\}$ est non-vide et puisque \n$\\text{ann}_M(r)$ est fini, il s'\u00e9crit comme $A=\\{x_1,\\dots, x_k\\}$. Posons\n$\\gamma=\\max_i\\{v(x_i)\\}$ et prenons $x\\in A$ de valuation $\\gamma$. Alors\n$y \\in M_{>\\gamma \\cdot r} \\setminus \\left(M_{>\\gamma}\\right).r$. On va montrer que ceci conduit \u00e0 une contradiction. Puisque $\\gamma$ est un saut, il est limite inf\u00e9rieure dans $v(M)$  des \u00e9l\u00e9ments $\\delta \\in v(M)\\setminus Saut_M(r)$ par le lemme 2.11 \\cite{Onay2018a} et par la finitude de $Saut_M(r)$; donc il est limite des \u00e9l\u00e9ments $\\delta$ tel que  $v(M_{>\\delta}.r)$ et $v(M_{>\\delta \\cdot r})$ d\u00e9finissent le m\u00eame segment final de $v(M)$. Cela impose que \n $M_{>\\gamma \\cdot r}=\\left(M_{>\\gamma}\\right).r$. Contradiction.\n \\end{proof}\n\\end{prop}\n\nEnfin, on donne une cons\u00e9quence du r\u00e9sultat ci-dessus dans le cas des corps valu\u00e9s: par  le lemme 2.11 \\cite{Onay2018a}, il suit qu'une $R$-cha\u00eene $o$-minimale est dense et pleine et il suit par le corollaire \\cite{Onay2018a} 2.26, qu'une $R$-cha\\^ine pleine et dense est $o$-minimale. De plus,\n  par la remarque 2.29 \\cite{Onay2018a},  un groupe ab\u00e9lien ordonn\u00e9 plein comme $R$-cha\u00eene est divisible. Ensuite, par l'exemple 2.28(2) \\cite{Onay2018a},  la $R$-cha\\^ine $v(\\mathcal{K})$, d'un corps aux difference valu\\'e $(\\mathcal{K},\\sigma, v)$, o\\`u $R=\\mathcal{K}[t;\\sigma]$ est pleine et dense que $v(\\mathcal{K})$ est un $\\mathbb{Z}[\\sigma]$-module (de nouveau par \\cite{Onay2018a}, 2.28). Enfin par le corollaire \\cite{Onay2018a}, un corps   valu\u00e9 hens\\'elien $(K,v)$ de caract\u00e9ristique $p>0$ est alg\u00e9briquement maximal si et seulement s'il est affinement maximal comme $K[t;x\\mapsto x^p]$-module valu\u00e9. \n\n\\begin{corr} On a:\n\\begin{itemize}\n\t\n\\item Soit $(K,v)$  un corps   valu\u00e9 infini hens\\'elien et de caract\u00e9ristique $p>0$. \n Alors $(K,v)$  est $C$-minimal comme $K[t;x\\mapsto x^p]$-module valu\u00e9 si et seulement s'il est alg\u00e9briquement maximal avec  un corps r\u00e9siduel $p$-clos\net un groupe de valuation divisible.\n\n\\item Soit $\\mathcal{K}$ l'ultraproduit  des corps alg\\`ebriquement valu\\'es $\\mathcal{K}_{p^n}$,  de caract\\'eristique  $p>0$, \\'equipp\\'e chacun du morphisme $x\\mapsto x^{p^n}$, selon un ultrafiltre $U$ sur $\\{p^n \\;\\: | \\:\\: n\\in \\mathbb{N}, \\; \\text{et} \\; p \\; \\text{premier}\\}$. $\\mathcal{K}_{p^n}$   muni de l'automorphisme  limite, {\\it Frobenius non-standard}, i.e., $\\sigma_{U}:=\\lim_{U} x \\mapsto x^{p^n}$, est $C$-minimal comme $\\mathcal{K}[t;\\sigma]$-module valu\\'e.\n\\end{itemize}\n\\end{corr}\n\\begin{remarque}\nNoter que en prenant un ultrafiltre trivial, le second point ci-dessus implique le premir.\n\\end{remarque}\n\\bibliographystyle{apalike}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s:intro}\n\\textcolor{black}{Geometric median-of-means (GMOM) has been widely used for robust estimation of multivariate means, and it has been broadly adopted in machine learning \\citep{Minsker2015,Hsu2016,Prasad2020}. The idea of GMOM is to first divide the data into disjoint subsamples and calculate the empirical means of each of the subsamples. Then the GMOM estimator is computed as the spatial median (also called geometric median) of the obtained empirical means. The previous studies on the GMOM focused on establishing its non-asymptotic error bounds under certain heavy-tailed assumptions. Its distributional properties, which are essential for statistical inference, remain unknown.}\n\nHigh-dimensional data with the dimension increases to infinity as the number of observations goes to infinity have been encountered in many scientific disciplines.\nThere is a growing evidence of the multivariate normal distribution is problematic to model high-dimensional data due to the presents of heavy-tailedness and inadequate to accommodate tail dependence.\nFor example, the distributions of the microarray expression are observed to be non-normal and have heavy tails even after log transformation in many gene expression data \\citep{Purdom2005,Wang2015}.\nAs another example, elliptical distributions, in particular the multivariate $t$-distribution and symmetric multivariate normal inverse Gaussian distribution, provided far superior models to the multivariate normal for daily and weekly US stock-return data \\citep{McNeil2005}.\nIn such cases, the sample spatial median is favored against the sample mean for estimating the location parameter. \n\\textcolor{black}{The above discussions strongly motivate studying the spatial median under high-dimensionality, especially its distributional properties and the implementation in statistical inference for high-dimensional location parameter.}\n\nLet $X_1,\\ldots,X_n$ be a sequence of independent and identically distributed (i.i.d.) $p$-dimensional random vectors from a population $X$ with cumulative distribution function $F_X$ in $\\mathbb{R}^p$. \nIn this paper, we work on a general multivariate model where $X$ admits the following stochastic representation:\n\\begin{eqnarray}\\label{eq:Model_X}\n\tX = {\\boldsymbol \\theta} + \\nu\\Gamma U\\,,\n\\end{eqnarray}\nwhere ${\\boldsymbol \\theta}$ is the location parameter, $\\nu$ is a nonnegative univariate random variable and $U$ is a $p$-dimensional random vector with independent components. Model \\eqref{eq:Model_X} covers many commonly used multivariate models and distribution families, including the independent components model \\citep{Yao2015} and the elliptical distribution family \\citep{Fang1990}. We refer to Section \\ref{sec:note} for more detailed discussions.\n\nSpatial median, an extension of the univariate median to multivariate distributions, was proposed for robust inference of the location parameter \\citep{Haldane1948,Weber1929}. \nThe sample spatial median $\\hat{{\\boldsymbol \\theta}}_{n}\\in\\mathbb{R}^{p}$ minimizes the empirical criteria function $L_n({\\boldsymbol \\beta}) =\\sum_{i=1}^{n}(\\|X_i-{\\boldsymbol \\beta}\\|-\\|X_i\\|)$, where $\\|\\cdot\\|$ is the Euclidean norm. Equivalently,\n\\begin{eqnarray}\\label{eq:btheta_criteria}\n\t\\hat{{\\boldsymbol \\theta}}_{n} = \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^p}L_{n}({\\boldsymbol \\beta}) = \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^p} \\sum_{i=1}^{n}(\\|X_i-{\\boldsymbol \\beta}\\|-\\|X_i\\|)\\,. \n\\end{eqnarray}\nThe function $L_{n}({\\boldsymbol \\beta})$ is convex, and $\\hat{{\\boldsymbol \\theta}}_{n}$ is unique if the observations $\\{X_i\\}_{i=1}^{n}$ are not concentrated on a line in $\\mathbb{R}^{p}$ when $p> 2$ \\citep{Milasevic1987}.\nWhen the dimension $p$ is fixed, the spatial median has been well studied in the literature. \nWe refer to Chapter 6.2 of \\citet{Oja2010} for a nice review. \n\nIn the high-dimensional setting, where the dimension $p$ diverges to infinity as the number of observations $n\\to\\infty$, there are several existing works that study the asymptotic properties of the sample spatial median.\n\\citet{Zou2014} offered an expansion of $\\hat{{\\boldsymbol \\theta}}_{n}$ under elliptical distributions with identical shape matrix, and \\citet{Cheng2019} extended the result to a general shape matrix. \nAs a recent work, \\citet{LiXu2022} improved the expansion in \\citet{Cheng2019} with a smaller order remainder term under stronger conditions, and established a central limit theorem for the squared Euclidean distance $\\|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}\\|^2$.\nIn \\citet{Zou2014} and \\citet{Cheng2019}, they both require that $p=O(n^2)$.\nIn addition, it is required in \\citet{LiXu2022} that $p$ diverges at the same rate as $n$.\nHowever, in modern areas such as genomics and proteomics, the dimension of the data may grow exponentially with the sample size, which lies in the ``ultrahigh dimensional'' region \\citep{FanLv2008}.\nThe previous works with restrictions on the polynomial dimensionality limit the usage of the spatial median under ultrahigh-dimensionality.\nMoreover, the previous results are all under elliptical distributions.\nThus, it is of great importance to establish asymptotic properties of the spatial median and investigate its applications under ultrahigh dimensionality and beyond elliptical distributions.\n\nIn this paper, we first establish Gaussian and bootstrap approximations hit hyperrectangles for the sample spatial median under the general model \\eqref{eq:Model_X} beyond elliptical distributions, which are valid when the dimension diverges exponential with the sample size.\nThey serve as the theoretical foundations of statistical inference for the location parameter based on the sample spatial median under ultrahigh dimensionality. \nConsistent simultaneous confidence intervals (SCIs) and global tests for the location parameters are established.\nWe also study multiple testing for every component of ${\\boldsymbol \\theta}$ based on $\\hat{{\\boldsymbol \\theta}}_n$.\nMotivated by simultaneous inference of ${\\boldsymbol \\theta}$, we define a high-dimensional asymptotic relative efficiency of the sample spatial median relative to the sample mean.\nMost importantly, our theoretical results guarantee the validity of the proposed inferential methods for exponentially divergent $p$.\nThe advantages of our proposed approaches have been justified by simulations and a real data analysis.\n\nThe main contributions of this paper are summarized as follow. \nFirstly, we establish SCIs for the location parameter ${\\boldsymbol \\theta}$ based on the sample spatial median $\\hat{{\\boldsymbol \\theta}}_n$, which is new in the literature. \nThe consistency of bootstrap approximation guarantees that the probability that the SCIs cover all components of the location parameter approaches the nominal confidence level under ultrahigh dimensionality.\nWe also propose a novel test for ultrahigh dimensional location parameter based on the maximum-norm of the sample spatial median. \nThe proposed test not only maintains nominal significance level asymptotically for exponentially divergent $p$, but also is more powerful under sparse alternatives compared to those based on $L_2$-norms \\citep{LiXu2022,Wang2015}.\nAs another major inference, we study multiple testing for every component of the location parameter, and the false discovery rate (FDR) can be well controlled combined with the Benjamini-Hochberg procedure based on the sample spatial median, which extends the existing methods based on the sample mean \\citep{Liu2014}. \nIn all inferential methods, the procedures based on the sample spatial median advances those based on the sample mean for heavy-tailed distributions.\n\nSecondly, this paper serves as the first work that provides Gaussian and bootstrap approximations for the sample spatial median under ultrahigh dimensionality. \nGaussian and bootstrap approximations for high-dimensional sample mean have received extensive attraction in the last decade. \n\\citet{Cher2013} and \\citet{Cher2017} established Gaussian and bootstrap approximations for the maxima of a sum of centered independent random vectors under Kolmogorov distance and on hyperrectangles, respectively. \nSee also \\citet{Chen2018}, \\citet{Cher2019} and \\citet{Cher2020} for related works.\nCompared to the sample mean, which has a simple linear form, \nthe theoretical difficulty for the sample spatial median lies in that it does not enjoy an explicit form.\nThis issue is addressed by deriving a novel Bahadur representation of the sample spatial median with a maximum-norm bound on the remainder term, which extends the results of \\citet{Zou2014}, \\citet{Cheng2019} and \\citet{LiXu2022} under elliptical distributions and polynomial dimensionality.\n\\textcolor{black}{Moreover, our results can be applied to the GMOM under reasonable conditions, and thus enhance the practice usage of GMOM.}\n\nThirdly, we propose a novel multiplier bootstrap method for the sample spatial median.\nInstead of multiplying on the loss function, which is generally the case for M-estimator \\citep{Imaizumi2021}, the multiplier is applied on the centralized $X_i$. Specifically, the bootstrap version of $\\hat{{\\boldsymbol \\theta}}_n$ is defined as $\\tilde{{\\boldsymbol \\theta}}_{n}= \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^d}\\sum_{i=1}^n\\|Z_i(X_i-\\hat{{\\boldsymbol \\theta}}_n)-{\\boldsymbol \\beta}\\|$, where $Z_1,\\ldots,Z_n$ are the multipliers.\nThe multiplier bootstrap is consistent under ultrahigh dimensionality thanks to this novel formulation.\nThis is, however, different from the multiplier bootstrap method for the sample mean, which again has an explicit form \\citep{Cher2013,Cher2017}.\n\n\nThe rest of the paper is organized as follows. \nSection \\ref{sec:note} introduces model and assumptions. \nSection \\ref{sec:main} establishes Gaussian and bootstrap approximations to the distribution of the sample spatial median. \nStatistical inference for the location parameter based on the sample spatial median is presented in Section \\ref{sec:applications}.\nSection \\ref{sec:numerical} reports numerical results including simulations and a real data analysis. \nPreliminary lemmas and proofs of main results are presented in Appendix A of the supplementary material.\nProofs of preliminary lemmas and additional simulations are given in Appendices B and C of the supplementary material.\n\n\n\\textbf{Notation:}\nDenote\n$|x|_{\\infty}=\\max(|x_1|,\\ldots,|x_d|)$ as the maximum-norm of $x=(x_1,\\ldots,x_d)^{\\top}$.\nDenote $a_n\\lesssim b_n$ if $a_n\\leq Cb_n$ for a positive constant $C$, and $a_n\\asymp b_n$ means $a_n\\lesssim b_n$ and $b_n\\lesssim a_n$. \nFor $\\alpha>0$, let $\\psi_{\\alpha}(x)=\\exp(x^{\\alpha})-1$ be a function defined on $[0,\\infty)$. Then the Orlicz norm $\\|\\cdot\\|_{\\psi_{\\alpha}}$ of a random variable $X$ is defined as\n$\n\\|X\\|_{\\psi_{\\alpha}}=\\inf\\left\\{t>0, {\\mathbb{E}}\\{\\psi_{\\alpha}\\left(|X|\/t\\right)\\}\\leq 1\\right\\}.\n$\nWe use ${\\rm tr}(\\cdot)$ to denote the trace operator for square matrices.\nMoreover, we denote $I_p$ as the $p\\times p$ identity matrix.\nFor $a,b\\in\\mathbb{R}$, we write $a \\wedge b=\\min(a,b)$.\n\n\n\n\\section{Model and assumptions}\\label{sec:note}\n\nIn this paper, we consider a general multivariate model for the distribution $F_X$ such that $X_i$ admits the following stochastic representation:\n\\begin{eqnarray}\\label{eq:model}\n\tX_i = {\\boldsymbol \\theta} +\n\t\\nu_i\\Gamma U_i\\,,\n\\end{eqnarray}\nwhere ${\\boldsymbol \\theta}$ is the location parameter, $\\Gamma$ is a nonrandom and invertible $p\\times p$ matrix, \n$U_i$ is a $p$-dimensional random vector with independent~standardized components, and $\\nu_i$ is a nonnegative univariate random variable independent with the spatial sign of $U_i$.\nThe distribution of $X_i$ depends on $\\Gamma$ through the shape matrix $\\Omega=\\Gamma\\Gamma^{\\top}$.\n\n\\begin{remark}\n\tModel \\eqref{eq:model} covers many commonly used multivariate models and distribution families. \n\tFirst, the independent components model \\citep{Yao2015} follows \\eqref{eq:model} with $\\nu_i$ being a nonnegative constant.\n\tSecond, model \\eqref{eq:model} also includes elliptical distributions by choosing $U_i\\sim N(0, I_{p})$ and $\\nu_i=\\xi_i\/\\|U_{i}\\|$ for some nonnegative random variable $\\xi_i$ independent of $U_{i}$. In this case, $\\nu_{i}$ is independent of the spatial sign of $U_{i}$, but not $U_{i}$.\n\tThe independent components model has received great extension in high-dimensional data analysis as well as signal processing and machine learning \\citep{Oja2001}.\n\tIn addition, the elliptical distribution family covers many non-Gaussian distributions such as multivariate $t$-distribution, multivariate logistic distribution, and so on. It is commonly adopted in the literature on studying the sample spatial median\n\t\\citep{Cheng2019,LiXu2022,Zou2014}. \n\t\\textcolor{black}{In terms of the GMOM, if the data are from the independent components model, the subsample means satisfy model \\eqref{eq:model} clearly. In addition, some subfamilies of elliptical distributions are closed under convolution, and thus the subsample means also follow model \\eqref{eq:model}. Our results can be applied to the GMOM estimator directly in those cases.}\n\n\n\\end{remark}\n\nFor $i=1,\\ldots,n$, and  $k=1,2,3,4$, denote\n\\begin{eqnarray}\\label{eq:nota}\n\tW_i = S(X_i-{\\boldsymbol \\theta}) \\text{~~and~~} R_i=\\|X_i-{\\boldsymbol \\theta}\\|\\,\n\\end{eqnarray}\nas the spatial-sign and radius of $X_i-{\\boldsymbol \\theta}$, where $S(X)=\\|X\\|^{-1}X{\\mathbb{I}}(X\\neq 0)$ is the multivariate sign function with ${\\mathbb{I}}(\\cdot)$ being the indicator function. \nThus,\n$\\hat{{\\boldsymbol \\theta}}_n$ satisfies\n$\\sum_{i=1}^{n}S(X_i-\\hat{{\\boldsymbol \\theta}}_{n}) = 0\\,. $\n\nDenote $U_{i}=(U_{i,1},\\ldots,U_{i,p})^{\\top}$,\nwe impose the following three conditions.\n\n\n\\begin{condition}\\label{c1}\n\t$U_{i,1},\\ldots,U_{i,p}$ are i.i.d.~symmetric random variables with ${\\mathbb{E}}(U_{i,j})=0$, ${\\mathbb{E}}(U_{i,j}^2)=1$, and\n\t$\\|U_{i,j}\\|_{\\psi_{\\alpha}}\\leq c_0$ with some constant $c_0>0$ and $1\\leq \\alpha\\leq 2$.\n\\end{condition}\n\n\t\\begin{condition}\\label{c2}\n\t\n\t\n\t\n\t\n\t\tThe moments $\\zeta_{k}={\\mathbb{E}}(R_i^{-k})$ for $k=1,2,3,4$ exist for large enough $p$. In addition, there exist two positive constants $\\underline{b}$ and ${\\bar{B}}$ such that\n\t\t$\n\t\t{\\underline b}\\leq  \\limsup_{p}{\\mathbb{E}}(R_i\/\\sqrt{p})^{-k}\\leq {\\bar{B}}\n\t\t$ for $k=1,2,3,4$.\n\t\\end{condition}\n\t\n\t\\begin{condition}\\label{c3}\n\t\tThe shape matrix $\\Omega=(\\omega_{j\\ell})_{p\\times p}$ satisfies ${\\rm tr}(\\Omega)=p$ and it belongs to the following class:\n\t\t\\begin{eqnarray}\n\t\t\t\\notag & & {\\mathcal U}(a_0(p), \\underline{m}, \\bar{M})  = \\left\\{\\Omega: \\underline{m} \\leq \\omega_{jj}\\leq \\bar{M},~  \\sum_{\\ell=1}^{p}|\\omega_{j\\ell}|\\leq a_0(p), \\text{~~for all~} j=1,\\ldots,p\\right\\}\\,,\n\t\t\\end{eqnarray}\n\t\twhere $\\underline{m}\\leq \\bar{M}$ are bounded positive constants.\n\t\\end{condition}\n\t\n\t\\begin{remark}\n\t\tIn Condition \\ref{c1}, the symmetric assumption is to ensure that ${\\boldsymbol \\theta}$ in model \\eqref{eq:model} coincides with the population spatial median, which minimizes $L({\\boldsymbol \\beta})={\\mathbb{E}}(\\|X-{\\boldsymbol \\beta}\\|-\\|X\\|)$.\n\t\tIt is obvious that Condition \\ref{c1} is satisfied by elliptical distributions with $U_i\\sim N(0,I_p)$.\n\t\tThe condition $\\|U_{i,j}\\|_{\\psi_{\\alpha}}\\leq c_0$ implies that $U_{i,j}$ has a sub-exponential distribution.\n\t\tIt is worth highlighting that with slight modification of the proofs of main theorems, the i.i.d.~condition on $U_{i,1},\\ldots,U_{i,p}$ can be weaken by replacing Condition \\ref{c1} with the following assumption:\n\t\t$U_{i,1},\\ldots,U_{i,p}$ are independent symmetric random variables with ${\\mathbb{E}}(U_{i,j})=0$, ${\\mathbb{E}}(U_{i,j}^2)=1$ for all $j=1,\\ldots,p$, and\n\t\t$\\sup_{1\\leq j\\leq p}\\|U_{i,j}\\|_{\\psi_{\\alpha}}\\leq c_0$ with some constant $c_0>0$ and $1\\leq \\alpha\\leq 2$.\n\t\\end{remark}\n\t\n\t\\begin{remark}\n\t\tThe condition ${\\underline b}\\leq  \\limsup_{p}{\\mathbb{E}}(R_i\/\\sqrt{p})^{-k}\\leq {\\bar{B}}$ indicates that $\\zeta_{k}\\asymp p^{-k\/2}$ for $k=1,2,3,4$. It is introduced to avoid $X_i$ from concentrating too much near ${\\boldsymbol \\theta}$. For elliptical distributions, it is a generalization of Assumption 1 of \\citet{Zou2014}, which is satisfied by many common distributions.\n\t\tFor the independent components model, Condition \\ref{c2} is equivalent to that\n\t\t$\n\t\t{\\underline b}\\leq  \\limsup_{p}{\\mathbb{E}}(\\|\\Gamma U_{i}\\|\/\\sqrt{p})^{-k}\\leq {\\bar{B}}\\,.\n\t\t$\n\t\tAccording to Lemma \\ref{lemma:SM_moments} in Appendix A, ${\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{k}) = p^{k\/2}\\{1+o(1)\\}$ for $k=1,2,3,4$. Then the Cauchy-Schwarz inequality implies that\n\t\t${\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{-k}) \\geq \\{{\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{k})\\}^{-1} = p^{-k\/2}\\{1+o(1)\\}\\,,$\n\t\tfrom which we know ${\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{-k})\\gtrsim p^{-k\/2}$. Furthermore, denote $\\Gamma_{j}$ as the $j$th row of $\\Gamma$, then by the inequality of harmonic and quadratic means,\n\t\t\\begin{eqnarray}\n\t\t\t\\notag \\notag p^{2}\\|\\Gamma U_{i}\\|^{-4} = \\left\\{\\frac{p}{(\\Gamma_{1}U_{i})^2+\\cdots+(\\Gamma_{p}U_{i})^2}\\right\\} \\leq \\frac{(\\Gamma_{1}U_{i})^{-4}+\\cdots+(\\Gamma_{p}U_{i})^{-4}}{p}\\,.\n\t\t\\end{eqnarray}\n\t\tIt follows that ${\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{-4})\\lesssim p^{-2}$ if ${\\mathbb{E}}\\{(\\Gamma_{1}U_{i})^{-4}\\},\\ldots,{\\mathbb{E}}\\{(\\Gamma_{p}U_{i})^{-4}\\}$ are uniformly bounded, and from which ${\\mathbb{E}}(\\|\\Gamma U_{i}\\|^{-k})\\lesssim p^{-k\/2}$ by Jensen's inequality. Thus, Condition \\ref{c2} is satisfied by the independent components models as long as $\\Gamma_{1}U_{i},\\ldots,\\Gamma_{p}U_{i}$ are not concentrating too much near $0$.\n\t\tSee also discussions in\n\t\t\\citet{Cardot2013} on similar conditions.\n\t\\end{remark}\n\t\n\t\\begin{remark}\n\t\tIt is noticed that the shape matrix $\\Omega$ is only well defined up to a scalar multiple, the condition ${\\rm tr}(\\Omega)=p$ is used to regularize $\\Omega$ to make model \\eqref{eq:model} identifiable.\n\t\tThe class ${\\mathcal U}(a_0(p), \\underline{m}, \\bar{M})$\n\t\tcovers a wide range of symmetric square matrices, and it is commonly adopted in the literature on high-dimensional analysis. \n\t\tFor example, a similar matrix class is introduced in \\citet{Bic2008}. The condition $\\underline{m} \\leq \\omega_{jj}\\leq \\bar{M}$ requires bounded diagonal elements. \n\t\tThe order of $a_0(p)$, which will be specified later, controls the orders of the off-diagonal elements of $\\Omega$.\n\t\\end{remark}\n\t\n\t\\section{Gaussian and bootstrap approximations}\\label{sec:main}\n\t\n\t\\subsection{Bahadur representation and Gaussian approximation}\\label{sec:02}\n\t\n\n\tIn this section, we establish Gaussian approximation for $\\hat{{\\boldsymbol \\theta}}_n$, which is valid when $p$ diverges exponentially over $n$.\n\tThe following lemma offers a Bahadur representation of $\\hat{{\\boldsymbol \\theta}}_n$, and it severs as the foundation of the Gaussian approximation result in Theorem \\ref{theo1}.\n\t\n\t\\begin{lemma}\\label{lem:Br}\n\t\t(Bahadur representation) Assume Conditions \\ref{c1}, \\ref{c2} and \\ref{c3} with $a_0(p)\\asymp p^{1-\\delta}$ for some positive constant\n\t\t$\\delta\\leq 1\/2$ hold. If\n\t\t$\\log p=o(n^{1\/3})$ and $\\log n=o(p^{1\/3 \\wedge \\delta})$, then\n\t\t\\[\n\t\tn^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta})=n^{-1\/2}\\zeta_{1}^{-1}\\sum_{i=1}^nW_i+C_n\\,,\n\t\t\\]\n\t\twhere $|C_n|_{\\infty}=O_p\\{n^{-1\/4}\\log^{1\/2}(np)+ p^{-(1\/6 \\wedge \\delta\/2)}\\log^{1\/2}(np)\\}$.\n\t\\end{lemma}\n\t\n\t\\begin{remark}\n\t\tTo the best of our knowledge, Lemma \\ref{lem:Br} serves as the first result that offers the Bahadur representation of the sample spatial median with a maximum-norm bound on the remainder term.\n\t\tIn \\citet{Zou2014} and \\citet{Cheng2019}, the same expansion with the remainder term $C_n$ satisfies $\\|C_n\\|=o_{p}(\\zeta_{1}^{-1})$ was obtained, and their result was improved to $\\|C_n\\|=o_p(1)$ in \\citet{LiXu2022}, by replacing $\\zeta_{1}$ with $n^{-1}\\sum_{i=1}^{n}R_i^{-1}$ in the linear term, but under a more restricted condition that $p$ and $n$ are of the same order.\n\t\tIt is worth noticing that the previous results \\citep{Cheng2019,LiXu2022,Zou2014} are all derived under elliptical distributions.\n\t\n\t\\end{remark}\n\t\n\tLet ${\\mathcal A}^{\\mathrm{re}}=\\{\\prod_{j=1}^{p}[a_j, b_j]: -\\infty\\leq a_j\\leq b_j\\leq \\infty, j=1,\\ldots,p\\}$ be the class of rectangles in $\\mathbb{R}^{p}$.\n\tWith the Bahadur representation in Lemma \\ref{lem:Br} on hand, we establish the following Gaussian approximation result for $\\hat{{\\boldsymbol \\theta}}_n$ over hyperrectangles.\n\n\t\\begin{theorem}\\label{theo1}\n\t\t\\textit{(Gaussian approximation)}\n\t\tAssume Conditions \\ref{c1}, \\ref{c2} and \\ref{c3} with $a_0(p)\\asymp p^{1-\\delta}$ for some positive constant\n\t\t$\\delta\\leq 1\/2$ hold. If\n\t\t$\\log p=o(n^{1\/5})$ and $\\log n=o(p^{1\/3 \\wedge \\delta})$,\n\t\n\t\n\t\n\t\tthen\n\t\t\\[\n\t\t\\rho_n({\\mathcal A}^{\\mathrm{re}}) = \\sup_{A \\in {\\mathcal A}^{{\\rm re}}}\\left|{\\mathbb{P}}\\{{n}^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}) \\in A \\}-{\\mathbb{P}}\\left(G \\in A\\right)\\right|\\rightarrow 0\\,\n\t\t\\]\n\t\tas $n\\to\\infty$, where $G\\sim N(0, \\zeta_{1}^{-2}{\\mathbb{B}})$ with ${\\mathbb{B}}={\\mathbb{E}}(W_1W_1^{\\top})$.\n\t\n\\end{theorem}\n\nThe Gaussian approximation for $\\hat{{\\boldsymbol \\theta}}_n$\nindicates that the probabilities ${\\mathbb{P}}\\{n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}) \\in A\\}$ can be approximated by that of a centered Gaussian random vector with covariance matrix $\\zeta_{1}^{-2}{\\mathbb{B}}$ for hyperrectangles $A\\in{\\mathcal A}^{\\mathrm{re}}$.\nTheorem \\ref{theo1} allows for an exponentially divergent $p$, which fits the ultrahigh dimensional setting. Compared to the asymptotic normality of\n$\\|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}\\|^2$ in \\citet{LiXu2022}, in which $p$ is assumed to have the same order as $n$, the Gaussian approximation result in Theorem \\ref{theo1} requires much weaker conditions on the rates of $n$ and $p$.\n\n\\begin{remark}\\label{remark:B}\n\tLet ${\\mathbb{B}}_{j\\ell}$ be the $(j,\\ell)$th element of ${\\mathbb{B}}$. According to Lemma \\ref{lemma:05} (iii) in Appendix A, $\\zeta_{1}^{-2}{\\mathbb{B}}_{j\\ell} = \\zeta_{1}^{-2}p^{-1}\\omega_{j,\\ell} + O(p^{-\\delta\/2})$ for all $1\\leq j,\\ell\\leq p$. Thus, the covariance matrix of $G$ in Theorem \\ref{theo1} is asymptotically proportional to the shape matrix $\\Omega$.\n\\end{remark}\n\n\\begin{remark}\\label{remark:M-estimator}\n\tAs the sample spatial median is a special M-estimator, \n\tGaussian approximation for M-estimator in \\citet{Imaizumi2021} is potentially applicable to the spatial median under high-dimensionality.\n\tHowever, it is worth highlighting that the results in \\citet{Imaizumi2021} cannot be applied to our framework. \n\tTo be precise, Assumption 1 (A3) in \\citet{Imaizumi2021} assumes that there exist constants $C>0$ and $\\alpha\\in(0,2)$ such that $\\log \\mathcal{N}(\\varepsilon, \\Theta, \\|\\cdot\\|)\\leq C\\varepsilon^{-\\alpha}$ holds for all $\\varepsilon\\in(0,1)$, where $\\Theta$ is the parameter space, and $\\mathcal{N}(\\varepsilon, \\Theta, \\|\\cdot\\|)$ is the $\\varepsilon$-covering number of $\\Theta$ under the Euclidean norm $\\|\\cdot\\|$ \\citep{VanderVaart1996}. \n\tWhen $\\Theta$ is a compact subset of $\\mathbb{R}^{p}$, $\\mathcal{N}(\\varepsilon, \\Theta, \\|\\cdot\\|)$ is of order $O(\\varepsilon^{-p})$. \n\tIn this case, $\\log \\mathcal{N}(\\varepsilon, \\Theta, \\|\\cdot\\|)\\leq C\\varepsilon^{-\\alpha}$ cannot be satisfied when $p\\to\\infty$. \n\tThus, our theoretical findings are independent of those in \\citet{Imaizumi2021}.\n\\end{remark}\n\n\\iffalse\n\\begin{remark}\\label{remark:04}\n\tWhen $p$ is fixed, the classical CLT for $\\hat{{\\boldsymbol \\theta}}_n$ \\citep{Oja2010} is\n\t$ n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}) \\to N(0, {\\mathbb{C}}^{-1}{\\mathbb{B}} {\\mathbb{C}}^{-1}) $\n\tin distribution, where ${\\mathbb{C}}={\\mathbb{E}}\\{R_i^{-1}(I_p-W_iW_i^{\\top} )\\}$. Compared to Theorem \\ref{theo1}, the asymptotic covariance matrix changes from ${\\mathbb{C}}^{-1}{\\mathbb{B}} {\\mathbb{C}}^{-1}$ to $\\zeta_{1}^{-2}{\\mathbb{B}}$ under high-dimensionality. \n\tIn fact, \n\n\t$\n\t\\left\\|{\\mathbb{C}}-\\zeta_{1}I_p \\right\\|_2= \\left\\|{\\mathbb{E}}(R_i^{-1}W_iW_i^{\\top})\\right\\|_2 \\lesssim p^{-1}{\\mathbb{E}}(R_i^{-1})=p^{-1}\\zeta_{1}\\,\n\t$\n\tin probability, where the last inequality can be shown similar to Lemmas A.5 and A.6 of \\citet{Wang2015}.\n\tNote that $p^{-1}\\zeta_{1}\/ \\|\\zeta_{1}I_p\\|_2\\to0$ under Condition \\ref{c2}, the distance between ${\\mathbb{C}}$ and $\\zeta_{1}I_p$, so as ${\\mathbb{C}}^{-1}{\\mathbb{B}} {\\mathbb{C}}^{-1}$ and $\\zeta_{1}^{-2}{\\mathbb{B}}$, is negligible under the matrix 2-norm as $p\\to\\infty$.\n\\end{remark}\n\\fi\n\nTheorem \\ref{theo1} immediately implies the following corollary\nsince the Kolmogorov distance of sup-norm is a subset of $\\mathcal{A}^{\\mathrm{re}}$ corresponding to max-hyperrectangles in $\\mathbb{R}^{p}$. \n\\begin{corollary}\\label{co1}\n\tUnder the conditions assumed in Theorem \\ref{theo1}, as $n\\to\\infty$,\n\t\\[\n\t\\rho_n=\\sup_{t\\in{\\mathbb{R}}}\\left|{\\mathbb{P}}(n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty}\\leq t)- {\\mathbb{P}}(|G|_{\\infty}\\leq t)\\right|\\rightarrow  0.\n\t\\]\n\\end{corollary}\n\\subsection{Multiplier bootstrap approximation}\\label{sec:boot}\n\nTheorem \\ref{theo1} allows us to approximate the distribution of $n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta})$ by that of $G$ hit hyperrectangles, where $G\\sim N(0, \\zeta_{1}^{-2}{\\mathbb{B}})$. \nHowever, it cannot be used directly in statistical inference for ${\\boldsymbol \\theta}$ as\nthe quantity $\\zeta_{1}$ and the matrix ${\\mathbb{B}}$ depend on the underlying distribution $F_X$ and are thus unknown. \nTo solve this issue, we propose an easy-to-implement bootstrap method to approximate the distribution of $n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta})$.\n\nLet $Z_1,\\ldots,Z_n$ be a sequence of i.i.d.~random variables with mean zero and unit variance. Define the bootstrap version of the sample spatial median as\n\\begin{align}\\label{btheta}\n\t\\tilde{{\\boldsymbol \\theta}}_{n}= \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^d}\\sum_{i=1}^n\\|Z_i(X_i-\\hat{{\\boldsymbol \\theta}}_n)-{\\boldsymbol \\beta}\\|\\,.\n\\end{align}\nThen, the distribution of $n^{1\/2}\\tilde{{\\boldsymbol \\theta}}_{n}$ conditional on $X_1,\\ldots,X_n$ is used to approximate that of $n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta})$.\nThis algorithm is called the multiplier bootstrap, and $Z_1,\\ldots,Z_n$ are the multiplier weights.\n\nRegarding the proof of Lemma \\ref{lem:SM_boot_br} in Appendix B, it is preferred that the multiplier weights $Z_1,\\ldots,Z_n$ are bounded and satisfy ${\\mathbb{E}}(Z_{i}^{-2})<\\infty$. \nThus, we choose the Rademacher variables as the multipliers \\citep{Cher2019}, that is, ${\\mathbb{P}}(Z_i=1)={\\mathbb{P}}(Z_i=-1)=1\/2$.\n\\begin{theorem}\\label{theo:boot}\n\t\\textit{(Bootstrap approximation)}\n\tUnder the conditions assumed in Theorem \\ref{theo1},\n\t\\[\n\t\\rho_n^{\\mathrm{MB}}({\\mathcal A}^{\\mathrm{re}})=\\sup_{A \\in {\\mathcal A}^{\\mathrm{re}}}\\left|{\\mathbb{P}}\\{n^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta})\\in A\\}- {\\mathbb{P}}^{*}(n^{1\/2}\\tilde{{\\boldsymbol \\theta}}_{n} \\in A)\\right|\\rightarrow 0\n\t\\]\n\tin probability as $n\\to\\infty$,\n\twhere ${\\mathbb{P}}^{*}$ denotes the conditional probability given $X_1,\\ldots,X_n$.\n\\end{theorem}\n\nUnder the same conditions on the divergence rates of $n$ and $p$ as in Theorem \\ref{theo1}, Theorem \\ref{theo:boot} validates that conditional on $X_1,\\ldots,X_n$, the distribution of the bootstrap sample spatial median $\\tilde{{\\boldsymbol \\theta}}_n$ approximates that of $\\hat{{\\boldsymbol \\theta}}_n$ consistently over hyperrectangles.\n\n\\begin{remark}\n\tThe proof of Theorem \\ref{theo:boot} is nontrivial and does not follow directly from existing results since $\\tilde{{\\boldsymbol \\theta}}_n$ has no explicit form, which is different from the multiplier bootstrap methods for high-dimensional sample mean that have been analysed in the literature.\n\tThe key step in the proof is to obtain a Bahadur representation of $\\tilde{{\\boldsymbol \\theta}}_n$ similar as $\\hat{{\\boldsymbol \\theta}}_n$ in Lemma \\ref{lem:Br}. Specifically, we show that $n^{1\/2}\\tilde{{\\boldsymbol \\theta}}_{n}=n^{-1\/2}\\zeta_{1}^{-1}\\sum_{i=1}^n Z_i W_i+{\\tilde C}_n$ with $|{\\tilde C}_n|_{\\infty}=O_p\\{n^{-1\/4}\\log^{1\/2} (np)+p^{-(1\/6\\wedge\\delta\/2)}\\log^{1\/2} (np)\\}$ in Lemma \\ref{lem:SM_boot_br} in Appendix A.\n\n\\end{remark}\n\nThe next corollary is an immediate consequence of Theorem \\ref{theo:boot}.\n\\begin{corollary}\\label{co2}\n\tUnder the conditions assumed in Theorem \\ref{theo:boot}, as $n\\to\\infty$,\n\t\\[\n\t\\rho_n^{\\mathrm{MB}}=\\sup_{t\\in{\\mathbb{R}}}\\left|{\\mathbb{P}}\\{n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty}\\leq t\\}- {\\mathbb{P}}^{*}(n^{1\/2}|\\tilde{{\\boldsymbol \\theta}}_{n}|_{\\infty}\\leq t)\\right|\\rightarrow  0 \\text{~~in probability.}\n\t\\]\n\t\n\\end{corollary}\n\n\n\\section{Statistical inference}\\label{sec:applications}\n\n\nThe Gaussian and multiplier bootstrap approximations for the sample spatial median\nenable many statistical inferential methods for ultrahigh dimensional  population location parameter.\nIn this section, we present the following statistical inferences: simultaneous confidence intervals (SCIs) and global tests for the population location parameter, multiple testing for every component of ${\\boldsymbol \\theta}$, and high-dimensional asymptotic relatively efficient of the sample spatial median compared to the sample mean.\n\n\n\\subsection{Simultaneous confidence intervals}\\label{sec:SCIs}\n\nWe are interested in building SCIs for all components of ${\\boldsymbol \\theta}=(\\theta_1,\\ldots,\\theta_p)^{\\top}$.\nCorollary \\ref{co2} motivates the following way of constructing SCIs for ${\\boldsymbol \\theta}$.\nGiven a nominal confidence level $1-\\tau$,\ndefine the set ${\\mathcal C}_{\\tau}$ as\n\\[\n{\\mathcal C}_{\\tau}=\\left\\{{\\boldsymbol \\theta}\\in {\\mathbb{R}}^p, n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty}<q^{B}_{1-\\tau}\\right\\},\n\\]\nwhere $q^{B}_{1-\\tau}$ is the $(1-\\tau)$th quantile of $n^{1\/2}|\\tilde{{\\boldsymbol \\theta}}_{n}|_{\\infty}$ given $X_1,\\ldots,X_n$.\nDenote $\\hat{{\\boldsymbol \\theta}}_n=(\\hat{\\theta}_{n,1},\\ldots,\\hat{\\theta}_{n,p})^{\\top}$, the confidence intervals are $[\\theta_{n,j}^{-},\\theta_{n,j}^{+}]$ for $j=1,\\ldots,p$, where\n$$\n\\theta_{n,j}^{-}=\\hat{\\theta}_{n,j}-n^{-1\/2}q^{B}_{1-\\tau} \\text{~~and~~} \\theta_{n,j}^{+}=\\hat{\\theta}_{n,j}+n^{-1\/2}q^{B}_{1-\\tau}.\n$$\nThe next theorem shows that $\\mathcal{C}_{\\tau}$ preserves the nominal simultaneous confidence level $1-\\tau$ asymptotically under ultrahigh dimensionality.\n\n\\begin{theorem}\\label{theo:SCIs}\n\tSuppose the conditions of Theorem \\ref{theo:boot} hold, then ${\\mathbb{P}}({\\boldsymbol \\theta}\\in {\\mathcal C}_{\\tau})\\rightarrow 1-\\tau$ as $n\\to\\infty$. Equivalently,\n\t${\\mathbb{P}}(\\theta_j\\in [\\theta_{n,j}^{-},\\theta_{n,j}^{+}] \\text{~~for all~} 1\\leq j\\leq p)\\rightarrow 1-\\tau$ as $n\\to\\infty$.\n\\end{theorem}\n\n\\begin{remark}\n\tUnlike the fixed dimensional setting, $n^{1\/2}|\\tilde{{\\boldsymbol \\theta}}_{n}|_{\\infty}$ and $n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty}$ are maxima of divergent numbers of variables, and their quantiles are generally divergent as $p\\to\\infty$.\n\tThus, Theorem \\ref{theo:SCIs} is not a direct consequence of Corollary \\ref{co2}.\n\tTo ascertain the consistency of $\\mathcal{C}_{\\tau}$ theoretically, we show that, with probability approaching one, $q_{1-\\tau}^{B}$ is bounded by two quantiles of $n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty}$ with quantile levels close enough to $1-\\tau$ using an anti-concentration inequality for divergent random sequences.\n\\end{remark}\n\n\\iffalse\nWe summarise the approach of building SCIs for ${\\boldsymbol \\theta}$ in Algorithm \\ref{al1}.\n\\RestyleAlgo{ruled}\n\\begin{algorithm}\\label{al1}\n\t\\caption{Construction of SCIs for ${\\boldsymbol \\theta}$}\n\t\\SetAlgoLined\n\t\\KwData{Data $X_1, \\ldots, X_n$; $\\alpha$, confidence level; $B$, number of bootstrap iterations}\n\t\\KwResult{Simultaneous confidence intervals: $[\\theta_{n,j}^{-},\\theta_{n,j}^{+}]$ for $j=1,\\ldots,p$}\n\t\\BlankLine\n\t\n\tCompute $\\hat{{\\boldsymbol \\theta}}_n=(\\hat{\\theta}_{n,1},\\ldots,\\hat{\\theta}_{n,p})^{\\top}$ by minimizing $L_n({\\boldsymbol \\beta})$ defined in \\eqref{eq:btheta_criteria};\n\t\n\t\n\t\\For{{$b\\leftarrow 1$ \\KwTo $B$}}{\n\t\tgenerate independent Rademacher random variables $Z_1^{(b)}, \\ldots, Z_n^{(b)}$;\n\t\t\n\t\tcompute $\\tilde{{\\boldsymbol \\theta}}_{n}^{(b)}$ by minimizing \\eqref{btheta} using $Z_1^{(b)}, \\ldots, Z_n^{(b)}$ as the multiplier weights;\n\t\t\n\t\tcompute $n^{1\/2}|\\tilde{{\\boldsymbol \\theta}}_{n}^{(b)}|_{\\infty},$;\n\t}\n\t\n\tCompute $q^{B}_{1-\\alpha}$ as the $(1-\\alpha)$-th quantiles of $n^{1\/2}|\\tilde{{\\boldsymbol \\theta}}_{n}^{(1)}|_{\\infty}, \\ldots, |n^{1\/2}\\tilde{{\\boldsymbol \\theta}}_{n}^{(B)}|_{\\infty}$;\n\t\n\t\\For{{$j\\leftarrow 1$ \\KwTo $p$}}{\n\t\tcompute $\\theta_{n,j}^{-}=\\hat{\\theta}_{n,j}-n^{-1\/2}q^{B}_{1-\\alpha}$;\n\t\t\n\t\tcompute $\\theta_{n,j}^{+}=\\hat{\\theta}_{n,j}+n^{-1\/2}q^{B}_{1-\\alpha}$;\n\t}\n\\end{algorithm}\n\\fi\n\n\n\\begin{remark}\\label{remark:SCIs_mean}\n\tThe Gaussian approximation for  the sample mean $\\bar{X}_n=n^{-1}\\sum_{i=1}^{n}X_i$ \\citep{Cher2013,Cher2017,Cher2019} indicate that if $\\log p = o(n^{1\/5})$,\n\t\\begin{eqnarray}\\label{eq:gaussian_approx_mean}\n\t\n\t\t\\sup_{t\\in{\\mathbb{R}}}\\left|{\\mathbb{P}}(n^{1\/2}|\\bar{X}_{n}-{\\boldsymbol \\theta}|_{\\infty}\\leq t)- {\\mathbb{P}}(|G_0|_{\\infty}\\leq t)\\right|\\rightarrow  0\n\t\\end{eqnarray}\n\tas $n\\to\\infty$ under some moderate conditions, where $G_0\\sim N(0, \\Sigma)$ with $\\Sigma={\\mathbb{E}}(XX^{\\top})$.\n\tDefine $X_{i}^{*} = Z_i(X_i-\\bar{X}_n)$ for $i=1,\\ldots,n$, where $Z_1,\\ldots,Z_n$ are the Rademacher weights. Denote $\\bar{X}_{n}^{*}=n^{-1}\\sum_{i=1}^{n}X_{i}^{*}$,\n\tit has been shown in \\citet{Cher2019} that\n\t\\begin{eqnarray}\\label{eq:boot_mean}\n\t\t& \\sup_{t\\in{\\mathbb{R}}}\\left|{\\mathbb{P}}(n^{1\/2}|\\bar{X}_{n}-{\\boldsymbol \\theta}|_{\\infty}\\leq t)- {\\mathbb{P}}^{*}(n^{1\/2}|\\bar{X}_{n}^{*}|_{\\infty}\\leq t)\\right|\\rightarrow  0 &\n\t\\end{eqnarray}\n\tin probability as $n\\to\\infty$ when $\\log p = o(n^{1\/5})$.\n\tBased on \\eqref{eq:boot_mean}, define\n\t$$\n\t{\\mathcal C}_{\\tau}^{\\prime}=\\left\\{{\\boldsymbol \\theta}\\in {\\mathbb{R}}^p, n^{1\/2}|\\bar{X}_{n}-{\\boldsymbol \\theta}|_{\\infty}<q^{B\\prime}_{1-\\tau}\\right\\},\n\t$$\n\twhere $q^{B\\prime}_{1-\\tau}$ is the $(1-\\tau)$th quantile of $n^{1\/2}|\\bar{X}_{n}^{*}|_{\\infty}$ conditional on $X_1,\\ldots,X_n$.\n\tThen $\\mathcal{C}_{\\tau}^{\\prime}$ is also an asymptotic $1-\\tau$ SCIs for ${\\boldsymbol \\theta}$.\n\tBased on the discussion in Section \\ref{sec:ARE}, $\\mathcal{C}_{\\tau}$ has advantage (relative shorter intervals) over $\\mathcal{C}_{\\tau}^{\\prime}$ under heavy-tailed distributions.\n\tWe refer to Section \\ref{sec:simulations} for finite-sample justifications on this.\n\\end{remark}\n\n\n\\subsection{Global tests for high-dimensional location parameters}\\label{sec:global_tests}\n\nIn this section, we propose a novel approach for global tests on high-dimensional location parameters.\nLet ${\\boldsymbol \\theta}_0$ be a known $p$-dimensional vector, we are interested in testing\n\\begin{eqnarray}\\label{eq:one_sample}\n\tH_0: {\\boldsymbol \\theta} = {\\boldsymbol \\theta}_0 \\text{~~versus~~} H_1:{\\boldsymbol \\theta}\\neq{\\boldsymbol \\theta}_0.\n\\end{eqnarray}\n\nTheorems \\ref{theo1} and \\ref{theo:boot} motivate us proposing a maximum-norm type test statistic.\nDefine\n\\begin{align}\\label{test}\n\tT_{n} = n^{1\/2}|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}_0|_{\\infty}\n\\end{align}\nas the test statistic, and $H_0$ is rejected when $T_{n}$ is larger than a critical value.\nWe can use the multiplier bootstrap to approximate the distribution of $T_n$ under $H_0$.\nSpecifically, with a nominal significance level $\\tau$, the null hypothesis is rejected if $T_n>q_{1-\\tau}^{B}$.\nTheorem \\ref{theo:SCIs} guarantees that the test based on $T_n$ maintains nominal significance level asymptotically under ultrahigh dimensionality, that is, ${\\mathbb{P}}(T_n>q^B_{1-\\tau}\\mid H_0)\\rightarrow \\tau$ as $n\\to\\infty$ when $\\log p = o(n^{1\/5})$.\n\n\\begin{remark}\\label{remark:06}\n\n\tAn alternative test for \\eqref{eq:one_sample} can be constructed based on $\\bar{X}_n$ by defining the test statistic as\n\n\t$ T_{\\mathrm{Mean}} = n^{1\/2}|\\bar{X}_{n}-{\\boldsymbol \\theta}_0|_{\\infty}$.\n\n\tThen, the null hypothesis is rejected if $T_{\\mathrm{Mean}}>q_{1-\\tau}^{B\\prime}$.\n\tThe test based on $T_n$ can be deemed as a nonparametric extension of the test based on $T_{\\mathrm{Mean}}$ .\n\tAs $\\hat{{\\boldsymbol \\theta}}_n$ is more efficient than $\\bar{X}_n$ for simultaneous inference of ${\\boldsymbol \\theta}$ under heavy-tailed distributions as discussed in Section \\ref{sec:ARE}, we expect that the proposed test based on $T_n$ is more powerful than that based on $T_{\\mathrm{Mean}}$ in those cases.\n\tThis has been reflected by the simulation results in Appendix C of the supplementary material.\n\\end{remark}\n\nThe next theorem summarises the asymptotic power of the proposed test based on $T_n$.\n\\begin{theorem}\\label{theo:power}\n\tSuppose the conditions of Theorem \\ref{theo:boot} hold.\n\tFor any given $0<\\tau<1$, if\n\t$|{\\boldsymbol \\theta}-{\\boldsymbol \\theta}_0|_{\\infty}\\geq C\\log^{1\/2}(\\tau^{-1}) n^{-1\/2}\\log^{1\/2}(np)$ for some large enough constant $C>0$,\n\tthen ${\\mathbb{P}}(T_n>q^B_{1-\\tau}\\mid H_1)\\rightarrow 1$ as $n\\rightarrow \\infty$.\n\\end{theorem}\n\nTheorem \\ref{theo:power} indicates that the test based on $T_n$ achieves consistency when the maximum element of $n^{1\/2}|{\\boldsymbol \\theta}-{\\boldsymbol \\theta}_0|$ has a magnitude much large than $\\log^{1\/2}(\\tau^{-1})\\log^{1\/2}(np)$ for a fixed significant level $\\tau$.\n\n\\begin{remark}\\label{remark:07}\n\t\\citet{Wang2015} proposed a $L_2$-norm type test (WPL test) for \\eqref{eq:one_sample} with ${\\boldsymbol \\theta}_0=0$ based on\n\n\t$ T_{\\mathrm{WPL}} = \\sum_{i=1}^{n}\\sum_{j=1}^{i-1}W_{i}^{\\top}W_i$.\n\n\tIt has been argued in \\citet{Wang2015} and \\citet{LiXu2022} that the\n\n\n\n\n\tsignal of the WPL test is determined by the magnitude of $\\|{\\boldsymbol \\theta}\\|$, which is the $L_2$-norm of ${\\boldsymbol \\theta}$.\n\tAs a contrast, the power of the test based on $T_n$  depends on $|{\\boldsymbol \\theta}|_{\\infty}$.\n\tThus, the proposed test based on $T_n$ is expected to be more powerful under sparse alternatives, when ${\\boldsymbol \\theta}$ contains only a limited number of non-zero components and its maximum element has certain order of magnitude.\n\tIn such cases, $\\|{\\boldsymbol \\theta}\\|$ is not big enough for the rejection of the WPL test.\n\tSee Appendix C in the supplementary material and Section \\ref{sec:real_data} for numerical justifications.\n\\end{remark}\n\\subsection{Multiple testing with FDR control in large-scale tests}\\label{sec:FDR}\n\nMultiple testing with false discovery rate (FDR) control has been applied to many real problems, such as detecting differentially expressed genes in genomic study.\nIn this section, we study multiple testing for every component of ${\\boldsymbol \\theta}$ based on the spatial median with the Benjamini and Hochberg (B-H) method for FDR control.\nFor $j=1,\\ldots,p$, we are interested in testing\n\\begin{eqnarray}\n\t\\notag H_{0j}:  \\theta_j=\\theta_{0,j}  \\text{~~versus~~} H_{1j}:\\theta_{j} \\neq\\theta_{0,j}\n\\end{eqnarray}\nsimultaneously, where $\\theta_{0,1},\\ldots,\\theta_{0,p}$ are given values.\n\nDefine the test statistics as\n\\begin{eqnarray}\n\t\\notag T_{n,j} = n^{1\/2}(\\hat{\\theta}_{n,j}-\\theta_{0,j})\/s_{n,j}\n\\end{eqnarray}\nfor $j=1,\\ldots,p$, where $s_{n,j}^2=\\hat{\\zeta}_1^{-2}\\hat{{\\mathbb{B}}}_{jj}$ with $\\hat{\\zeta}_1=n^{-1}\\sum_{i=1}^n\\|X_i-\\hat{{\\boldsymbol \\theta}}_n\\|^{-1}$, and $\\hat{{\\mathbb{B}}}_{jj}$ is the $j$th diagonal element of ${\\hat {\\mathbb{B}}}=n^{-1}\\sum_{i=1}^n\\|X_i-\\hat{{\\boldsymbol \\theta}}_n\\|^{-2}(X_i-\\hat{{\\boldsymbol \\theta}}_n)(X_i-\\hat{{\\boldsymbol \\theta}}_n)^{\\top}$.\n\nAccording to the proof of Theorem \\ref{theo:FDR} in Appendix A, $T_{n,j}$ converges in distribution to a standard normal under $H_{0j}$ for $j=1,\\ldots,p$.\nThus, we utilise the standard normal distribution to estimate the marginal $p$-values.\nFor $j=1,\\ldots,p$, define the $p$-value for $H_{0j}$ as\n$P_{j} = 2-2\\Phi(|T_{n,j}|)$.\nDenote $P_{(1)}\\leq \\cdots\\leq P_{(p)}$ be the ordered $p$-values, and define\n\\begin{eqnarray}\n\t\\notag \\hat{k} = \\max\\left\\{j=0,\\ldots,p: P_{(j)}\\leq \\tau j\/p\\right\\}\n\\end{eqnarray}\nfor a pre-specific significance level $\\tau$.\nThen, the B-H procedure rejects the null hypotheses for which $P_{j}\\leq P_{(\\hat{k})}$. Denote $\\mathcal{H}_{R}=\\{j:P_{j}\\leq P_{(\\hat{k})}\\}$ as the set of indices $j$ such that $H_{0j}$ is rejected by the B-H method, and let $|\\mathcal{H}_{R}|$ be the cardinality of $\\mathcal{H}_{R}$ that equals the total number of rejected null hypotheses.\n\nLet $\\mathcal{H}_0\\subset\\{1,\\ldots,p\\}$ be the set of indices $j$ corresponding to the true null hypotheses $H_{0j}$.\nThe false discovery proportion (FDP) and false discovery rate (FDR) of the B-H method are defined as\n\\begin{eqnarray}\n\t\\notag \\mathrm{FDP}_{M}=\\frac{|\\mathcal{H}_{0}\\cap\\mathcal{H}_{R}|}{|\\mathcal{H}_{R}|\\vee 1} \\text{~~and~~} \\mathrm{FDR}_{M} = \\mathbb{E}(\\mathrm{FDP}_{M}).\n\\end{eqnarray}\n\nRegarding that $T_{n,1},\\ldots,T_{n,p}$ are dependent, we impose the following condition on the weak dependence between any two components of $W_i$. Define $(r_{j\\ell})_{p\\times p}=\\{\\mathrm{diag}({\\mathbb{B}})\\}^{-1\/2}{\\mathbb{B}}\\{\\mathrm{diag}({\\mathbb{B}})\\}^{-1\/2}$ as the correlation matrix, where $\\mathrm{diag}({\\mathbb{B}})$ is the diagonal matrix of ${\\mathbb{B}}$.\n\n\\begin{condition}\\label{c5}\n\tSuppose $\\max_{1\\leq j , \\ell \\leq p} |r_{j\\ell}|\\leq r$ with some constant $0< r< 1$. In addition, $\\sum_{j=1}^p{\\mathbb{I}}(r_{j\\ell}=0)=O(p^{\\eta})$ for some constant $0<\\eta<(1-r)\/(1+r)$.\n\\end{condition}\n\nSimilar conditions are assumed in \\citet{Liu2014} and \\citet{Bell2018}.\nLet $p_0=|\\mathcal{H}_0|$ be the number of true null hypotheses and ${\\mathbb{B}}_{jj}$ be the $j$th diagonal element of ${\\mathbb{B}}$.\n\n\\begin{theorem}\\label{theo:FDR}\n\tSuppose Condition \\ref{c5} and the conditions of Theorem \\ref{theo1} hold. In addition, there exists $\\mathcal{H}\\subset\\{1,\\ldots,p\\}$ such that\n\t${\\mathcal H}=\\big\\{j: \\zeta_{1}{\\mathbb{B}}_{jj}^{-1\/2}n^{1\/2}|\\theta_{j}-\\theta_{0,j}|\\geq 2\\log^{1\/2} (p)\\big\\}$\n\tand $|\\mathcal H|\\geq \\log \\log p \\rightarrow \\infty$ as $p\\rightarrow \\infty$.  Assume that the number of false null hypotheses $p_1\\leq p^{\\varpi}$ for some $0<\\varpi<1$. Then, $\\mathrm{FDR}_{M}\/(\\tau p_0\/p)\\to1$ as $n \\rightarrow \\infty$.\n\\end{theorem}\n\nTheorem \\ref{theo:FDR}  shows the B-H procedure based on $P_1, \\ldots, P_p$ controls the FDR asymptotically, and it extends Theorem 4.1 in \\citet{Liu2014} to spatial median-based test statistic.\n\n\n\n\n\\subsection{High-dimensional asymptotic relative efficiency}\\label{sec:ARE}\n\n\nAs two candidate estimators of the location parameter ${\\boldsymbol \\theta}$, it is of interest to study the asymptotic relative efficiency (ARE) of the sample spatial median $\\hat{{\\boldsymbol \\theta}}_n$  relative to the sample mean $\\bar{X}_n$.\nWhen $p$ is fixed, for spherical multivariate normal distribution, \\citet{Brown1983} showed that the asymptotic efficiency of $\\hat{{\\boldsymbol \\theta}}_n$ relative $\\bar{X}_n$, denoted as $\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n},\\bar{X}_n)$, exceeds the usual univariate case $2\/\\pi$. \nIn addition, $\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n},\\bar{X}_n)$ increases as the dimension increases, and it approaches to $1$ as $p$ tends to be sufficient large \\citep{Magyar2011}. \nHowever, when $p\\to\\infty$, the ARE is not straightforward to quantify as there are no obvious ``final'' limit distributions for $\\hat{{\\boldsymbol \\theta}}_{n}$ and $\\bar{X}_n$.\nMotivated by the discussions in Sections \\ref{sec:SCIs} and \\ref{sec:global_tests}, we compare $\\hat{{\\boldsymbol \\theta}}_n$ and $\\bar{X}_n$ in terms of their efficiencies in simultaneous inference for ${\\boldsymbol \\theta}$, which are determined by the variations of $|\\hat{{\\boldsymbol \\theta}}-{\\boldsymbol \\theta}|_{\\infty}$ and $|\\bar{X}_n-{\\boldsymbol \\theta}|_{\\infty}$.\nAccording to Corollary \\ref{co1} and \\eqref{eq:gaussian_approx_mean}, we define the high-dimensional ARE of $\\hat{{\\boldsymbol \\theta}}_n$ compared to $\\bar{X}_n$ in simultaneous inference for ${\\boldsymbol \\theta}$ as\n\\begin{align}\\label{AE}\n\t\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)=\\mathrm{Var}(|G_0|_{\\infty})\/\\mathrm{Var}(|G|_{\\infty})\\,,\n\\end{align}\nwhich approximates $\\mathrm{Var}(|\\bar{X}_n-{\\boldsymbol \\theta}|_{\\infty})\/\\mathrm{Var}(|\\hat{{\\boldsymbol \\theta}}_{n}-{\\boldsymbol \\theta}|_{\\infty})$. If $\\lim_{p\\to\\infty}\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)>1$, we say that $\\hat{{\\boldsymbol \\theta}}_n$ is more efficient than $\\bar{X}_n$ in simultaneous inference for ${\\boldsymbol \\theta}$ under high-dimensionality.\n\nAs discussed in Remark \\ref{remark:B}, $G\\sim N(0, \\zeta_{1}^{-2}{\\mathbb{B}})$ with $\\zeta_{1}^{-2}{\\mathbb{B}}_{j\\ell}=\\zeta_{1}^{-2}p^{-1}\\omega_{i\\ell}$ for all $1\\leq j,\\ell\\leq p$.\nMoreover, we can show that $\\Sigma_{j\\ell} = {\\mathbb{E}}(\\nu_{i}^{2})\\omega_{j\\ell} + O(p^{-1\/2})$ similar to the proof of Lemma \\ref{lemma:SM_Q} in Appendix B of the supplementary material, where $\\Sigma_{j\\ell}$ is the $(j,\\ell)$th element of $\\Sigma$.\nThus, both the covariance matrix $\\Sigma$ and $\\zeta_{1}^{-2}{\\mathbb{B}}$ are proportional to $\\Omega$ asymptotically, and $\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)$ is approximately ${\\mathbb{E}}(\\nu_{i}^{2})\\zeta_{1}^{2}p$.\n\nAs $\\Sigma$ and $\\zeta_{1}^{-2}{\\mathbb{B}}$ are rarely known in practice, we use bootstrap approximation to estimate the value of $\\mathrm{Var}(|G_0|_{\\infty})\/\\mathrm{Var}(|G|_{\\infty})$.\nCombining Corollary \\ref{co2} and \\eqref{eq:boot_mean}, we propose using\n$$\n\\mathrm{Var}^{*}(|\\bar{X}_{n}^{*}|_{\\infty})\/ \\mathrm{Var}^{*}(|\\tilde{{\\boldsymbol \\theta}}_{n}|_{\\infty}),\n$$\nto estimate $\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)$.\n\n\\begin{example}\n\tSuppose $X_1, \\ldots X_n$ are i.i.d.~from $N({\\boldsymbol \\theta}, I_p)$, then $\\nu_i^2$ follows a chi-square distribution with $p$ degrees of freedom. \n\tIt follows that ${\\mathbb{E}}(\\nu_i^2)=p$ and ${\\mathbb{E}}(\\nu_i^{-1})=\\Gamma(p\/2-1\/2)\/\\{2^{1\/2}\\Gamma(p\/2)\\}$, where $\\Gamma(\\cdot)$ is the gamma function.. So the ARE is\n\t$\n\t\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)=p\\{\\Gamma(p\/2-1\/2)\\}^2\/\\{2^{1\/2}\\Gamma(p\/2)\\}^2.\n\t$\n\tUsing Stirling's formula,\n\t$ \\lim_{p\\to\\infty} \\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n) = 1. $\n\tThus, for high-dimensional Gaussian data, the sample spatial median has the same asymptotically efficiency as the sample mean in simultaneous inference for ${\\boldsymbol \\theta}$.\n\\end{example}\n\n\n\\begin{example}\n\tWhen the data are from the multivariate $t$-distribution with degrees of freedom $v>2$ and shape matrix $\\Omega=I_{p}$, $\\nu_i^2\/p\\sim F_{p, v}$, where $F_{p,v}$ is the $F$ distribution with parameters $p$ and $v$.\n\tThen, ${\\mathbb{E}}(\\nu_i^2)=pv\/(v-2)$ and ${\\mathbb{E}}(\\nu_i^{-1})=\\Gamma(v\/2+1\/2)\\Gamma(p\/2-1\/2)\/\\{v^{1\/2}\\Gamma(v\/2)\\Gamma(p\/2)\\}$. Thus, the ARE is\n\t$\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n)=(v-2)^{-1}p\\{\\Gamma(v\/2+1\/2)\\Gamma(p\/2-1\/2)\\}^2\/\\{\\Gamma(v\/2)\\Gamma(p\/2)\\}^2. $\n\tIt is clear that $\\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n},~ \\bar{X})>1$ for large enough $p$.~ In addition, \t\n\t$\n\t\\lim_{p\\to\\infty} \\mathrm{ARE}(\\hat{{\\boldsymbol \\theta}}_{n}, \\bar{X}_n) = 2(v-2)^{-1}\\{\\Gamma(v\/2+1\/2)\\}^2\/\\{\\Gamma(v\/2)\\}^2>1\\,.\n\t$\n\tThus, for high-dimensional $t$-distribution, the sample spatial median is asymptotically more efficient than the sample mean in simultaneous inference for ${\\boldsymbol \\theta}$.\n\\end{example}\n\nFigure \\ref{Fig1} plots the simulated values of $\\mathrm{Var}(|\\bar{X}_n|_{\\infty})\/\\mathrm{Var}(|\\hat{{\\boldsymbol \\theta}}_n|_{\\infty})$ with a range of dimensions and sample sizes under different models.\nFor Gaussian data, the relative efficiency kept increasing in $p$, and it approached $1$ as $p$ getting larger.\nFor the data simulated from multivariate $t$-distribution, the relative efficiency was greater than $1$ for all combinations of $n$ and $p$. This indicates that the sample spatial median is more efficiency than the sample mean for $t$-distribution.\nThe results were consistent under different covariance structure considered in the simulation.\n\n\\begin{figure}[hbt]\n\t\\centerline{\n\t\t\\includegraphics[width=14cm]{p2.pdf}\n\t}\n\t\\caption{Finite sample relative efficiency of $|\\hat{{\\boldsymbol \\theta}}_{n}|_{\\infty}$ compared to $|\n\t\t\\bar{X}_n|_{\\infty}$ based on $5000$ replications,  the data are generated from multivariate normal distribution (Gaussian) and $t$-distribution with $5$ degrees of freedom ($t_{5}$). The shape matrix $\\Omega=(\\rho^{|j-\\ell|})_{p\\times p}$ with $\\rho=0$ and $0.8$.}\n\t\\label{Fig1}\n\\end{figure}\n\n\\iffalse\n\\section{Extension to two-sample Gaussian and bootstrap approximations}\\label{sec:two-sample}\n\nIn this section, we extend the one-sample Gaussian and bootstrap approximations for the sample spatial median to the two-sample case. \nLet $X_1, \\ldots, X_{n_1}$ and $Y_1, \\ldots, Y_{n_2}$ be two independent sequences of i.i.d.~$p$-dimensional random vectors from two populations with cumulative distribution functions $F_X$ and $F_Y$, respectively. Assume that $X_{i_1}$ and $Y_{i_2}$ admit the following representations:\n\\begin{eqnarray}\n\t\\notag X_{i_1} = {\\boldsymbol \\theta}^X+ \\nu_{X,i_1}\\Gamma_XU_{X,i_1}, \\text{~~and~~} Y_{i_2} = {\\boldsymbol \\theta}^Y+ \\nu_{Y,i_2}\\Gamma_YU_{Y,i_2}\\,,\n\\end{eqnarray}\nwhere $\\nu_{X,i_1}$ is independent of the spatial sign of $U_{X,i_1}$, and $\\nu_{Y,i_2}$ is independent of the spatial sign of $U_{Y,i_2}$.\nThe shape  matrices are $\\Omega_X=\\Gamma_X\\Gamma_X^{\\top}$ and $\\Omega_Y=\\Gamma_Y\\Gamma_Y^{\\top}$.\nSimilarly, denote $W_{X,i_1}, R_{X,i_1}$ and $W_{Y,i_2}, R_{Y,i_2}$ as in \\eqref{eq:nota} with additional subscripts $X$ or $Y$ to distinguish populations.\nLet $U_{X,i_1}=(U_{X,i_1,1},\\ldots,U_{X,i_1,p})^{\\top}$ and $U_{Y,i_2}=(U_{X,i_2,1},\\ldots,U_{X,i_2,p})^{\\top}$.\nThe following condition is parallel to Conditions \\ref{c1} and \\ref{c2}.\n\\begin{condition}\\label{c4}\n\t$U_{X,i_1,1},\\ldots,U_{X,i_1,p}$ are i.i.d.~symmetric random variables satisfies ${\\mathbb{E}}(U_{X,i_1,j_1})=0$, ${\\mathbb{E}}(U_{X,i_1,j_1}^2)=1$ for $j_1=1,\\ldots,p$, and\n\t$U_{Y,i_2,1},\\ldots,U_{Y,i_2,p}$ are i.i.d.~symmetric with ${\\mathbb{E}}(U_{Y,i_2,j_2})=0$, ${\\mathbb{E}}(U_{Y,i_2,j_2}^2)=1$ for $j_2=1,\\ldots,p$,\n\tIn addition, $\\|U_{X,i_1,j_1}\\|_{\\psi_{\\alpha}}\\leq c_0$ and $\\|U_{Y,i_2,j_2}\\|_{\\psi_{\\alpha}}\\leq c_0$ with some constant $c_0>0$ and $1\\leq \\alpha\\leq 2$.\n\tThe moments $\\zeta_{X,k}={\\mathbb{E}}(R_{X,i_1}^{-k})$ and $\\zeta_{Y,k}={\\mathbb{E}}(R_{Y,i_2}^{-k})$ for $k=1,2,3,4$ exist for large enough $p$. In addition, there exist two positive constants $\\underline{b}$ and ${\\bar{B}}$ such that\n\t$\n\t{\\underline b}\\leq \\limsup_{p}{\\mathbb{E}}(R_{X,i_1}\/\\sqrt{p})^{-k} \\leq {\\bar{B}}\n\t$ and $\n\t{\\underline b}\\leq \\limsup_{p}{\\mathbb{E}}(R_{Y,i_2}\/\\sqrt{p})^{-k} \\leq {\\bar{B}}\n\t$ for $k=1,2,3,4$.\n\\end{condition}\n\nLet $\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}$ and $\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y}$ be the spatial medians\nbased on $X_1, \\ldots, X_{n_1}$ and $Y_1, \\ldots, Y_{n_2}$, respectively.\n\\begin{theorem}\\label{theo:two}\n\tUnder Condition \\ref{c4}, assume that $\\Omega_X$ and $\\Omega_Y$ belong to $\\mathcal{U}(a_0(p), \\underline{m}, \\bar{M})$ with $a_0(p)\\asymp p^{1-\\delta}$ for some positive constant $\\delta \\leq 1\/2$. Let $N=n_1+n_2$, assume that $b_1\\leq n_1\/N\\leq b_2$ for some universal constants $0<b_1<b_2<1$. In addition, if\n\t$\\log p=o(N^{1\/5})$ and $\\log N=o(p^{1\/3\\wedge\\delta})$,\n\n\n\n\tthen\n\t\\begin{eqnarray\n\t\t\\notag & & \\sup_{A\\in {\\mathcal A}^{{\\rm re}}}\\left|{\\mathbb{P}}\\left\\{n_1^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-{\\boldsymbol \\theta}^{X})-\\gamma_{n_1,n_2}n_2^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y}-{\\boldsymbol \\theta}^{Y})\\in A\\right\\}  - {\\mathbb{P}}(G_X+\\gamma_{n_1,n_{2}}G_Y\\in A)\\right|\n\t\t\\rightarrow 0\n\t\\end{eqnarray}\n\tas $N\\to\\infty$, where $\\gamma_{n_1,n_2}=n_1^{1\/2}n_2^{-1\/2}$, $G_X\\sim N(0, \\zeta_{X,1}^{-2}{\\mathbb{B}}_{X})$ with ${\\mathbb{B}}_X={\\mathbb{E}}(W_{X,1}W_{X,1}^{\\top})$ and \n\t$G_Y\\sim N(0, \\zeta_{Y,1}^{-2}{\\mathbb{B}}_{Y})$ with ${\\mathbb{B}}_Y={\\mathbb{E}}(W_{Y,1}W_{Y,1}^{\\top})$ are independent.\n\\end{theorem}\n\nThe condition $b_1\\leq n_1\/N\\leq b_2$ assumes that the sample sizes $n_1$ and $n_2$ are comparable.\nNote that \n$$n_1^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-{\\boldsymbol \\theta}^{X})-\\gamma_{n_1,n_2}n_2^{1\/2}(\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y}-{\\boldsymbol \\theta}^{Y})=n_1^{1\/2}\\big\\{(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-\\hat{{\\boldsymbol \\theta}}^{Y}_{n_2})-({\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y})\\big\\},$$\nthe distribution of $G_X+\\gamma_{n_1,n_{2}}G_Y$ can be used to approximation that of $n_1^{1\/2}\\big\\{(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-\\hat{{\\boldsymbol \\theta}}^{Y}_{n_2})-({\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y})\\big\\}$ over hyperrectangles. As the covariance matrices of $G_X$ and $G_Y$ are unknown and inaccessible in practice, we consider the following bootstrap approximation.\n\nLet $Z_{X,1},\\ldots,Z_{X,n_1}; Z_{Y,1},\\ldots,Z_{Y,n_2}$ be i.i.d.~Rademacher random variables. \nDefine the bootstrap version of the $\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}$ and $\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y}$ as\n\\begin{align}\n\t\\notag \\tilde{{\\boldsymbol \\theta}}_{n_1}^{X}= \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^p}\\sum_{i=1}^{n_1}\\|Z_{X,i}(X_i-\\hat{{\\boldsymbol \\theta}}_{n_1}^{X})-{\\boldsymbol \\beta}\\|, {\\rm ~and~}\n\t\\tilde{{\\boldsymbol \\theta}}_{n_2}^{Y}= \\argmin_{{\\boldsymbol \\beta}\\in {\\mathbb R}^p}\\sum_{j=1}^{n_2}\\|Z_{Y,j}(Y_j-\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y})-{\\boldsymbol \\beta}\\|\\,.\n\\end{align}\nThen, the distribution of $n_1^{1\/2}(\\tilde{{\\boldsymbol \\theta}}_{n_1}^{X}-\\tilde{{\\boldsymbol \\theta}}_{n_2}^{Y})$ conditional on $X_1, \\ldots, X_{n_1}$ and $Y_1, \\ldots, Y_{n_2}$ is used to approximate that of $n_1^{1\/2}\\big\\{(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y})-({\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y})\\big\\}$. \n\n\\begin{theorem}\\label{theo:boottwo}\n\tUnder the conditions assumed in Theorem \\ref{theo:two}, as $N\\to\\infty$,\n\t\\begin{eqnarray}\n\t\t\\notag & & \\sup_{A\\in {\\mathcal A}^{{\\rm re}}}\\left|{\\mathbb{P}}\\left(n_1^{1\/2}\\big\\{(\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y})-({\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y})\\big\\}\\in A\\right)  - {\\mathbb{P}}^{*}\\left\\{n_1^{1\/2}(\\tilde{{\\boldsymbol \\theta}}_{n_1}^{X}-\\tilde{{\\boldsymbol \\theta}}_{n_2}^{Y})\\in A\\right\\}\\right| \\to 0\\, \n\t\\end{eqnarray}\n\tin probability.\n\\end{theorem}\n\nSame to the one-sample case, the multiplier bootstrap is valid under an exponentially divergent $p$.\nLet $q^{\\mathrm{TS},B}_{1-\\alpha}$ be the $(1-\\alpha)$th quantile of $\\sqrt{n_1}|\\tilde{{\\boldsymbol \\theta}}_{n_1}^{X}-\\tilde{{\\boldsymbol \\theta}}_{n_2}^{Y}|_{\\infty}$ given $X_1,\\ldots,X_{n_1}$ and $Y_1,\\ldots,Y_{n_2}$.  Similar to the one-sample case, an asymptotic $1-\\alpha$ simultaneous confidence intervals for ${\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y}$ is\n\\begin{eqnarray\n\t\\notag {\\mathcal C}_{\\alpha}^{\\mathrm{TS}}=\\left\\{{\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y}\\in {\\mathbb{R}}^p, n_1^{1\/2}|(\\tilde{{\\boldsymbol \\theta}}_{n_1}^{X}-\\tilde{{\\boldsymbol \\theta}}_{n_2}^{Y})-({\\boldsymbol \\theta}^{X}-{\\boldsymbol \\theta}^{Y})|_{\\infty}<q^{\\mathrm{TS},B}_{1-\\alpha}\\right\\}.\n\\end{eqnarray}\n\nFor the two-sample global testing problem, suppose we are interested in testing\n\\begin{eqnarray\n\t\\notag H_0^{\\mathrm{TS}}: {\\boldsymbol \\theta}^{X} = {\\boldsymbol \\theta}^{Y} \\text{~~versus~~} H_1^{\\mathrm{TS}}:{\\boldsymbol \\theta}^{X}\\neq{\\boldsymbol \\theta}^{Y}.\n\\end{eqnarray}\nA direct test statistic is\n$T_{N}^{\\mathrm{TS}} = n^{1\/2}\\left|\\hat{{\\boldsymbol \\theta}}_{n_1}^{X}-\\hat{{\\boldsymbol \\theta}}_{n_2}^{Y}\\right|_{\\infty}$,\nand the null hypothesis $H_{0}^{\\mathrm{TS}}$ is rejected when $T_{N}^{\\mathrm{TS}}>q_{1-\\alpha}^{\\mathrm{TS},B}$.\nThe size and power analysis of the two-sample test based on $T_{N}^{\\mathrm{TS}}$ is similar to the one-sample case and is thus straightforward.\n\\fi\n\n\\section{Numerical studies}\\label{sec:numerical}\n\nIn this section, we report Monte Carlo simulations on simultaneous confidence intervals and multiple testing with FDR control, along with a real data analysis, to demonstrate the performance of the proposed approaches. Additional simulations on global tests can be found in Appendix C of the supplementary material.\nIn the simulations, all results were based on $2500$ replications. In the bootstrap implementation, the number of bootstrap iterations was set to $B=400$. \n\n\\subsection{Simulations on simultaneous confidence intervals}\\label{sec:simulations}\n\nWe first examine the performance of the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$, and compare it with the SCIs based on $\\bar{X}_n$.\nThe sample size $n$ is taken to be $100$ or $200$, and the dimensions $p= 100$ and $1000$ are considered for each sample size. \nTwo types of commonly used elliptical distributions are considered: (I) the multivariate normal distribution $N({\\boldsymbol \\theta},\\Sigma)$; (II) the multivariate $t$-distribution with $3$ degrees of freedom, mean vector ${\\boldsymbol \\theta}$, and covariance matrix $\\Sigma$.\nIn addition, we include the following independent components model: (III) $X_i={\\boldsymbol \\theta} + \\Sigma^{1\/2}Z_i$, where each component of $Z_i$ are i.i.d.~from the standard Laplace distribution.\nWe set $\\Sigma=(\\rho^{|j-\\ell|})$ with $\\rho=0, 0.2, 0.5$ and $0.8$.\nTo save space, we present the results for $\\rho=0$ and $0.8$ here. \nThe results for $\\rho\\in\\{0.2,0.5\\}$ are similar and are reported in the supplementary material. \nWe consider both sparse and dense case scenarios for ${\\boldsymbol \\theta}$: (i) ${\\boldsymbol \\theta}_{1}=(2,-2,3,0,\\ldots, 0)$; (ii) ${\\boldsymbol \\theta}_{2}=(0.2,\\ldots,0.2_{\\lfloor p\/4\\rfloor},0,\\ldots,0)$. Here $\\lfloor\\cdot\\rfloor$ is the floor function.\n\nTable \\ref{tab1} reports the coverage probability and median length of the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$, the results of the SCIs based on $\\bar{X}_n$ are presented in parentheses. \nFor Models I and II from elliptical distributions, we observe that the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$ and $\\bar{X}_n$ both achieve satisfying coverage probability for different choices for $\\rho$, ${\\boldsymbol \\theta}$, $n$ and $p$. \nFor the data simulated from the multivariate normal distribution, the median length of the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$ is very close to that of the the SCIs based on $\\bar{X}_n$. \nThese results indicate that $\\hat{{\\boldsymbol \\theta}}_n$ has similar asymptotic efficiency as $\\bar{X}_n$ in simultaneous inference for ${\\boldsymbol \\theta}$ under high-dimensional Gaussian model as discussed in Section \\ref{sec:ARE}.\nFor the multivariate $t$-distribution, the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$ is much narrower than the SCIs based on $\\bar{X}_n$.\nThese results suggest that the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$ is more efficient than the SCIs based on $\\bar{X}_n$ for multivariate $t$-distribution, which is heavy-tailed.\nThis is consistent with the asymptotic analysis in Section \\ref{sec:ARE}.\nMoreover, the results for Model III, which does not belong to the elliptical distribution family, shows the robustness of the SCIs based on the spatial median, and it performs similar to the SCIs based on the sample mean.\nWe also note that the median length of the SCIs decreases when $n$ increases or $p$ decreases for each model.\n\n\\begin{table}[ht]\n\t\\footnotesize\n\t\\centering\n\t{\n\t\t\\caption{Coverage probability (in $\\%$) and median length of the SCIs based on $\\hat{{\\boldsymbol \\theta}}_n$, the results of the SCIs based on $\\bar{X}_n$ are in parentheses.}\n\t\t\\label{tab1}\n\t\t\\setlength\\tabcolsep{3pt}\n\t\t\\resizebox{\\textwidth}{!}{\n\t\t\t\\begin{tabular}{@{}cccccccccccccc@{}}\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t&&&&\\multicolumn{4}{c}{${\\boldsymbol \\theta}={\\boldsymbol \\theta}_1$}& &\\multicolumn{4}{c}{${\\boldsymbol \\theta}={\\boldsymbol \\theta}_2$}\\cr\n\t\t\t\t&&&&\\multicolumn{2}{c}{Coverage probability}&\\multicolumn{2}{c}{Median length}&&\\multicolumn{2}{c}{Coverage probability}&\\multicolumn{2}{c}{Median length}\\cr\n\t\t\t\t\n\t\t\t\tModel & $\\rho$ & $n$&$p$& 90\\% & 95\\% & 90\\% & 95\\%& &90\\% & 95\\% & 90\\% & 95\\%\\cr\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\tI & 0 &100&~$100$ & 89.6 (89.9) & 94.4 (94.4) & 0.65 (0.65) & 0.69 (0.69)\n\t\t\t\t&& 88.9 (88.8) & 94.1 (93.9) & 0.65 (0.65) & 0.69 (0.69) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 89.5 (89.6) & 94.7 (94.4) & 0.77 (0.77) & 0.80 (0.80)\n\t\t\t\t&& 89.5 (89.5) & 94.0 (94.0) & 0.77 (0.77) & 0.81 (0.80) \\cr\n\t\t\t\t\n\t\t\t\t&&200&~$100$& 89.8 (89.8) & 95.1 (95.1) & 0.46 (0.46) & 0.49 (0.49)\n\t\t\t\t&& 88.6 (88.8) & 94.4 (94.7) & 0.46 (0.46) & 0.49 (0.49) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 89.7 (89.7) & 94.4 (94.6) & 0.55 (0.55) & 0.57 (0.57)\n\t\t\t\t&& 89.1 (89.2) & 94.7 (94.6) & 0.55 (0.55) & 0.57 (0.57) \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\cline{2-13}\n\t\t\t\t\n\t\t\t\t& 0.8 &100&~$100$ & 89.1 (88.7) & 94.6 (94.6) & 0.64 (0.63) & 0.68 (0.67)\n\t\t\t\t&& 88.4 (88.6) & 93.7 (94.1) & 0.64 (0.63) & 0.68 (0.67) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 88.4 (88.4) & 93.8 (93.7) & 0.76 (0.76) & 0.80 (0.79)\n\t\t\t\t&& 89.0 (89.2) & 94.6 (94.6) & 0.76 (0.76) & 0.80 (0.79) \\cr\n\t\t\t\t\n\t\t\t\t&&200&~$100$& 90.5 (90.1) & 95.2 (94.9) & 0.45 (0.45) & 0.48 (0.48)\n\t\t\t\t&& 89.6 (89.6) & 94.0 (94.1) & 0.45 (0.45) & 0.48 (0.48) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 90.4 (90.4) & 94.5 (94.4) & 0.54 (0.54) & 0.56 (0.56)\n\t\t\t\t&& 88.4 (88.5) & 93.6 (93.8) & 0.54 (0.54) & 0.56 (0.56) \\cr\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\tII & 0 &100&~$100$ & 89.7 (88.6) & 94.7 (93.7) & 0.71 (1.05) & 0.75 (1.11)\n\t\t\t\t&& 88.8 (88.8) & 94.5 (94.2) & 0.71 (1.05) & 0.75 (1.11) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 89.4 (91.0) & 95.8 (95.0) & 0.84 (1.25) & 0.88 (1.30) \n\t\t\t\t&& 89.1 (89.0) & 94.4 (94.5) & 0.84 (1.25) & 0.88 (1.31) \\cr\n\t\t\t\t\n\t\t\t\t&&200&~$100$& 88.6 (89.1) & 94.2 (95.1) & 0.50 (0.76) & 0.53 (0.81)\n\t\t\t\t&& 89.5 (89.7) & 94.4 (94.8) & 0.50 (0.76) & 0.53 (0.80) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 89.6 (88.7) & 94.8 (94.6) & 0.59 (0.90) & 0.62 (0.94)\n\t\t\t\t&& 90.1 (89.5) & 94.8 (93.9) & 0.59 (0.90) & 0.62 (0.94) \\cr\n\t\t\t\t\n\t\t\t\t\\cline{2-13}\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.8 &100&~$100$ & 89.1 (90.7) & 94.4 (94.9) & 0.69 (1.02) & 0.74 (1.09)\n\t\t\t\t&& 89.4 (89.7) & 94.2 (94.4) & 0.69 (1.02) & 0.74 (1.09) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 89.3 (89.1) & 94.6 (94.4) & 0.83 (1.23) & 0.87 (1.29)\n\t\t\t\t&& 89.8 (88.8) & 94.7 (94.4) & 0.83 (1.23) & 0.87 (1.29) \\cr\n\t\t\t\t\n\t\t\t\t&&200&~$100$& 87.6 (87.7) & 93.4 (93.6) & 0.49 (0.73) & 0.52 (0.78)\n\t\t\t\t&& 90.3 (90.1) & 94.9 (95.2) & 0.49 (0.73) & 0.52 (0.78) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$& 88.7 (89.7) & 94.7 (94.6) & 0.59 (0.88) & 0.61 (0.92)\n\t\t\t\t&& 90.2 (90.8) & 94.7 (95.7) & 0.59 (0.89) & 0.61 (0.93) \\cr\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\tIII&0&100&$100$& 89.8 (89.4) & 94.6 (94.5) & 0.65 (0.66) & 0.69 (0.70)\n\t\t\t\t&& 89.1 (89.0) & 94.4 (94.4) & 0.65 (0.66) & 0.69 (0.70) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$ & 88.3 (88.2) & 93.6 (93.7) & 0.78 (0.78) & 0.82 (0.82)\n\t\t\t\t&& 89.1 (89.0) & 94.2 (93.8) & 0.78 (0.78) & 0.82 (0.82) \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t&&200&~$100$ & 90.6 (91.1) & 95.0 (95.0) & 0.46 (0.46) & 0.49 (0.49)\n\t\t\t\t&& 90.6 (90.1) & 95.2 (95.2) & 0.46 (0.46) & 0.49 (0.49) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$ & 90.1 (90.4) & 95.0 (94.6) & 0.55 (0.55) & 0.57 (0.58)\n\t\t\t\t&& 88.7 (89) & 93.6 (93.8) & 0.55 (0.55) & 0.57 (0.58)\\cr\n\t\t\t\t\n\t\t\t\t\\cline{2-13}\n\t\t\t\t\n\t\t\t\t&0.8&100&$100$& 90.4 (89.7) & 95.0 (94.8) & 0.63 (0.63) & 0.68 (0.68)\n\t\t\t\t&& 89.0 (88.9) & 95.0 (94.9) & 0.63 (0.63) & 0.67 (0.68) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$ & 88.7 (88.9) & 93.8 (94.0) & 0.77 (0.77) & 0.80 (0.80)\n\t\t\t\t&& 89.0 (89.0) & 94.6 (94.3) & 0.76 (0.76) & 0.80 (0.80) \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t&&200&~$100$ & 88.8 (89.1) & 94.2 (94.0) & 0.45 (0.45) & 0.48 (0.48)\n\t\t\t\t&& 90.2 (89.7) & 94.8 (95.0) & 0.45 (0.45) & 0.48 (0.48) \\cr\n\t\t\t\t\n\t\t\t\t&&&$1000$ & 90.0 (90.3) & 95.0 (95.0) & 0.54 (0.54) & 0.57 (0.57)\n\t\t\t\t&& 88.8 (89.1) & 94.2 (94.1) & 0.54 (0.54) & 0.57 (0.57) \\cr\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t}\n\\end{table}\n\n\\subsection{Simulations on multiple testing with FDR control}\n\nIn this section, we examine the performance of the sample spatial median-based B-H method introduced in Section \\ref{sec:FDR}, and compare it to the B-H procedure based on the sample mean with p-values calculated from $N(0,1)$ in \\citet{Liu2014}. We set $\\theta_{0,j}=0$ for all $j=1,\\ldots,p$. The data are generated from Models I and II with $p=1000$. For ${\\boldsymbol \\theta}=(\\theta_1,\\ldots,\\theta_{p})^{\\top}$, let $\\theta_j=2(\\log p\/n)^{1\/2}$ for $1\\leq j\\leq p_1$ and $\\theta_j=0$ for $(p_1+1)\\leq j\\leq p$, where\n$p_1=0.1p$.\n\nTable \\ref{tab2} reports the empirical FDR and power for the sample spatial median-based ($\\mathrm{FDR}_{M}$ and $\\mathrm{power}_{M}$) and the sample mean-based ($\\mathrm{FDR}_{A}$ and $\\mathrm{power}_{A}$) B-H procedures \\citep{Liu2014} with nominal level $\\alpha=0.1$ and $0.2$. The results indicate that the FDR are well controlled by both methods. For the multivariate normal distribution, the B-H procedures based on the spatial median and the sample mean have similar performance. However, the sample spatial median-based B-H method outperforms the sample mean-based B-H procedure in terms of empirical power under multivariate $t$-distribution, which is heavy-tailed.\n\n\\begin{table}[ht]\n\t\\centering\n\t{\n\t\t\\caption{Empirical FDR and power for the spatial median-based ($\\mathrm{FDR}_{M}$ and $\\mathrm{power}_{M}$) and the sample mean-based ($\\mathrm{FDR}_{A}$ and $\\mathrm{power}_{A}$) in \\citet{Liu2014} via B-H procedures.}\n\t\t\\label{tab2}\n\t\t\\setlength\\tabcolsep{3pt}\n\t\n\t\t\t\\begin{tabular}{@{}ccccccccccccc@{}}\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\n\t\t\t\t&&&\\multicolumn{4}{c}{$\\alpha=0.1$}& &\\multicolumn{4}{c}{$\\alpha=0.2$}\\cr\n\t\t\t\n\t\t\t\t\n\t\t\t\tModel & $\\rho$ & $n$& $\\mathrm{FDR}_{M}$ & $\\mathrm{FDR}_{A}$ & $\\mathrm{power}_{M}$ & $\\mathrm{power}_{A}$ & & $\\mathrm{FDR}_{M}$ & $\\mathrm{FDR}_{A}$ & $\\mathrm{power}_{M}$ & $\\mathrm{power}_{A}$ \\cr\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\n\t\t\t\tI & 0 &50& 0.124 & 0.124 & 0.996 & 0.996\n\t\t\t\t&& 0.224 & 0.222 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.107 & 0.106 & 0.997 & 0.997\n\t\t\t\t&& 0.202 & 0.201 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.2 &50& 0.125 & 0.124 & 0.996 & 0.996\n\t\t\t\t&& 0.224 & 0.223 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.107 & 0.106 & 0.997 & 0.997\n\t\t\t\t&& 0.202 & 0.201 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.5 &50& 0.125 & 0.124 & 0.996 & 0.996\n\t\t\t\t&& 0.225 & 0.223 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.107 & 0.105 & 0.997 & 0.997\n\t\t\t\t&& 0.202 & 0.201 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.8 &50& 0.127 & 0.124 & 0.996 & 0.996\n\t\t\t\t&& 0.227 & 0.223 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.108 & 0.105 & 0.997 & 0.997\n\t\t\t\t&& 0.204 & 0.199 & 0.999 & 0.999 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\tII & 0 &50& 0.117 & 0.099 & 0.984 & 0.728\n\t\t\t\t&& 0.215 & 0.193 & 0.992 & 0.805 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.103 & 0.088 & 0.987 & 0.710\n\t\t\t\t&& 0.197 & 0.179 & 0.994 & 0.795 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.2 &50& 0.117 & 0.098 & 0.984 & 0.727\n\t\t\t\t&& 0.215 & 0.194 & 0.992 & 0.805 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.103 & 0.087 & 0.987 & 0.709\n\t\t\t\t&& 0.198 & 0.179 & 0.994 & 0.795 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.5 &50& 0.118 & 0.099 & 0.984 & 0.727\n\t\t\t\t&& 0.216 & 0.194 & 0.992 & 0.803 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.103 & 0.087 & 0.987 & 0.708\n\t\t\t\t&& 0.198 & 0.178 & 0.994 & 0.794 \\cr\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t& 0.8 &50& 0.120 & 0.098 & 0.984 & 0.724 \n\t\t\t\t&& 0.218 & 0.192 & 0.992 & 0.800 \\cr\n\t\t\t\t\n\t\t\t\t&&100& 0.104 & 0.087 & 0.987 & 0.705\n\t\t\t\t&& 0.199 & 0.177 & 0.994 & 0.791 \\cr\n\t\t\t\t\n\t\t\t\t\\hline \n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t\n\t}\n\\end{table}\n\n\n\n\\subsection{Real data analysis}\\label{sec:real_data}\n\nType 2 diabete\nis a disease in which the body becomes resistant to normal effects of insulin and gradually loses the capacity to produce enough insulin.\nBecause skeletal muscle is the main tissue for insulin-stimulated glucose disposal, skeletal muscle insulin resistance is commonly viewed as the critical component of whole-body insulin resistance, and thus is critical to the pathogenesis of Type 2 diabetes.\nTo investigate the effects of insulin on gene expression in skeletal muscle, a microarray study was performed in 15 diabetic patients using the Affymetrix Hu95A chip of muscle biopsies both before and after insulin treatment \\citep{Wu2007}. \nIn this paper, we are interested in the gene expression alteration, that is, the change of the gene expression level, due to the treatment. \nThe data are available at \\texttt{https:\/\/www.ncbi.nlm.nih.gov\/geo\/query\/acc.cgi?acc=GSE22309}.\nThe data were normalized by quantile normalization by the \\texttt{normalizeQuantiles} function in the \\texttt{limma} R package.\nFollow \\citet{Wang2015}, we focused on $2547$ curated gene sets with at least 15 genes, which are from the C2 collection of the GSEA online pathway databases. \nThe gene expression values are consolidated by taking the average when multiple probes are associated with the same gene.\n\nWe implemented the Median test based on $T_n$ on the $2519$ gene sets.\nThis is equivalent to testing whether the median change vector of gene expression levels is equal to 0.\nThe number of bootstrap iterations is $B=10^5$.\nWith the Bonferroni correction, there are $1242$ gene sets identified as significant at $5\\%$ level.\nFor comparison, we applied the WPL test \\citep{Wang2015} and the CQ test \\citep{Chen2010} on the same gene sets. \nFor the WPL test, $1060$ gene sets are selected as significant; and for the CQ test, $630$ gene sets are identified as significant. \nOut of the $630$ gene sets selected by the CQ test, $605$ of them are also identified by our proposed method, and $629$ of them are identified by the WPL test.\nIt has been argued in \\citet{Wang2015} that some gene expression levels have heavy tails as their kurtosises are much larger than the kurtosis of a normal distribution, 3. \nThus, the methods based on the spatial median (Median test and the WPL test) are expected to be more robust and efficient than those based on moments (CQ test).\nIn addition, out of the $1060$ gene sets identified by the WPL test, $958$ of them are significant based on our proposed approach.\n\n\nAs argued in Remark \\ref{remark:07}, the Median test based on $T_n$ is more powerful in detecting strong sparse signal compared to the WPL test.\nTo see this, we look into the following three gene sets: \\\\\n(1) ZHAN\\_MULTIPLE\\_MYELOMA\\_UP; \\\\\n(2) MIKKELSEN\\_MEF\\_HCP\\_WITH\\_H3K27ME3; \\\\\n(3) JAZAG\\_TGFB1\\_SIGNALING\\_VIA\\_SMAD4\\_UP. \\\\\nThe p-values of the WPL test for these three gene sets are $0.41$, $0.31$, $0.27$, respectively.\nHowever, the p-values of the Median test are all less than $1.0\\times 10^{-5}$ with $B=10^5$ bootstrap iterations for these three gene sets.\nFigure \\ref{Fig_real_data} \nplots the SCIs for the spatial median vectors of the change of gene expression levels for these three gene sets.\nThe confidence intervals that do not cover $0$ are colored in red.\nIt is very clear that the only one or two big values in the spatial median results in a rejection of the Median test, while the signals from other dimensions are not strong enough to land a rejection by the the WPL test.\n\n\\begin{figure}[htp]\n\t\\centerline{\n\t\t\\includegraphics[width=14cm,height=16cm]{genePlot.pdf}\n\t}\n\t\\caption{Simultaneous Confidence intervals (SCIs) for spatial medians of three gene sets. \n\t\n\t}\n\t\\label{Fig_real_data}\n\\end{figure}\n\n\nFinally, we use the spatial median-based B-H procedure to perform multiple testing with FDR control on the three gene sets to detect differentially expressed genes (DEG), which is one of the most important targets in genomic analysis.\nTable \\ref{tab3} reports the detected differentially expressed genes (DEG) in each gene set with nominal level $\\alpha=0.1$, along with the corresponding marginal p-value $P_{j}=2-2\\Phi(|T_{n,j}|)$ and the confidence interval in the SCIs for the selected genes.\nIt can be seen that for all the selected genes, the marginal p-values are very small, and the corresponding confidence intervals do not cover $0$.\n\n\n\\begin{table}[htp]\n\t\\centering\n\t{\n\t\t\\caption{Detected differentially expressed genes (DEG) by the spatial median-based B-H procedure for three gene sets with $\\alpha=0.1$; ``$p$-value'' refers to the marginal $p$-value $P_{j}=2-2\\Phi(|T_{n,j}|)$, and ``CI'' refers to the confidence interval in the SCIs for the selected genes.}\n\t\t\\label{tab3}\n\t\t\\setlength\\tabcolsep{3pt}\n\t\t\\resizebox{\\textwidth}{!}{\n\t\t\t\\begin{tabular}{@{}lccc@{}}\n\t\t\t\t\\hline\n\t\t\t\t\n\t\t\t\tGene set & DEG & p-value & CI \\\\\n\t\t\t\t\n\t\t\t\t\\hline \n\t\t\t\t\n\t\t\t\tZHAN\\_MULTIPLE\\_MYELOMA\\_UP & CDKN1A & 0.00082 & (0.234, 0.550) \\\\\n\t\t\t\t\n\t\t\t\tMIKKELSEN\\_MEF\\_HCP\\_WITH\\_H3K27ME3 & MYOD1 & $<0.00001$ & (0.433, 0.791) \\\\\n\t\t\t\t\n\t\t\t\tJAZAG\\_TGFB1\\_SIGNALING\\_VIA\\_SMAD4\\_UP & HDAC4 & 0.00058 & (0.254, 0.644) \\\\\n\t\t\t\t\n\t\t\t\t\\hline \n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t}\n\t}\n\\end{table}\n\n\\section{Discussion}\\label{sec:conclusion}\n\nIn this paper, we established one-sample and two-sample Gaussian and bootstrap approximations for ultrahigh dimensional sample spatial median under a general model beyond elliptical distributions.\nIt is of interest to study whether our results are potentially extendable to some other distribution families. \nWe leave this to a future work.\nIn addition, the proposed test based on the maxima of the sample spatial median is more powerful under sparse alternatives compared to those based on $L_2$-norms.\nIt is well known that the $L_2$-norm type tests are more powerful under dense alternatives. Thus, it is of interest to consider combining the test based on the maximum-norm and $L_2$-norm, which could be potentially powerful under both sparse and dense alternatives. We also leave this to a future study.\n\n\n\n\n\\section*{Supplementary material}\nThe supplementary material\nincludes all the technical proofs and  some additional numerical results.\n\n\n\\bibliographystyle{agsm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec-intro}\n\\nobreak\nThe Gemini NICI Planet-Finding Campaign was a direct-imaging survey of\n\\new{about} 250 nearby stars for substellar and planetary-mass companions\nconducted at Gemini South Observatory between 2008 and 2012\n\\citep{2010SPIE.7736E..53L}.  The Campaign used the Near-Infrared\nCoronagraphic Imager \\citep[NICI,][]{2008SPIE.7015E..49C}, which\ncombines adaptive optics, coronagraphy, angular differential imaging\n\\citep{2004Sci...305.1442L, 2006ApJ...641..556M}, and dual-channel\nmethane-band infrared imaging to achieve an H-band contrast\ndetection limit of 14.4~magnitudes at 1$''$ radius\n\\citep{2013ApJ...779...80W}.  The principal scientific results have\nbeen published by \n\\citet{2010ApJ...720L..82B}, \n\\citet{2011ApJ...729..139W},\n\\citet{2012ApJ...750...53N},\n\\citet{2013ApJ...773..179W},\n\\citet{2013ApJ...776....4N}, and \n\\citet{2013ApJ...777..160B}; \nthe pipeline processing algorithm is described by\n\\citet{2013ApJ...779...80W}.\n\nThe multi-epoch imaging data acquired for the Campaign require\naccurate astrometric calibration in order to detect common proper\nmotion and parallax between the host star and any candidate companions and\ntherefore distinguish true companions from background stars.  In this\npaper, we describe the process used to establish the alignment of the NICI science\ndetectors relative to the celestial coordinate system and calibrate the World\nCoordinate System (WCS) contained in the data headers.\n\n\n\\section{Instrument Design and Data Format}\n\\label{sec-instdesign}\n\\nobreak\nNICI consists of two parts: an adaptive optics (AO) system and a\ndual-channel science camera \\citep{2008SPIE.7015E..49C}.  The\nAO-corrected wavefront entering the science camera first passes\nthrough a focal plane coronagraph mask and a pupil\/spider mask, then is divided\nbetween the two science channels by the\n{\\it Dichroic Wheel} (DW).  Light transmitted through the DW enters the\n{\\it Red} channel while reflected light enters the {\\it Blue}. \nThe DW can be set to one of the following elements: {\\it H-50\/50}\nbeamsplitter, which directs 50\\% of H-band (1.65~\\micron) light to\neach science channel; {\\it H\/K Dichroic}, which divides light between the H and\nK (2.2~\\micron) bands; {\\it Mirror}, which reflects all light to the\nBlue channel; or {\\it Open}, which passes all light to the Red channel.\n\nAfter the DW, each science channel contains a filter wheel, reimaging\noptics, and an ALADDIN~II 1024$\\times$1024 InSb array detector.  The\nCampaign most frequently used 4\\% bandpass filters centered in and out\nof the $\\lambda = 1.63~\\mu$m methane absorption feature (named\nCH4-H4\\%L and CH4-H4\\%S, respectively) to search for methane-bearing\nplanets very close to the primary star, and a broadband H filter for\ndeeper searches at larger separations.  The on-sky pixel size is\n$\\approx 18$~milli-arcseconds (mas) and the field size is $\\approx\n18\\farcs4 \\times 18\\farcs4$.  The mapping of the sky onto each of the\ntwo science detectors is different and changes with the DW element and\nfilter.\n\nNICI data are processed by the Gemini Data-Handling System (DHS) and\nare written to Multi-Extension FITS (MEF) files\n\\citep{1981A&AS...44..363W, 2010A&A...524A..42P} which contain a\nPrimary Header followed by two extensions, one for each of the two\nscience channels.  The header for each extension contains its own WCS\nspecifying the astrometric parameters for that channel\n\\citep{2002A&A...395.1061G}.  The calibration of the WCS for each channel is a\nprincipal subject of this work.\n\nDetails on the data format, WCS, and other header data are specified\nin the Appendix, and more information on NICI is available at\n\\url{http:\/\/www.gemini.edu\/sciops\/instruments\/nici}.\n\n\n\\section{Astrometric Requirements}\n\\label{sec-astreq}\n\\nobreak\nThe astrometric calibration requirements for the NICI Planet-Finding\nCampaign arise from three aspects of the observing strategy.\n\nFirst, most Campaign data were taken in a mode called \n{\\it Angular Differential Imaging} \n\\citep[ADI,][]{2004Sci...305.1442L, 2006ApJ...641..556M}, \nin which the Cassegrain rotator is held fixed and the sky \nallowed to rotate on the science detectors so that astronomical\nobjects surrounding the primary star under observation move relative\nto the fixed speckles in the NICI point-spread function (PSF).  \nThe individual $t \\approx 1$ minute exposures taken over a range\nof field angles (in some cases $> 90^\\circ$) were \nderotated and registered before being co-added by the \npipeline-processing software.  To achieve accurate alignment over the \nentire field, image distortion must be corrected before derotation.\n\nSecond, targets were imaged simultaneously in the two science channels\nin and out of the $\\lambda = 1.63~\\mu$m methane absorption feature, in a mode\nknown as {\\it Angular Spectral Differential Imaging} (ASDI), to\nsuppress the stellar continuum further and enhance potential\nmethane-bearing planets.  The distortion, rotation and scale\ndifference between the two channels must be corrected before the\npipeline can accurately register and subtract the two images,\nespecially when they are added in order to find non-methane-bearing\nplanets.  In practice, sufficient accuracy can be achieved because\nthese differences are static properties of the instrument that can be\nmeasured once and applied to all data.\n\nThird, the Campaign strategy required measuring the relative proper\nmotion (PM) between a target star and any candidate companions\ndetected in the surrounding field in order to distinguish true\ncompanions (with common PM) from background stars.  Because only the\nrelative PM is required, we are not concerned with absolute astrometry\n(i.e. the absolute celestial coordinate zero points), but only with\nthe image distortion, field rotation, and pixel scale.  \\new{In\norder to assess proper motion and parallax robustly, this calibration\nhad to be maintained over multi-year timescales}, during which time NICI\nwas dismounted and remounted on the telescope, sometimes on different\ninstrument ports with different numbers of reflections in the optical\npath to the instrument.  Therefore, the astrometric calibration\nrequired observations of astrometric fields during each instrument\nmounting.  In addition, because a field observed at multiple epochs was\nnot typically observed at the same hour angle each time, the field\nrotation was often significantly different, again requiring correction of the\nimage distortion before the two datasets could be compared.\n\nThe required astrometric accuracy is dictated by the PM of the target\nstars.  In the Campaign's target list of nearby stars, the lowest PMs\nare \\new{ $\\approx 20$~mas yr$^{-1}$, or $\\approx 1$~NICI pixel yr$^{-1}$};\nmost are at least a few times larger.  \\new{An accuracy of 1~pixel\n  from the center of the NICI field (the usual location of a primary\n  star) to the edge} therefore permits most candidate companions to be\nchecked for common PM within one year, especially since accuracy near\nthe field center will be considerably better.\n\nFor the pixel scale, 1~pixel out of \\new{512 is 0.2\\%}, while a\nrotation error of \\new{$0\\fdg112$ corresponds to 1~pixel across half the\nfield.}  These constraints will serve as guidelines throughout this\npaper.\n\n\n\\section{Astrometric Calibration}\n\\label{sec-astromcalib}\n\\nobreak\nOur astrometric calibration process has three principal components:\n\n\\begin{enumerate}\n\\item correcting the image distortion introduced by the instrument optics;\n\\item measuring the field rotation and setting the instrument alignment angle;\n\\item constructing the WCS.\n\\end{enumerate}\n\n\\subsection{Image Distortion}\n\\label{sec-distortion}\n\\nobreak\nThe two NICI channels suffer from image distortion arising from the\noff-axis reflective optics in both the AO system and science camera.\nTo measure the distortion, we imaged a pinhole grid mask mounted in\nthe {\\em Fiber Optic Calibration Source} (FOCS), which can be deployed\ninto the telescope focal plane at the entrance to the AO\nsystem.  The FOCS mask distortion at the science detectors represents\nthe combined distortions of the AO system and science camera optics.\n\\new{We preferred to use the mask rather than an astronomical field due\nto the grid spots' very high signal-to-noise, their uniform size and\ndistribution over almost the entire field, and the freedom from seeing\nand anisoplanatism effects.}\n\nFOCS Grid Mask images are shown in Figure~\\ref{fig-gridmask}.  The\ngrid is rotated by $22\\fdg3$ relative to the detector.  The\nBlue channel image is reflected left-right relative to the Red due to\nthe extra reflection at the DW optic in the path to the Blue channel.\n\n\n\\subsubsection{Fitting Procedure}\n\\label{sec-fitproc}\n\\nobreak\nTo determine the distortion, a grid mask image is first\nbackground-subtracted and the centroid of each spot\nmeasured.  A synthetic rectilinear grid is then fitted to the spot\npositions within 256 pixels of the image center, the region\nof lowest distortion, by minimizing the $\\chi^2$ statistic\n\\begin{equation}\n\\label{eq-chisq}\n\\chi^2 = \\sum_{i=1}^{N} \\frac{|{\\bf r}_i - {\\bf r}[x_i(x_0,y_0,d,\\theta),\\, y_i(x_0,y_0,d,\\theta)]|^2}{\\sigma_i^2},\n\\end{equation}\nwhere ${\\bf r}_i$ is the measured centroid of spot $i$,\n${\\bf r}[x_i(x_0,y_0,d,\\theta),\\, y_i(x_0,y_0,d,\\theta)]$ are\nthe spot positions in the synthetic grid, and \n$\\sigma_i$ is the uncertainty of the centroid measurement.  \nThere are four free parameters: $x_0$ and $y_0$ represent the overall \ngrid position, $d$ is the spot separation in pixels (assumed to be \nthe same in X and Y), and $\\theta$ is the grid rotation.  \nThe $\\chi^2$ value is minimized using the Nelder-Mead simplex method\n\\citep{1965TCJ..7..308P,Press:1992:NRC:148286} as implemented in the\nIDL built-in routine {\\tt AMOEBA} (Exelis Visual Information Solutions, \nBoulder, Colorado).  \\new{The simplex algorithm begins by computing $\\chi^2$ for a set of\ntrial parameters, then finds the trajectory through parameter space\nthat steadily reduces $\\chi^2$ until a minimum is reached.}  The uncertainty $\\sigma$ is \nset to 1 for all points because we are only interested in the best-fit\nrectangular grid at this first stage, not the positional uncertainties.\n\n\\new{We used the IDL routine {\\tt POLY\\_2D} to transform the images\n  geometrically to correct the distortion.  The transformation is\n  defined by polynomials of degree $N$:\n\\begin{equation}\n\\label{eq-distrans-x}\nx' = \\sum_{i=0}^{N}\\sum_{j=0}^{N} \\, P_{i,j} \\, x^j \\, y^i\n\\end{equation}\n\\begin{equation}\n\\label{eq-distrans-y}\ny' = \\sum_{i=0}^{N}\\sum_{j=0}^{N} \\, Q_{i,j} \\, x^j \\, y^i\n\\end{equation}\nwhere $x$ and $y$ are the initial pixel coordinates and $x'$ and $y'$\nare the transformed coordinates.  $P_{i,j}$ and $Q_{i,j}$ are\ncoefficients determined from a least-squares fit performed by the\ncompanion routine {\\tt POLY\\_WARP}, based on the measured grid mask\n($x, y$) positions and the rotated rectilinear grid ($x', y'$)\npositions.  After computing ($x', y'$) for all the pixels, \n{\\tt POLY\\_2D} generates a transformed image using cubic convolution.}\nThe residual errors of the distortion correction were computed by applying\nthe correction to the grid mask images, then measuring the difference\nbetween the rectilinear points and the mask spots.\n\n\\subsubsection{Distortion Corrections}\n\\label{sec-distcorr}\n\\nobreak\nWe derived distortion corrections for six pairs of images taken in\nfive different instrument configurations as shown in\nTable~\\ref{tab-dcparams}.  By experimentation, we determined that a\nfifth-order ($N = 5$) fit had a residual error of 0.5 -- 0.6 pixels\nroot-mean-square (rms) for both channels, and that higher-order\npolynomials did not reduce the error.  The initial distortion is as\nhigh as 12~pixels in the array corners.\n\nGiven the high signal-to-noise images of the spots, the uncertainty of\nthe individual centroid measurements is much smaller than 0.5 pixels,\nso the quality of the fits is limited by some source of systematic\nerror.  (In statistical terms, if $\\sigma$ is set to 0.1 pixel, the\nresultant $\\chi^2$ per degree of freedom of the perfect grid relative\nto the distortion-corrected images is $\\approx 100$, much higher than\nthe value of 1 expected when random errors dominate.)  The main source\nof systematic error is most likely the use of a polynomial to\napproximate the distortion; an optical raytrace model may be required\nto improve the fit.  The accuracy of the polynomial fit is within our\n1~pixel goal, so we accept it as satisfactory for the Campaign.\n\nWe fitted six different images to evaluate the consistency of the\ndistortion across multiple instrument configurations.  The consistency\nis a concern because the grid position on the Blue channel array\nvaries by up to a few dozen pixels (Table~\\ref{tab-spotpos}) due to\nthe slightly different mounting angles of the optical elements in the\nDichroic Wheel, and in some cases the failure of DW to seat\nconsistently in its detent.  The Blue channel variations are much\nlarger than the Red because the reflected beam from DW is deviated\nmore by optical element tilts than the transmitted beam.  For the\nH-50\/50 beamsplitter, the Blue channel ($X, Y$) positions vary by up to\n25 pixels between different positionings (two extremes ``min''\nand ``max'' are listed in Table~\\ref{tab-spotpos}); evidently the DW\ndoes not fully settle into its detent at the H-50\/50 position.  Such\nlarge variations are not detected for the other DW elements.\n\nAcross the six datasets, there are only small variations in each of\nthe parameters for scale and rotation (see Table~\\ref{tab-dcparams}).\nThe spot separations vary by 0.05\\% for the Red channel and 0.10\\% for\nthe Blue, while the rotations are consistent to $0\\fdg096$ for the Red\nchannel and $0\\fdg088$ for the Blue.  If the Open position is omitted,\na reasonable action given that DW substrates probably introduce\naberrations into the transmitted beam that would\nbe absent for the Open position, the Red channel rotation range is\nonly $0\\fdg027$.\n\nThe Blue channel exhibits DW-dependent rotations which appear to be\ncorrelated with the largest position offsets listed in\nTable~\\ref{tab-spotpos}, indicating that the irreproducible positions\nof the DW are causing both effects.  The rotations are smaller than\nour $0\\fdg1$ precision goal, and not much larger than the rms error of\nthe fit, but nevertheless appear to be systematic.\n\nThe fitting results do reveal significant differences in scale and\nrotation between the Red and Blue channels.  How these differences are\nmanaged is described in the WCS section below.\n\nTo evaluate distortion variations between the instrument\nconfigurations, we applied the correction for the H-50\/50 + CH4-H4\\%\nimages to the other four datasets.  Table~\\ref{tab-dch5050} shows that\nin all cases the errors are $< 1$~pixel rms, or $\\leq 0.1\\%$.\n\n\\new{\nGiven the acceptable repeatability of the H-50\/50 + CH4-H4\\%\ndistortion correction across all the instrument configurations, we\nadopted it as the standard correction for all Campaign data.  The\ndistortion correction coefficients for this mode are listed\nin Table~\\ref{tab-dccoeffs}, and maps of the distortion correction\nvectors and the post-correction residual errors are displayed in\nFigure~\\ref{fig-distmap}.}  More than half of\nthe Campaign data were taken in the dual-channel ASDI mode with this\nconfiguration.  Applications which require the highest possible\naccuracy, however, may wish to use more detailed calibration datasets\ntaken in a particular configuration.\n\n\n\\subsection{Instrument Alignment Angle}\n\\label{sec-iaa}\n\\nobreak\nThe Telescope Control System (TCS) uses a simple formula to control\nthe Cassegrain Rotator (CR) so that the desired position angle on sky,\n{$\\theta_{\\rm IPA}$}, the {\\em Instrument Position Angle}, is oriented\nparallel to the detector columns toward the top of the image.  A \nquantity {$\\theta_{\\rm IAA}$}, the {\\em Instrument Alignment\n  Angle}, is defined to represent the rotation between\nthe CR and science detector reference frames.  The CR angle\n$\\theta_{\\rm CR}$\\ is related to the other angles by\n\\begin{equation}\n   \\theta_{\\rm CR} = \\theta_{\\rm IPA} - \\theta_{\\rm par} - \\theta_{\\rm IAA} + 180^\\circ,\n\\end{equation}\nwhere $\\theta_{\\rm par}$\\ is the parallactic angle of the target under\nobservation.\n\nFor example, if a detector is mounted exactly aligned with the CR\nframe, $\\theta_{\\rm IAA} = 0$.  To achieve $\\theta_{\\rm IPA} = 0$\n(North up on detector) on the meridian where $\\theta_{\\rm par} = 0$,\n$\\theta_{\\rm CR}$\\ would be 180$^\\circ$.  Note that $\\theta_{\\rm IAA}$\\ is dependent on the\nmechanical orientation of the instrument on the telescope; small\nrotations or other changes in the mounting may in turn require a\ndifferent $\\theta_{\\rm IAA}$\\ setting in order to achieve an accurate $\\theta_{\\rm IPA}$.\n\nObservations may be defined with CR in two modes: {\\em Follow\\\/} mode, in\nwhich case the TCS sets and continuously updates the CR to keep the\nspecified $\\theta_{\\rm IPA}$\\ vertical on the detector, and {\\em Fixed\\\/} mode where the TCS\nsets the CR to a fixed value, allowing the sky (and therefore $\\theta_{\\rm IPA}$) to\nrotate on the detector during the observation.  Follow mode is used\nfor most observations with other science instruments, but Fixed is\nused for NICI ADI and ASDI observations.\n\nFor NICI, $\\theta_{\\rm IAA}$\\ can represent only one of the two science\nchannels.  The natural choice is the Red channel, due to its smaller\nDW-dependent systematic variations in scale and rotation than the\nBlue's.  With this calibration, at $\\theta_{\\rm IPA} = 0$, north is exactly\nvertical on the Red channel, while it is offset by $-1\\fdg1$ on the\nBlue channel.\n\nSetting $\\theta_{\\rm IAA}$\\ requires observing an astrometric standard\ntarget or field, an exercise which must be performed each time NICI is\nmounted on the telescope.  Table~\\ref{tab-iaa} lists the individual\nNICI mountings between 2008 and 2012, the astrometric standard\nobserved, and the $\\theta_{\\rm IAA}$\\ and pixel scale results.  Over the course of the\nCampaign, our calibration technique grew more sophisticated as we\nadded distortion corrections and switched from binary stars to a more\nprecise astrometric field.\n\n\n\\subsubsection{Binary Stars}\n\\label{sec-binstars}\n\nFor the Campaign's first two years, from 2008 August to 2010 October, we\nused two binary stars for astrometric calibration.  An ideal binary\nwould have a separation between 5$''$ and 10$''$ (small enough to fit\ninto NICI's 18$''$ field yet large enough to provide a sufficient lever\narm for an accurate measurement), a separation precision $<$ a few mas,\nand a position angle precision $< 0\\fdg1$.  Such high precision requires\nmodern speckle measurements; orbits based on older historical data\nare not sufficiently precise.  Several well-known systems with\nprecise orbits were rejected because their stars were too bright,\nsaturating on NICI even with the shortest possible exposure times.\n\nWe adopted \\object{70 Oph} (STF~2272AB, WDS~18055+0230), which has an\naccurate speckle-based orbit \\citep{2000A&AS..145..215P,\n  2008A&A...482..631E} and 5\\farcs5 separation in mid-2008, as our\ninitial astrometric standard.  The $V = 4.2$ primary component was\nbright enough to provide sufficient S\/N when placed under the\ncoronagraph mask.  The off-mask image of the secondary component, with\n$V = 6.2$, saturated slightly in the PSF core, but its position could\nstill be measured to $\\pm 0.5$~pixel based on the unsaturated halo.\nIn this way, $\\theta_{\\rm IAA}$\\ for mounting 1 was measured (see\nTable~\\ref{tab-iaa}), with an estimated uncertainty of $\\pm 0\\fdg09$.\nWhen NICI changed ISS ports (from side-looking to up-looking) in late\n2009, \\object{70 Oph} was not accessible, so we observed the binary \n\\objectname[CCDM~J22398]{HDO~171~B-C}\n(CCDM~J22398-1942~B, WDS 22398-1942) instead.  \\object{70 Oph} was\nobserved again for mounting 3 in 2010 March.  At this time we had not\ndeveloped the distortion-correction algorithm, so these data were\nreduced and $\\theta_{\\rm IAA}$\\ set with no distortion correction.\n\nFollow-up checks and corrections of these early binary star\ncalibrations are described in \\S\\ref{sec-earlycheck}.\n\n\n\\subsubsection{Trapezium Cluster}\n\\label{sec-trapezium}\n\\nobreak At the start of mounting 4 in 2010 October, we observed the\nTrapezium Cluster in Orion as an astrometric field.  We used the {\\em\n  HST}-derived coordinates from \\citet{2008AJ....136.2136R}, applied\nthe grid mask distortion correction procedure described in\n\\S\\ref{sec-distortion}, and fitted the measured position centroids to\nthe celestial coordinates as described for the LMC field in\n\\S\\ref{sec-lmc}.  Although the technique appeared to be superior to\nthe binary star calibrations, the fitting errors were unacceptably\nlarge: $\\approx 2$~pixels rms, \\new{considerably larger than\n  what we eventually achieved for the LMC field using the same\n  observing and reduction techniques.  The large errors\n  may be caused by systematic errors in the\n  {\\em HST} optical astrometry or the underlying Two Micron All Sky\n  Survey (2MASS) catalog \\citep{2003tmc..book.....C} absolute astrometry against\n  which the {\\em HST} data were registered.}  The magnitude of these\nerrors led us to develop the LMC-based calibration described in the\nnext section.\n\nNote that during this period in 2010 October-December, while new\nsoftware was being developed to apply the distortion correction,\nmeasure the alignment, and compute $\\theta_{\\rm IAA}$\\ changes, a number of\nerrors were made computing the direction of the $\\theta_{\\rm IAA}$\\ \ncorrections and the WCS parameters in the headers, as indicated in\nTable~\\ref{tab-iaa}.  Methods for correcting these errors are\ndescribed in \\S\\ref{sec-wcs}.\n\n\n\\subsubsection{LMC Astrometric Field}\n\\label{sec-lmc}\n\\nobreak\nIn 2010 November we began to observe a field in the Large Magellanic\nCloud (LMC) that was the subject of a detailed {\\em HST}-based\nastrometric calibration in support of {\\em JWST}\\\/\n\\citep{2007ASPC..364...81D}.  For {\\em JWST}, the field was chosen to\nbe relatively free of bright stars, but fortunately it contains three\n$R = 11-12$~mag stars bright enough to serve as natural guide stars\nfor NICI's AO system.  The data file containing high-precision\ncoordinates was supplied to us by J.\\,Anderson \\citetext{private\ncommunication}.\n\nThe subfield with the brightest guide star, named ``f606w\\_11.341'' in\nthe STScI documentation but which we refer to as ``LMC-11mag,''\ncontains 12 stars with $H < 17$ which can be detected by NICI\nin about 5~minutes observing time.  For consistency, the field was\nobserved in Cass Rotator Follow mode at {$\\theta_{\\rm IPA}$}~=~0, always with\nthe Clear focal-plane mask (no coronagraph), the H-50\/50 beamsplitter,\nand the {CH4-H4\\%L} and S filters.  Image pairs taken with a 4$''$\ndither were subtracted to cancel the background emission\n(Fig.~\\ref{fig-lmc11mag}), then the standard distortion correction\nwas applied and the stars' centroid positions in the Red channel\nmeasured.\n\nThe centroid position probable errors were estimated by synthesizing\nimages with Gaussian PSF's over a range of S\/N combined with\nbackgrounds with random variations.  After the centroid for a given\nS\/N PSF was measured for each of the random backgrounds, the standard\ndeviation $\\sigma$ could be computed directly, and a lookup table of\nS\/N vs. $\\sigma$ constructed.  The probable error of each star in the\nreal image was then estimated by measuring its S\/N and finding the\ncorresponding $\\sigma$ in the lookup table.\n\nThe centroid positions were fitted with a $\\chi^2$ minimization\ntechnique similar to the grid mask fitting described by\nEqn.\\ref{eq-chisq} to determine the pixel scale and the $\\theta_{\\rm IAA}$\\ \ncorrection, using the probable errors $\\sigma$ estimated for each\nstar.\n\nTable~\\ref{tab-lmc} displays the results of an LMC-11mag astrometric\nfit taken for mounting~10 on 2012 December 22 UTC.  The $R = 11$~mag guide star\nconsistently shows a large 5 - 6 pixel error in all images, so it is\nnot included in the fit.  (The exact reason for the error is unknown;\nthe star is certainly a foreground star, but the error appears to be\ntoo large to be explained by proper motion, and no proper motion was\ndetected between the different NICI astrometric images over a period\nof 3 years.)  As with the pinhole-grid images, the value\nof $\\chi^2$ per degree of freedom $> 30$ indicates that the errors are\ndominated by systematic effects.  In addition to the distortion\ncorrection errors described in \\S\\ref{sec-fitproc}, additional sources\nof error on a star field are errors in the provided celestial\ncoordinates, proper motions, atmospheric refraction, and\nanisoplanicity effects which cause stars far from the center of the\nfield to become elongated.\n\nThe residual error between the expected and measured positions \\new{of\n  all the LMC field stars is 0.68~pixels rms, with no errors $\\gtrsim\n  1$~pixel, better than our goal of $< 1$ pixel across half the field.\n  This 2012 December dataset was taken in good seeing and is one of\n  our best results, but in all our calibrations the residual errors\n  are $< 0.8$~pixels across the field, or $\\approx 0.08$\\% in scale\n  and 0\\fdg046 in rotation.  This corresponds to a center-to-edge\n  accuracy of $< 0.4$~pixels or 7~mas, better than our 20~mas\n  requirement.  The errors are similar for both up-looking and \n  side-looking port mountings, even with the extra reflection \n  on the side port.}\n\nAfter 2010 the LMC-11mag field became the primary NICI astrometric\nreference, observed regularly after most instrument mountings and port\nchanges and periodically during observing semesters.  We find that\n$\\theta_{\\rm IAA}$\\ varies by a few tenths degree between mountings, but\nperiodic checks within a mounting are consistent to $<0\\fdg03$,\nexceeding our calibration requirements.  We calibrated a secondary\nfield, \\object{HIP 62403}, relative to LMC-11mag for use during the southern\nwinter months when the LMC is inaccessible; our derived astrometric\ndata for this field are listed in Table~\\ref{tab-HIP62403}.\n\nThe LMC fits also yield the image pixel scale which, unlike {$\\theta_{\\rm IAA}$},\nshould not vary between mountings.  Over 12 separate LMC\nmeasurements, we find a Red channel scale of\n17.958$\\pm$0.022~mas\/pixel.  The relative probable error is 0.12\\%, or\nabout 1 pixel across the detector, again within our requirements.\n\n\\subsubsection{Checks and Corrections of Early Calibrations}\n\\label{sec-earlycheck}\n\\nobreak\nAfter developing the distortion correction and LMC-based calibration\nin late 2010, we checked the earlier binary star calibrations\nin two ways.  First, the binary star centroids were remeasured\nafter applying the grid mask distortion correction.  We found that the\nmounting~1 and 2 calibrations were unaffected by the distortion\ncorrection to a level of $\\approx 0\\fdg1$, because both binary components\nwere placed within the low-distortion central region.\nFor mounting~3, however, 70~Oph~B was placed in the lower left corner\nof the field and suffered a distortion of $\\approx 2.5$~pixels, causing a\nrotation error of $0\\fdg7$.\n\nAs a second check method, we utilized several Campaign target fields\nwith multiple background stars that had been observed both with binary\nstar and later LMC-based astrometric calibration.  The results are\nsummarized in Table~\\ref{tab-astromcorr}.  This method verified the\naccuracy of the mounting 1 and 2 calibrations and the error with\nmounting 3.  Two good comparison fields for mounting 3 indicated an\nerror of $0\\fdg5$ and a true {$\\theta_{\\rm IAA}$}~=~112.2, which we judge to be\nmore reliable than the 70~Oph derived value due to saturation of\n70~Oph~B and its position in the corner of the field.  A detailed\nprocedure for correcting the $\\theta_{\\rm IAA}$\\ error is presented in\n\\S\\ref{sec-wcs}; these corrections were applied to the Campaign data\nreductions, but {\\em not\\\/} to the raw data stored in the Gemini\nScience Archive.\n\nThese checks revealed one range of dates, 2009 April 26-27 UTC, for\nwhich the field rotation is in error by $0\\fdg5$, an unusually large\namount.  \\object{Proxima Cen} and \\object{HD 196544} data taken on\nthese dates both exhibit the anomaly.  We suspect that an error in the\nCass Rotator position or datum sometime between April 8 and 26 caused\nthis error.  After April 27, NICI operations were suspended for the\nwinter, so we do not have futher data until 2009 August, by which time\nthe Cass Rotator position is correct.\n\n\n\\subsection{WCS}\n\\label{sec-wcs}\n\\nobreak\nThe WCS in each NICI MEF extension specifies the mapping between sky\nand detector for that channel, including the field rotation, pixel\nscale, and reflection between sky and detector.  It has\nlimitations, however, in that it describes the absolute astrometric position\nonly to low accuracy and does not represent the image distortion.  In\naddition, during NICI's operations there were many instances when $\\theta_{\\rm IAA}$\\ \nwas set incorrectly and therefore the WCS does not\nrepresent the true astrometry.  Therefore, in this section we will\nreview how the WCS is constructed, its limitations and errors, and how\nthe errors can be corrected.\n\nThe Gemini algorithm that builds the WCS for each MEF file begins with\na mapping file that contains a series of $X, Y$ positions in detector\ncoordinates along with their corresponding on-sky angular offsets from\nthe field center in the telescope coordinate system.  The algorithm\nfits a transformation to these points, then applies the current\ntelescope pointing and Cassegrain rotator angles to compute the on-sky\nWCS.  The original intent was to populate the mapping file with points\nmeasured by offsetting a star to different positions on the detector\nand recording both the detector and celestial positions.  For NICI's\nvery narrow field of view and small pixels, however, this technique is\ninsufficiently precise. (It relies on the mapping accuracy of the\nPeripheral Wavefront Sensor probe arm, which could contain errors of\nseveral tenths of an arcsec.)  Therefore, we developed a simple script\nto generate synthetic mapping files based on the pixel scale and $\\theta_{\\rm IAA}$\\ \nrotation angle determined from the calibrations.  This\ntechnique produces a WCS limited only by the quality of the\nastrometric field measurements as described in preceeding sections.\n\nThe Red channel mapping file is generated directly from the measured\nRed channel scale and {$\\theta_{\\rm IAA}$}.  The Blue channel file is generated\nusing the more precise scale and rotation angle relative to the Red\nchannel determined from the grid mask images.  For the LMC\ncalibrations after 2010 November, the Blue WCS rotation angle is set\n$-1\\fdg1$ from the Red angle, and the scale is 1\/0.9980 = 1.0020 $\\times$\nthe Red scale (with the scale, in mas\/pixel, being inversely\nproportional to the magnification).\n\nGemini's WCS standard does not include distortion parameters such as\nthe SIP system, which defines FITS header distortion coefficients to\nsupplement the WCS \\citep{2005ASPC..347..491S}.  Therefore, for NICI\nfiles after 2010 November, the WCS represents the coordinate system\n{\\em after}\\\/ the standard distortion correction (for H-50\/50 +\nCH4-H4\\%) described in \\S\\ref{sec-distortion} has been applied.\n\nIn addition, note that the WCS represents the orientation at the {\\em\n  beginning}\\\/ of the exposure.  If the CR mode is Follow, then this\norientation is correct throughout the exposure.  However, if the CR\nmode is Fixed, as is the case for ADI observations, a rotation must be\napplied to correct the change in field angle between the start and\nmid-point of the exposure.\n\n\\subsubsection{WCS  Corrections}\n\\label{sec-wcs-corr}\n\\nobreak\nAs we previously described in \\S\\ref{sec-binstars} and\n\\ref{sec-trapezium}, the $\\theta_{\\rm IAA}$\\ and pixel scale values measured\nduring several calibrations contain errors.  These\nerrors are also manifested in the WCS.  Known issues are:\n\\begin{enumerate}\n\\item{The WCS values for each mounting are based on the pixel scale derived from\n    a particular calibration dataset.  For highest\n    accuracy and consistency, we recommend using the mean LMC-based scales of\n    17.958~mas\/pixel for the Red channel and 17.994 for Blue \n    for all epochs.}\n\\item{The values of $\\theta_{\\rm IAA}$\\ and WCS may be incorrect for data taken\n    immediately after each mounting, until astrometric calibrations\n    could be taken and analyzed.  The affected dates are listed in Table~\\ref{tab-iaa}.\n    Mounting 4 is the most severe example: data from 2010 October\n    and November contain several errors due to problems with the\n    distortion-correction and astrometric fitting software used at\n    that time for the Trapezium and LMC fields.}\n\\item{Before 2010-12-14, the rotation of the Blue WCS relative to the\n    Red did not use the correct value of $-1\\fdg1$.}\n\\item{For mounting 3 from 2010 March to October, the $\\theta_{\\rm IAA}$\\ \n    value of 111.70 was too low by $0\\fdg5$.}\n\\item{Data taken on 2009 April 26-27 have a rotation error of\n    $0\\fdg5$, apparently caused by a short-term error in the Cass Rotator.  (See last\n    paragraph of \\S\\ref{sec-earlycheck}}\n\\end{enumerate}\n\nThe WCS rotation correction for the Red channel is listed in\nTable~\\ref{tab-iaa}.  In addition, we have developed an IDL script\nthat automatically corrects a MEF file's WCS according to the date\nof observation (see Appendix).  Note that the script does not modify\nthe image data; only the WCS is updated to represent the true\nastrometry of the data.  The updated file with the corrected WCS can\nthen be pipeline-processed as usual.\n\n\n\\section{Campaign Astrometry Statistics}\n\\label{sec-campstats}\n\\nobreak\nSeveral dozen of the Campaign's target fields contained candidate\ncompanions which required follow-up checks for common proper motion\nand parallax.  The vast majority of these candidates were determined\nto be background stars.  We can use these multi-epoch observations to\nevaluate the consistency of the astrometric calibration over the\ncourse of the Campaign.\n\nOur technique is simply to compare the relative positions of the\nprimary target and background stars at the available epochs, after\ncorrecting the primary's relative parallax and proper motion to the\nfirst epoch.  Differences in the corrected positions between epochs\nindicate errors in the position measurements themselves, the proper\nmotion correction, the astrometric calibrations, or a combination of\nall three effects.  Some dense fields contained multiple background\nstars, permitting more detailed checks independent of the proper\nmotion correction, but because many fields just contained the primary\nand one background star we will discuss only the primary-to-background\nposition differences here.\n\nFigure~\\ref{fig-offsetdistrib} shows the offset distributions for a\ntotal of 218 pairs of observations of 205 background stars in 113\nfields.  The offsets are indicated parallel and perpendicular to the\nradius vector from the primary to the background star, which is\nappropriate given the large derotations applied to the ADI frames\nbefore stacking and helps to identify systematic errors in scale and\nrotation.\n\nThe left panel of Figure~\\ref{fig-offsetdistrib} shows the separation\n$r$ and perpendicular $d = r\\, \\tan(\\theta)$ offsets vs. $r$ itself.\nThe distribution of the $r$ offsets shows a possible increase with\nradius, suggesting scale errors on the order of 0.1\\%.  The $d$\noffsets, however, maintain a consistent width from small to large $r$,\nindicating that the primary source of error is in the individual\nposition measurements of the stars.  In other words, if the dominant\nsource of errors were in the rotation calibration, the distribution\nwould be expected to increase proportional to $r$, which is not\nobserved.\n\nThe right panel shows the distribution histograms in the same pixel\nunits.  The distributions have a central core with $\\sigma =\n0.85$~pixels in $r$ and 1.20 in $d$; values that are only slightly\nlarger than the rms errors of the distortion correction and the LMC\nfield astrometric fits.  Above 3.5~pixels total offset lie 9 {\\it\n  outlier} points, all of which have $r > 7''$.  The large errors in\nthese cases are caused by inaccurate distortion correction near the\nedges of the field and very different positions of the stars on the detector\nbetween the two epochs -- due to either different field angles, or to\nthe two observations being taken on different telescope ports with an\nodd and even number of reflections, which reversed the image and\ncaused the same object to appear on opposite sides of the detector.\n\nThat the primary source of the $d$ offset errors appears to be due to\nindividual position errors does not have an obvious explanation.\nBecause the PSF core FWHM is usually 3-4~pixels, centroid measurement\nerrors as large as 3~pixels would typically be several $\\sigma$.  It\nshould be noted that the astrometry is measured from final images\nwhich are coadded from many derotated ADI frames, which tends to\nbroaden the individual stars.  Possible causes of systematic error are\nsmall-scale irregularities in the distortion correction, shifts in the\nposition of the primary star which is dimmed by more than 6 magnitudes\nby the partially transparent occulting mask, or other irregularities\nin the mask.\n\n\n\n\\section{Conclusions}\n\\label{sec-conclusions}\n\\nobreak \\new{We have developed an astrometric calibration for the\n  NICI Planet-Finding Campaign based on a grid mask distortion\n  correction and an LMC astrometric field.  The accuracy\n  achieved by the calibration is $\\lesssim0.08$\\% in scale and\n  $\\lesssim0\\fdg046$ in rotation, corresponding to $\\lesssim7$~mas\n  center-to-edge.}  The calibration for each channel is represented by\nthe MEF header WCS.  Before 2010 November the calibrations and WCS\ncontain known errors; we provide tables and an IDL routine to correct\narchival NICI data to the final calibration.\n\n\\acknowledgements\n\nThis work was supported in part by NSF grants AST-0713881 and\nAST-0709484.  The authors thank the NICI team members J.~Hinds and\nC.~Lockhart, and Gemini engineers R.~Galvez, G.~Gausachs, J.~Luhrs,\nG.~Perez, R.~Rogers, R.~Rojas, L.~Solis, G.~Trancho, and C.~Urrutia\nfor their expert support of NICI during commissioning and operations.\nF.~Rantakyro, E.~Christensen, and E.~Artigau provided valuable\noperational and software support.  We thank J. Anderson for supplying\nthe {\\em HST\\\/} data for the LMC fields, and the anonymous referee\nfor providing valuable suggestions.  This research has made use of\nthe SIMBAD database, operated at CDS, Strasbourg, France; the\nWashington Double Star Catalog maintained at the U.S. Naval\nObservatory; and NASA's Astrophysics Data System.\n\n{\\it Facilities:} \\facility{Gemini (NICI)}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nThe anchored $k$-core problem can be explained by the following illustrative example. We want\nto organize a workshop on Theory of Social Networks. We send invitations to most  distinguished researchers in the area and\nreceived many  replies of the following nature: ``Yes, in theory, I would be happy to come but my final decision  depends on\nhow many   people I know will be there.\"  Thus  we have a list of  tentative participants, but some of them can cancel their\nparticipation and we are afraid that the cancellation process\nmay escalate. On the other hand, we also have  limited funds to reimburse travel expenses for\na small number of participants, which we believe, will guarantee their participation. Thus we want to ``anchor\" a small subset\nof  participants whose guaranteed participation would prevent the unraveling process, and by\nfixing a small group   we hope to minimize the number of cancellations, or equivalently,\nmaximize the number of participants, or the core.\n\nUnraveling processes are common for social networks where the behavior of an individual is often influenced by the actions of\nher\/his friends. New events occur quite often in social networks: some examples are usage of a particular cell phone brand,\nadoption of a new drug within the medical profession, or the rise of a political movement in an unstable society. To estimate\nwhether these events or ideas spread extensively or die out soon, one has  to model and study the dynamics of \\emph{influence\npropagation} in social networks. Social networks are generally represented by making use of  undirected or directed graphs,\nwhere the edge set represents  the relationship between individuals in the network. Undirected graph  model works fine for\nsome networks, say Facebook, but the nature of interaction on some social networks such as Twitter is asymmetrical:  the fact\nthat  user $A$ follows  user $B$ does not imply that that user $B$ also follows $A$.\\footnote{The first author follows LeBron\nJames on Twitter (and so do 8,017,911 other people), but he only follows 302 people with the first author not being one of\nthem.}  In this case, it is more appropriate to model interactions in  the network by \\textbf{directed} graphs. We add a\ndirected edge $(u,v)$ if $v$ follows $u$.\n\n\nIn this work we are interested in the  model of \\emph{user engagement}, where each individual with less than $k$ people to\nfollow (or equivalently whose in-degree is less than $k$) drops out of the network. This\nprocess can be contagious, and may affect even those individuals who initially were linked to more than $k$ people, say follow\non Twitter. An extreme example of this was given by Schelling (see page 17 of ~\\cite{schelling2006micromotives}): consider a\ndirected path on $n$ vertices and let $k=1$. The left-endpoint has in-degree zero, it drops out and now the in-degree of its\nonly out-neighbor in the path becomes zero and it drops out as well. It is not hard to see that this way the whole network\neventually drops out as the result of a \\emph{cascade of iterated withdrawals}. In general at the end of all the iterated\nwithdrawals the remaining engaged individuals form a unique maximal induced subgraph whose minimum in-degree is at least $k$.\nThis is called as the \\emph{$k$-core} and is a well-known concept in the theory of social networks. It was introduced by\nSeidman~\\cite{seidman-k-core} and also been studied in various social sciences\nliterature~\\cite{chwe1999structure,chwe2000communication}.\n\n\\medskip\n\\noindent\\textbf{Preventing Unraveling:} The unraveling process described above in Schelling's example of a directed path can\nbe  highly undesirable in many scenarios.  How can one attempt to prevent this unraveling? In Schelling's example it is easy\nto see: if we ``buy\" the left end-point person into being engaged then the whole path becomes engaged. In general we overcome\nthe issue of unraveling by allowing some ``anchors\":  these are the vertices that remain engaged irrespective of their\npayoffs. This can be achieved by giving them extra incentives or discounts. The hope is that with a few \\emph{anchors} we can\nnow ensure a large subgraph remains engaged. This subgraph is  called as the \\emph{anchored $k$-core}: each non-anchor vertex\nin this induced subgraph must have in-degree at least $k$ while the anchored vertices can have arbitrary in-degrees. The\nproblem of identifying $k$-cores in a network also has the following  game-theoretical interpretation   introduced by\nBhawalkar et al.~\\cite{BhawalkarKLRS12}: each user in the social network pays a cost of $k$ to remain engaged. On the other\nhand, he\/she receives a profit of one from every neighbor who is engaged. The ``network effects\" come into play, and an\nindividual decides to remain engaged if has non-negative payoff, i.e., it has at least $k$ in-neighbors who are engaged. The\n$k$-core can be viewed as the unique maximal equilibrium in this model.\n\n\nBhawalkar et al.~\\cite{BhawalkarKLRS12} introduced  the \\textsc{Anchored $k$-Core} problem for (undirected) graphs. In the\n\\textsc{Anchored $k$-Core} problem the input is an undirected graph $G=(V,E)$ and integers $b, k$, and the task is to find  an\ninduced subgraph $H$ of maximum size with all vertices but at most $b$ (which are anchored)\nto be of degree at least $k$. In this work we extend the\nnotion of {anchored $k$-core} to directed graphs.  We are interested in the case, when  in-degrees of all but $b$ vertices of\n$H$ are at least  $k$.  More formally, we study the following parameterized version of the problem.\n\\begin{center}\n\\noindent\\framebox{\\begin{minipage}{4.50in} { \\textsc{Directed Anchored $k$-Core}  (\\textsc{Dir-AKC}\\xspace)}\\\\\n\\emph{Input}: A directed graph $G=(V,E)$ and integers $b, k,p$. \\\\\n\\emph{Parameter~1}: $b$.\\\\\n\\emph{Parameter~2}: $k$.\\\\\n\\emph{Parameter~3}: $p$.\\\\\n\\emph{Question}:  Do there exist sets of vertices $A\\subseteq H\\subseteq V(G)$ such that $|A|\\leq b$, $|H|\\geq p$, and every\n$v\\in H\\setminus a$ satisfies $d^{-}_{G[H]}(v)\\geq k$?\n\\end{minipage}}\n\\end{center}\nWe will call the set $A$ as the \\emph{anchors}, the graph $H$ as the \\emph{anchored $k$-core}. Note that the undirected\nversion of \\textsc{Anchored $k$-Core} problem can be modeled by the directed version: simply replace each edge $\\{u,v\\}$ by\narcs $(u,v)$ and $(v,u)$. Keeping the parameters $b,k,p$ unchanged it is now easy to see that the two instances are\nequivalent.\n\n\n\n\\medskip\n\\noindent\\textbf{Parameterized Complexity:} We are mainly interested in the parameterized complexity of  \\textsc{Anchored\n$k$-Core}. For the general background, we refer to the books by Downey and Fellows~\\cite{downey-fellows-book},  Flum and\nGrohe~\\cite{flum-grohe-book} and Niedermeier~\\cite{niedermeier-book}. Parameterized complexity is basically a two dimensional\nframework for studying the computational complexity of a problem. One dimension is the input size $n$ and another one is a\nparameter $k$. A problem is said to be \\emph{fixed parameter tractable} (or \\ensuremath{\\operatorClassFPT}) if it can be solved in time $f(k)\\cdot\nn^{O(1)}$ for some function $f$. A problem is said to be in \\ensuremath{\\operatorClassXP}, if it can be solved in time $O(n^{f(k)})$ for some\nfunction $f$. The $\\operatorClassW$-hierarchy is a collection of computational complexity classes: we omit the technical\ndefinitions here. The following relation is known amongst the classes in the $\\operatorClassW$-hierarchy:\n$\\ensuremath{\\operatorClassFPT}=\\classW{0}\\subseteq W[1]\\subseteq W[2]\\subseteq \\ldots$. It is widely believed that $FPT\\neq W[1]$, and hence if a\nproblem is hard for the class $W[i]$ (for any $i\\geq 1$) then it is considered to be fixed-parameter intractable.\n\n\\medskip\n\\noindent\\textbf{Previous Results:} Bhawalkar et al.~\\cite{BhawalkarKLRS12} initiated the algorithmic  study of\n\\textsc{Anchored $k$-Core}  on undirected graphs and obtained an interesting dichotomy result:  the decision version of the\nproblem is solvable in polynomial time for $k\\leq 2$ and is  \\ensuremath{\\operatorClassNP}-complete for all $k\\geq 3$. For $k\\geq 3$, they also\nstudied the problem from the viewpoint of parameterized complexity and approximation algorithms.\nThe current set of authors~\\cite{ChitnisFG13} improved and generalized these results by showing that for $k\\geq 3$ the problem\nremains \\ensuremath{\\operatorClassNP}-complete even on planar graphs.\n\n\n\\medskip\\noindent\\textbf{Our Results:} In this paper we provide a number of new results on the algorithmic complexity of\n\\textsc{Directed Anchored $k$-Core}  (\\textsc{Dir-AKC}\\xspace).\nWe start (Section~\\ref{sec:defs}) by showing that that the decision version of \\textsc{Dir-AKC}\\xspace{} is \\ensuremath{\\operatorClassNP}-complete for every $k\\geq\n1$  even if the input graph is restricted to be a planar directed acyclic graph (DAG) of maximum degree at most $k+2$. Note\nthat this shows that the directed version is in some sense strictly harder than the undirected version since it is known be in\n\\ensuremath{\\operatorClassP}\\ if $k\\leq 2$, and \\ensuremath{\\operatorClassNP}-complete if $k\\geq 3$~\\cite{BhawalkarKLRS12}. The \\ensuremath{\\operatorClassNP}-hardness result for \\textsc{Dir-AKC}\\xspace\nmotivates us to make a more refined analysis of the \\textsc{Dir-AKC}\\xspace problem via the paradigm of parameterized complexity. In\nSection~\\ref{sec:saved}, we  obtain the following dichotomy result: \\textsc{Dir-AKC}\\xspace is \\ensuremath{\\operatorClassFPT} \\, parameterized by $p$ if $k=1$, and\n\\classW1-hard if $k\\geq 2$. This fixed-parameter intractability result parameterized by $p$ forces us to consider the\ncomplexity on special classes of graphs such as bounded-degree directed graphs or directed acyclic graphs. In\nSection~\\ref{sec:bound-deg}, for graphs of degree upper bounded by $\\Delta$, we show that the \\textsc{Dir-AKC}\\xspace problem is FPT\nparameterized by $p+\\Delta$ if $k\\geq \\frac{\\Delta}{2}$. In particular, it implies that \\textsc{Dir-AKC}\\xspace is FPT parameterized by $p$ for\ndirected graphs of maximum degree at most four. We complement these results by showing in Section~\\ref{sec:concl} that  if $k<\n\\frac{\\Delta}{2}$ and $\\Delta\\geq 3$, then \\textsc{Dir-AKC}\\xspace is \\classW2-hard when parameterized by the number of anchors $b$ even for\nDAGs, but the problem is \\ensuremath{\\operatorClassFPT}\\ when parameterized by $\\Delta+p$ for DAGs of maximum degree at most $\\Delta$. Note that we\ncan always assume that $b\\leq p$, and hence any \\ensuremath{\\operatorClassFPT}\\ result with parameter $b$ implies \\ensuremath{\\operatorClassFPT}\\ result with parameter\n$p$ as well. On the other side, any hardness result with respect to $p$ implies the same hardness with respect to $b$.\n\n\\section{Preliminaries}\\label{sec:defs}\nWe consider finite directed and undirected graphs without loops or multiple arcs. The vertex set of a (directed) graph $G$ is\ndenoted by $V(G)$ and its edge set (arc set for a directed graph) by $E(G)$. The subgraph of $G$ induced by a subset\n$U\\subseteq V(G)$ is denoted by $G[U]$. For $U\\subset V(G)$ by $G-U$ we denote the graph $G[V(G)\\setminus U]$. For a directed\ngraph $G$, we denote by $G^*$ the undirected graph with the same set of vertices such that $\\{u,v\\}\\in E(G^*)$ if and only if\n$(u,v)\\in E(G)$. We say that $G^*$ is the \\emph{underlying} graph of $G$.\n\nLet $G$ be a directed graph. For a vertex $v\\in V(G)$, we say that $u$ is an \\emph{in-neighbor} of $v$ if $(u,v)\\in E(G)$. The\nset of all in-neighbors of $v$ is denoted by $N_G^-(v)$. The \\emph{in-degree} $d_G^-(v)=|N_G^-(v)|$. Respectively, $u$ is an\n\\emph{out-neighbor} of $v$ if $(v,u)\\in E(G)$, the set of all out-neighbors of $v$ is denoted by $N_G^+(v)$, and the\n\\emph{out-degree} $d_G^+(v)=|N_G^+(v)|$. The \\emph{degree} $d_G(v)$ of a vertex $v$ is the sum $d_G^-(v)+d_G^+$, and the\n\\emph{maximum degree} of $G$ is $\\Delta(G)=\\max_{v\\in V(G)}d_G(v)$. A vertex $v$ of $d_G^-(v)=0$ is called a \\emph{source},\nand if  $d_G^+(v)=0$, then $v$ is a \\emph{sink}. Observe that isolated vertices are sources and sinks simultaneously.\n\n\n\nLet $G$ be a directed graph. For $u,v\\in V(G)$, it is said that $v$ can be \\emph{reached} (or \\emph{reachable}) from $u$ if\nthere is a directed $u\\rightarrow v$ path in $G$. Respectively, a vertex $v$ can be reached from a set $U\\subseteq V(G)$ if\n$v$ can be reached from some vertex $u\\in U$. Notice that each vertex is reachable from itself. We denote by $R_G^+(u)$\n($R_G^+(U)$ respectively) the set of vertices that can be reached from a vertex $u$ (a set $U\\subseteq V(G)$ respectively).\nLet $R_G^-(u)$ denote the set of all vertices $v$ such that $u$ can be reached from $v$.\n\nFor two non-adjacent vertices $s,t$ of a directed graph $G$, a set $S\\subseteq V(G)\\setminus\\{s,t\\}$ is said to be a\n\\emph{$s-t$ separator} if $t\\notin R_{G-S}^+(s)$. An $s-t$ separator $S$ is \\emph{minimal} if no proper subset $S'\\subset S$\nis a $s-t$ separator.\n\nThe notion of important separators was introduced by Marx~\\cite{Marx06} and generalized for directed graphs\nin~\\cite{ChitnisHM12}. We need a special variant of this notion. Let $G$ be a directed graph, and let $s,t$ be non-adjacent\nvertices of $G$. An minimal $s-t$ separator is an \\emph{important $s-t$ separator} if there is no $s-t$ separator $S'$ with\n$|S'|\\leq |S|$ and $R_{G-S}^-(t)\\subset R_{G-S'}^-(t)$. The following lemma is a variant of Lemma~4.1 of~\\cite{ChitnisHM12}.\nNotice that to obtain it, we should replace the directed graph in Lemma~4.1 of~\\cite{ChitnisHM12} by the graph obtained from\nit by reversing direction of all arcs.\n\n\n\\begin{lemma}[\\cite{ChitnisHM12}]\\label{lem:imp-sep}\nLet $G$ be a directed graph with $n$ vertices, and let $s,t$ be non-adjacent vertices of $G$. Then for every $h\\geq 0$, there\nare at most $4^h$ important $s-t$ separators of size at most $h$. Furthermore, all these separators can be enumerated in time\n$O(4^{h}\\cdot n^{O(1)})$.\n\\end{lemma}\n\n\nAs further we are interested in the parameterized complexity of \\textsc{Dir-AKC}\\xspace, we show first \\ensuremath{\\operatorClassNP}-hardness of the problem. \n\n\\begin{theorem}\\label{thm:NPc}\nFor any $k\\geq 1$, \\textsc{Dir-AKC}\\xspace is \\ensuremath{\\operatorClassNP}-complete, even for planar DAGs of maximum degree at\nmost $k+2$.\n\\end{theorem}\n\n\n\\begin{proof}\nWe reduce {\\sc Satisfiability}:\n\\begin{center}\n\\noindent\\framebox{\\begin{minipage}{4.50in}{Satisfiability}\\\\\n\\emph{Input}: Sets of Boolean variables $x_1,\\ldots,x_n$ and clauses $C_1,\\ldots,C_m$. \\\\\n\\emph{Question}: Can the formula $\\phi=C_1\\vee\\ldots\\vee C_m$ be satisfied?\n\\end{minipage}}\n\\end{center}\nIt is known (see e.g.~\n\\cite{DahlhausJPSY94}) that this problem remains \\ensuremath{\\operatorClassNP}-hard even if each clause contains at most 3 literals (notice that clauses of size one or two are allowed), each variable\nis used in at most 3 clauses: at least once in positive and at least once in negation, and the graph that correspond to a\nboolean formula is planar. Consider an instance of {\\sc Satisfiability} with $n$ variables $x_1,\\ldots,x_n$ and $m$ clauses\n$C_1,\\ldots,C_m$ that satisfies these restrictions on planarity and the number of occurrences of the variables. We construct the graph $G$\nas follows.\n\n\n\\begin{figure}[ht]\n\\centering\\scalebox{0.7}{\\input{fig1.pdf_t}}\n\\caption{Construction of $G$ for $k=3$.\n\\label{fig:NPh}}\n\\end{figure}\n\n\n\n\\begin{itemize}\n\\item For each $i\\in\\{1,\\ldots,n\\}$,\n\\begin{itemize}\n\\item add vertices $x_i,\\overline{x}_i,r_i$ and add arcs $(x_i,r_i),(\\overline{x}_i,r_i)$;\n\\item add a set of $k-1$ vertices $Y_i$ and draw an arc from each of them to $r_i$;\n\\item for each vertex $y\\in Y_i$, add $k$ vertices and draw an arc from each of them to $y$, denote the set of these\n    $k(k-1)$ vertices $Z_i$.\n\\end{itemize}\n\\item For each $j\\in\\{1,\\ldots,j\\}$,\n\\begin{itemize}\n\\item add a vertex $v_j$, and for each literal $x_i$ ($\\overline{x}_i$ respectively) in the clause $C_j$, join the\n    vertex $x_i$ ($\\overline{x}_i$ respectively) with $v_j$ by an arc;\n\\item add a set of $k-1$ vertices $U_j$ and draw an arc from each of them to $v_j$;\n\\item for each vertex $u\\in U_j$, add $k$ vertices and draw an arc from each of them to $u$, denote the set of these\n    $k(k-1)$ vertices $W_j$.\n\\end{itemize}\n\\end{itemize}\nNotice that if $k=1$, then $Y_i=Z_i=U_j=W_j=\\emptyset$. The construction of $G$ is shown in Fig.~\\ref{fig:NPh}. We set\n$b=n(k(k-1)+1)+mk(k-1)$ and $p=n((k+1)(k-1)+2)+m((k+1)(k-1)+1)$. It is straightforward to see that $G$ is acyclic. Because\neach variable $x_i$ is used at most 2 times in positive and at most 2 times in negations, $d_G(x_i), d_G(\\overline{x}_i)\\leq\n3$ for all $i\\in\\{1,\\ldots,n\\}$, and $\\Delta(G)\\leq k+2$. \nBecause the graph of the boolean formula is a subcubic planar graph,  \n$G$ is planar.\n\nWe claim that all clauses $C_1,\\ldots,C_m$ can be satisfied if and only if there are a set $A\\subseteq V(G)$\nand an induced subgraph $H$ of $G$ such\nthat $A\\subseteq V(H)$, $|A|\\leq b, |V(H)|\\geq p$, and for every $v\\in V(H)\\setminus A$, we have $d^{-}_{H}(v)\\geq k$.\n\nSuppose that we have a YES-instance of {\\sc Satisfiability} and consider a truth assignment of $x_1,\\ldots,x_n$ such that\nall clauses are satisfied. We construct $A$ by including all the vertices $Z_1\\cup\\ldots\\cup Z_n\\cup W_1\\cup\\ldots\\cup W_m$ in\nthis set, and for each $i\\in\\{1,\\ldots,n\\}$, if $x_i=\\text{true}$, then $x_i$ is included in $A$ and $\\overline{x}_i$ is\nincluded otherwise. Clearly, $|A|=|Z_1|+\\ldots+|Z_n|+|W_1|+\\ldots+|W_m|+n=n(k(k-1)+1)+mk(k-1)=b$. Let $H=G[A\\cup\nY_1\\cup\\ldots\\cup Y_n\\cup U_1\\cup\\ldots U_m\\cup\\{r_1,\\ldots,r_n\\}\\cup\\{v_1,\\ldots,v_m\\}]$. Consider $w\\in V(H)\\setminus A$. If\n$w\\in Y_i$ for $i\\in\\{1,\\ldots,n\\}$, then $w$ has $k$ in-neighbors in $Z_i\\subseteq A$. If $w=r_i$ for $i\\in\\{1,\\ldots,n\\}$,\nthen $w$ has $k-1$ in-neighbors in $Y_i$ and either $x_i$ or $\\overline{x}_i$ is an in-neighbor of $w$ as well. If $w\\in U_j$\nfor $j\\in\\{1,\\ldots,m\\}$, then $w$ has $k$ in-neighbors in $W_j\\subseteq A$. Finally, if $w=v_j$ for some\n$j\\in\\{1,\\ldots,m\\}$, then $w$ has $k-1$ in-neighbors in $U_j$. As the clause $C_j$ is satisfied, it contains a literal $x_i$\nor $\\overline{x_i}$ that has the value $\\text{true}$. Then by the construction of $A$, the corresponding vertex $x_i$ or\n$\\overline{x}_i$ respectively is in $A$, and $w$ has one in-neighbor in $A$. It remains to observe that\n$|V(H)|=|A|+|Y_1|+\\ldots+|Y_n|+|U_1|+\\ldots+|U_m|=n(k(k-1)+1)+mk(k-1)+k(n+m)=p$.\n\nAssume now there are a set $A\\subseteq V(G)$\nand an induced subgraph $H$ of $G$ such\nthat $A\\subseteq V(H)$, $|A|\\leq b, |V(H)|\\geq p$ and for every $v\\in V(H)\\setminus A$ we have $d^{-}_{H}(v)\\geq k$.\n\nLet $S=\\{w\\in V(G)\\ |\\ d_G^-(w)=0\\}=(\\cup_{i=1}^n\\{x_i,\\overline{x_i}\\})\\cup (\\cup_{i=1}^nZ_i)\\cup(\\cup_{j=1}^mW_j)$ and\n$T=V(G)\\setminus S=\\{r_1,\\ldots,r_n\\}\\cup (\\cup_{i=1}^nY_i)\\cup(\\cup_{j=1}^mU_j)$. \n We claim that\n$A\\subseteq S$ and $T\\subseteq V(H)$. To show it, observe that any vertex $w\\in S$ is in $H$ if and only if $w\\in A$ as\n$d_G^-(w)=0$. Because $|V(G)|-|V(H)|\\leq n$, at least $|S|-n$ vertices of $S$ are in $A$. Since $|S|=b+n$, we conclude that\nexactly $b=|S|-n$ vertices of $S$ are in $A$ and $A\\subseteq S$. Moreover, $V(H)=T\\cup A$.\n\nLet $z\\in Z_i$ for some $i\\in\\{1,\\ldots,n\\}$ and assume that $z$ is adjacent to $y\\in Y_i$. If $z\\notin A$, then $y\\in T$ has\nat most $k-1$ in-neighbors in $H$, a contradiction. Hence, $Z_1\\cup\\ldots\\cup Z_n\\subseteq A$. By the same arguments we\nconclude that $W_1\\cup\\ldots\\cup W_m\\subseteq A$. Then we have exactly $n$ elements of $A$ in\n$\\cup_{i=1}^n\\{x_i,\\overline{x}_i\\}$. Consider a pair of vertices $x_i,\\overline{x}_i$ for $i\\in\\{1,\\ldots,n\\}$. If\n$x_i,\\overline{x}_i\\notin A$, then $r_i\\in T$ has at most $k-1$ in-neighbors in $H$, a contradiction. Therefore, for each\n$i\\in\\{1,\\ldots,n\\}$, exactly one vertex from the pair  $x_i, \\overline{x}_i$ is in $A$. For $i\\in\\{1,\\ldots,n\\}$, we set the\nvariable $x_i=\\text{true}$ if the vertex $x_i\\in A$, and $x_i=\\text{false}$ otherwise.\n\nIt remains to prove that we have a satisfying truth assignment. Consider a clause $C_j$ for $j\\in\\{1,\\ldots,m\\}$. The vertex\n$v_j\\in T$ has $k-1$ in-neighbors in $H$ that are vertices of $T$. Hence, it has at least one in-neighbor in $A$. It can be\neither a vertex $x_i$ or $\\overline{x}_i$ that correspond to a literal in $C_j$. It is sufficient to observe that if $x_i\\in\nA$, then the literal $x_i=\\text{true}$, and if  $\\overline{x}_i\\in A$, then the literal $\\overline{x}_i=\\text{true}$ by our\nassignment.\n\\end{proof}\n\n\nWe conclude this section by the simple observation that \\textsc{Dir-AKC}\\xspace is in \\ensuremath{\\operatorClassXP}\\ when parameterized by the number of anchors $b$.\nFor a directed graph $G$ with $n$ vertices, we can consider all the at most $n^b$ possibilities to\nchoose the anchors, and then recursively delete non-anchor vertices that have the in-degree at most $k-1$. Trivially, if we\nobtain a directed graph with at least $p$ vertices for some selection of the anchors, we have a solution and otherwise we can\nanswer NO.\n\n\\section{\\textsc{Dir-AKC}\\xspace parameterized by the size of the core}\n\\label{sec:saved}\n\\vskip-2mm\nIn this section we consider the \\textsc{Dir-AKC}\\xspace problem for fixed $k$ when $p$ is a parameter and obtain the following dichotomy: If\n$k=1$ then the \\textsc{Dir-AKC}\\xspace problem is FPT parameterized by $p$, otherwise for $k\\geq 2$ it is \\classW1-hard parameterized by $p$.\n\n\\begin{theorem}\\label{thm:fpt-d-1}\nFor $k=1$, the \\textsc{Dir-AKC}\\xspace problem is solvable in time $2^{O(p)}\\cdot n^2\\log n$ on digraphs with $n$ vertices.\n\\end{theorem}\n\n\\begin{proof}\nThe proof is constructive, and we describe an \\ensuremath{\\operatorClassFPT}~ algorithm for the problem. Without loss of generality, we assume that\n$b< p\\leq n$.\n\nWe apply the following preprocessing rule reducing the  instance to  an acyclic graph. Let  $C_1,\\ldots,C_r$ be strongly\nconnected components of $G$. By making use of Tarjan's algorithm~\\cite{Tarjan72}, the sets  $C_1,\\ldots,C_r$ can be found  in\nlinear time.  Let $R=R_G^{+}\\Big(\\bigcup_{i=1}^rV(C_i)\\Big)$ be the set of vertices reachable from strongly connected\ncomponents. Then every $v\\in R$ satisfies $d_{G[R]}^-(v)\\geq 1$. If $b\\geq p-|R|$, then we select in $V(G)\\setminus R$ any\narbitrary $b'=p-|R|$ vertices $a_1,\\ldots,a_{b'}$. In this case we  output the set of anchors $A=\\{a_1,\\ldots,a_{b'}\\}$ and\ngraph $H=G[A\\cup R]$. Otherwise, if $b< p-|R|$,  we set $G'=G-R$ and $p'=p-|R|$ and consider a new instance of\n\\textsc{Dir-AKC}\\xspace  with the graph $G'$ and the parameter $p'$.\n\nTo see that the rule is safe, it is sufficient to observe that a set of anchors $A$ and a subgraph $H'$ of size at least $p'$\nis a solution of the obtained instance if and only if $(A,H=G[V(H')\\cup R])$ is a solution for the original problem. Let us\nremark that the preprocessing rule can be easily  performed in  time $O(n^2)$.\n\nFrom now we can assume that $G$ has no strongly connected components, i.e., $G$ is a directed acyclic graph. Denote by\n$S=\\{s_1,\\ldots,s_h\\}$ the set of sources of $G$. If $|S|\\leq b$, then set $A=S$. In this case, we output the pair $(A,H=G)$.\nThe pair $(A,H)$ is a solution because every vertex $v\\in V(G)\\setminus S$ satisfies $d_G^-(v)\\geq 1$. It remains to consider\nthe case when $|S|>b$. For $i\\in\\{1,\\ldots,h\\}$, let $R_i=R_G^+(s_i)$. Then $V(G)=R_G^+(S)=\\bigcup_{i=1}^hR_i$. Without loss\nof generality, we can assume that every anchored vertex is from $S$. Indeed, if $s_i$ is an anchor, then each vertex of $R_i$\ncan be included in a solution. Hence for every anchor $a\\in R_j\\setminus \\{s_j\\}$, we can delete anchor from $a$ and anchor\n$s_j$, if it is not yet anchored.\nSince we  can choose anchors only from $S$, we are able to reduce the problem to {\\sc Partial Set Cover}.\n\n\\begin{center}\n\\noindent\\framebox{\\begin{minipage}{4.50in}{\\textsc{Partial Set Cover}\\xspace }\\\\\n\\emph{Input }: A collection $X=\\{X_1,\\ldots,X_r\\}$ of subsets of a finite $n$-element set $U$ and positive integers $p, b$. \\\\\n\\emph{Parameter}: $p$.\\\\\n\\emph{Question}: Are there at most $b$ subsets $X_{i_1},\\ldots,X_{i_{b}}$, $1\\leq i_1<\\ldots<i_{b}\\leq r$, covering at least\n$p$ elements of $U$, i.e., $|\\bigcup_{j=1}^{b} X_{i_j}|\\geq p$?\n\\end{minipage}}\n\\end{center}\n\nBl\\\"{a}ser~\\cite{Blaser03} showed that \\textsc{Partial Set Cover}\\xspace if \\ensuremath{\\operatorClassFPT} parameterized by $p$ and can be solved in time $O(2^{O(p)}\\cdot r\nn\\log n)$.\nFor \\textsc{Dir-AKC}\\xspace, \nwe consider the collection of subsets $\\{R_1,\\ldots,R_r\\}$ of $V(G)$. If we can select at most $b$\nsubsets $R_{i_1},\\ldots,R_{i_{b}}$ such that  $|\\cup_{j=1}^{b}R_{i_j}|\\geq p$, we return the solution with anchors\n$A=\\{s_{i_1},\\ldots,s_{i_{b}}\\}$ and $H=G[\\bigcup_{j=1}^{b}R_{i_j}]$. Otherwise, we return a NO-answer.\n\nBecause our preprocessing can be done in time $O(n^2)$ and  {\\sc Partial Set Cover} is solvable in time $2^{O(p)}\\cdot n^2\\log\nn$, we conclude that the total running time is $2^{O(p)}\\cdot n^2\\log n$.\n\\end{proof}\n\nNow we complement Theorem~\\ref{thm:fpt-d-1} by showing that for $k\\geq 2$,  \\textsc{Dir-AKC}\\xspace becomes hard\nparameterized by the core size.\n\n\\begin{theorem}\\label{thm:w-saved}\nFor any fixed $k\\geq 2$, the \\textsc{Dir-AKC}\\xspace problem is \\classW1-hard parameterized by $p$, even for DAGs.\n\\end{theorem}\n\n\\begin{proof}\nWe reduce from the {\\sc $b$-Clique} problem which is known to be \\classW1-hard~\\cite{downey-fellows-book}:\n\n\\begin{center}\n\\noindent\\framebox{\\begin{minipage}{4.50in}{$b$-Clique}\\\\\n\\emph{Input}: A undirected graph  $G$ and a positive integer $b$. \\\\\n\\emph{Parameter}: $b$\\\\\n\\emph{Question}: Is there a clique of size $b$ in $G$?\n\\end{minipage}}\n\\end{center}\n\n\nFrom a given graph $G$ we construct a directed graph $G'$ as follows.\n\\begin{itemize}\n\\item Construct a copy of $V(G)$.\n\\item For each edge $\\{u,v\\}\\in E(G)$, construct a new vertex $w_{uv}$ and join $u,v$  with $w_{uv}$  the copy of $V(G)$\n    by arcs $(u,w_{uv})$ and $(v,w_{uv})$.\n\\item Construct $k-2$ vertices $z_1,\\ldots,z_{k-2}$, and for each $e\\in E(G)$, join $z_1,\\ldots,z_{k-2}$ with $w_e$ by\n    arcs.\n\\end{itemize}\nIt is straightforward to see that $G'$ is a directed acyclic graph. We say that the vertex $w_{uv}$ for $\\{u,v\\}\\in E(G)$ is a\n\\emph{subdivision} vertex, and we say that $v\\in V(G)$  is a \\emph{branch} vertex. Let $b'=b+k-2$ and\n$p=\\frac{b(b+1)}{2}+k-2$. Let $Z=\\{z_1\\ldots,z_{k-1}\\}$. We claim that $G$ has a clique of size $b$ if and only if there is a\nset of at most $b'$ vertices $A\\subseteq V(G')$ such that there exists an an induced subgraph $H$ of $G'$ with at least $p$\nvertices, $A\\subseteq V(H)$ and for any $v\\in V(H)\\setminus A$ we have  $d_H^-(v)\\geq k$.\n\nSuppose that $K$ is a clique in $G$ of size $b$. We let $A=K\\cup Z$ and define $U=\\{w_{uv}|u,v\\in K\\}$. Notice that\n$|U|=\\frac{b(b-1)}{2}$ and each vertex of $U$ has two in-neighbors in $A\\cap K$ and $k-2$ in-neighbors in $Z$. We conclude\nthat $H=G'[A\\cup U]$ has $p$ vertices and  for any $v\\in V(H)\\setminus A$ satisfies $d_H^-(v)\\geq k$.\n\nAssume now that there is a set of at most $b'$ vertices $A\\subseteq V[G']$ such that there exists an induced subgraph $H$ of\n$G'$ with at least $p$ vertices, $A\\subseteq V(H)$ and for any $v\\in V(H)\\setminus A$ we have  $d_H^-(v)\\geq k$. Since\nsubdivision vertices of $G'$ are sinks, we can assume that $A$ contains only \\emph{branch} vertices and vertices from $Z$, as\notherwise we can replace an anchor $a\\in A$ that is a subdivision vertex of $G'$ by an arbitrary branch vertex or a vertex of\n$Z$. Because branch vertices of $G'$ and the vertices of $Z$ are sources, any such vertex $v$ is in $H$ if and only if $v\\in\nA$. Hence, $H$ has at most $b'$ sources of $G'$ and at least $\\frac{b(b-1)}{2}$ subdivision vertices. If there is a vertex\n$z_i\\in Z$ such that $z_i\\notin A$, then each subdivision vertex $w_e$ has at most $k-1$ in-neighbors and $H$ cannot contain\nsubdivision vertices. Therefore $Z\\subseteq A$ and $A$ has at most $b'-(k-2)=b$ branch vertices. It remains to observe that a\nsubdivision vertex $w_{uv}$ has $k$ in-neighbors in $H$ if and only if $u,v\\in A$. Then the claim follows.\n\\end{proof}\n\n\\section{\\textsc{Dir-AKC}\\xspace on graphs of bounded degree}\n\\label{sec:bound-deg} \nIn this section we show that \\textsc{Dir-AKC}\\xspace problem is \\ensuremath{\\operatorClassFPT}\\ parameterized by $\\Delta+p$ if\n$k\\geq\\frac{\\Delta}{2}$.\n\nIn our algorithms we need to check the existence of solutions for \\textsc{Dir-AKC}\\xspace that have bounded size. It can be observed that if we\nare interested in solutions $(A,H)$ such that $p\\leq |V(H)|\\leq q$, then for every positive $q$, we can express this problem\nin the first order logic. It was proved by Seese~\\cite{Seese96} that any graph problem expressible in the first-order logic\ncan be solved in linear time on (directed) graphs of bounded degree. Later this result was extended for much more rich graph\nclasses (see~\\cite{DvorakKT10} ). These meta theorems are very general, but do not provide\ngood upper bounds for running time for particular problems. Hence, we give the following lemma.\nOur algorithms use the random separation technique due to Cai et al.~\\cite{CaiCC06} (which is a variant of the color coding\nmethod introduced by Alon et al.~\\cite{AlonYZ95}) .\n\n\n\\begin{lemma}\\label{lem:bounded}\nThere is a randomized algorithm with running time $2^{O(\\Delta q)}\\cdot n$ that for an instance of \\textsc{Dir-AKC}\\xspace with an $n$-vertex\ndirected graph of maximum degree at most $\\Delta$ and a positive integer $q\\geq p$, either returns a solution $(A,H)$ with $V(H)\\geq p$ or gives the answer\nthat there is no solution with $|V(H)|\\leq q$. Furthermore, the algorithm can be derandomized, and the deterministic variant\nruns in time  $2^{O(\\Delta q)}\\cdot n\\log n$.\n\\end{lemma}\n\n\\begin{proof}\nConsider an instance of \\textsc{Dir-AKC}\\xspace with an $n$-vertex directed graph $G$ of maximum degree at most $\\Delta$. We assume that $b\\leq\np\\leq n$. For given $q\\geq p$, to decide if $G$ contains a solution of size at  most $q$, we do the following.\n\nWe color each vertex of $G$ uniformly at random with probability $\\frac{1}{2}$ by one of  two colors, say red or\nblue. Let  $R$ be the set of vertices  colored red. Observe that if there is a solution $(A,H)$ with $|V(H)|\\leq q$, then with\nprobability at least $\\frac{1}{2^q}$ all vertices of $H$ are colored red and with probability at least $\\frac{1}{2^{\\Delta q}}$\nall \nin- and out-neighbors of the vertices of $H$ that are outside of $H$ are colored blue. Using this observation, we assume that\n$H$ is the union of some weakly connected components of  the graph $G[R]$ induced by red vertices.\n\nIn time $O(\\Delta n)$ we find all weakly connected components of $G[R]$. If there is a component $C$ with at least $b+1$\nvertices of in-degree at most $k-1$ (in $C$), then we discard this component as it cannot be a part of any solution. Denote by\n$C_1,\\ldots,C_r$ the remaining components. For $i\\in\\{1,\\ldots,r\\}$, let $A_i=\\{v\\in V(C_i)|d_{C_i}^-(v)<k\\}$, $b_i=|A_i|$ and\n$p_i=|V(C_i)|$.\n\nThus everything boils down to the problem of finding a set $I\\subseteq \\{1,\\ldots,r\\}$ such that $\\sum_{i\\in I} b_i\\leq b$ and\n$\\sum_{i\\in I}p_i \\geq p$.  But this is  the well known {\\sc Knapsack} problem, which  is solvable in time $O(bn)$ by dynamic\nprogramming. If we obtain a solution $I$, then we output $(A,H)$, where $A=\\cup_{i\\in I} A_i$ and $H=G[\\cup_{i\\in I}V(C_i)]$.\nOtherwise, we return a NO-answer. Notice that this algorithm can also find a solution $(A,H)$  with $|V(H)|> q \\geq p$.\n\n\nIt remains to observe that for any positive number $\\alpha<1$, there is a constant $c_{\\alpha}$ such that after running our\nrandomized algorithm $c_{\\alpha}\\cdot 2^{\\Delta q}$ times, we either find a solution $(A,H)$ or can claim that with\nprobability $\\alpha$ that it does not exist.\n\n\nThis algorithm can be derandomized by the  technique proposed by Alon et al.~\\cite{AlonYZ95}: replace the random colorings by\na family of at most $2^{O(\\Delta q)}\\cdot \\log n$ hash functions which are known to be constructible  in time $2^{O(\\Delta\nq)}\\cdot n\\log n$.\n\\end{proof}\n\n\nOur next aim is to prove that for $k>\\Delta\/2$ the \\textsc{Dir-AKC}\\xspace problem is \\ensuremath{\\operatorClassFPT}\\ when parameterized by the number of anchors\n$b$.\n\n\\begin{lemma}\\label{lem:bound-deg-anchors}\nLet $\\Delta$ be a positive integer. If $k>\\Delta\/2$, then the \\textsc{Dir-AKC}\\xspace problem can be solved in time $2^{O(\\Delta^2 b)}\\cdot\nn\\log n$ for $n$-vertex directed graphs of maximum degree at most $\\Delta$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $(A,H)$ is a solution for the \\textsc{Dir-AKC}\\xspace problem. Let us observe that because $k>\\Delta\/2$, for every vertex $v\\in\nV(H)\\setminus A$,  we have $d_H^-(v)>d_H^+(v)$. Recall that for any directed graph, the sum of in-degrees equals the sum of\nout-degrees. Then\n$$  \\sum_{v\\in V(H)\\setminus A}(d_H^-(v)-d_H^+(v))=\\sum_{v\\in A}(d_H^+(v)-d_H^-(v)).$$\nSince for every vertex $v\\in V(H)\\setminus A$,  $d_H^-(v)-d_H^+(v)\\geq 1$, we have that\n$$|V(H)\\setminus A|\\leq \\sum_{v\\in V(H)\\setminus A}(d_H^-(v)-d_H^+(v)).$$\nOn the other hand,  $ d_H^+(v)-d_H^-(v)\\leq \\Delta$, and we arrive at\n$$|V(H)\\setminus A|\\leq \\sum_{v\\in V(H)\\setminus A}(d_H^-(v)-d_H^+(v))=\\sum_{v\\in A}(d_H^+(v)-d_H^-(v))\\leq \\Delta|A|.$$\nHence, $|V(H)|\\leq (\\Delta+1)|A|\\leq (\\Delta+1)b$. Using this observation, we can solve the \\textsc{Dir-AKC}\\xspace problem as follows. If\n$p>(\\Delta+1)b$, then we return a NO-answer. If $p\\leq(\\Delta+1)b$, we apply Lemma~\\ref{lem:bounded} for $q=(\\Delta+1)b$, and\nsolve that problem in time  $2^{O(\\Delta^2 b)}\\cdot n\\log n$.\n\\end{proof}\n\n\nNow we show that if $k=\\frac{\\Delta}{2}$ then the \\textsc{Dir-AKC}\\xspace problem is FPT parameterized by $\\Delta+p$. Due the space restrictions we only sketch the proof of the following lemma.\n\n\\begin{lemma}\\label{lem:bound-deg-saved}\nLet $\\Delta$ be a positive integer. If $k=\\Delta\/2$, then the \\textsc{Dir-AKC}\\xspace problem can be solved in time $2^{O(\\Delta^3 b+ \\Delta^2 b\np)}\\cdot n^{O(1)}$ for $n$-vertex directed graphs of maximum degree at most $\\Delta$.\n\\end{lemma}\n\n\\begin{proof}\nWe describe an \\ensuremath{\\operatorClassFPT}~ algorithm. Consider an instance of the \\textsc{Dir-AKC}\\xspace problem. Without loss of generality we assume that $b<p\\leq n$.\n\nWe apply the following preprocessing rule. Suppose that $G$ has a (weakly) connected component $C$ such that for any $v\\in\nV(C)$, $d_C^-(v)=d_C^+(v)=k$. If $b\\geq p-|V(C)|$, then we choose a set $A$ of $b'=p-|V(C)|$ vertices arbitrary in\n$V(G)\\setminus V(C)$. Then we return a YES-answer, as the anchors $A$ and $H=G[A\\cup V(C)]$ is a solution. Otherwise, if $b<\np-|V(C)|$, we let $G'=G-V(C)$ and $p'=p-|V(C)|$. Now we consider a new instance of the problem with the graph $G'$ and the\nparameter $p'$. To see that the rule is safe, it is sufficient to observe that a set of anchors $A$ and a subgraph $H'$ of\nsize at least $p'$ is a solution of the obtained instance if and only if $A$ and $H=G[V(H')\\cup V(C)]$ is a solution for the\noriginal problem. From now we assume that $G$ has no such components.\n\n\\medskip\n\nWe need the following claim.\n\n\\medskip\n\\noindent\n{\\bf Claim A.} {\\it\nIf an instance of the \\textsc{Dir-AKC}\\xspace problem has a core with at least $(\\Delta p+1)b+1$ vertices, then it has a solution\n$(A,H)$ with the following property: there is a vertex $t\\in V(H)\\setminus A$  reachable in $H$ from any vertex of $H$.\nMoreover, for each vertex $v$ of $H$, there is a path from $v$ to $t$ with all vertices except $v$ in  $ V(H)\\setminus A$.\n}\n\n\n\n\\begin{proof}[Proof of Claim~A]\nLet  $(A,H')$ be a solution with the set of anchors $A$ and such that $V(H')> (\\Delta p+1)b$.\n\nWe show that $V(H')=R_{H'}^+(A)$, i.e., all vertices of $H'$ are reachable from the anchors. To obtain a contradiction,\nsuppose that there is a vertex $u\\in V(H')$ such that $u\\notin R_{H'}^+(A)$. Let $U=R_{H'}^-(u)$, i.e., $U$ is the set of\nvertices from which we can reach $u$. Clearly, $A\\cap U=\\emptyset$. Therefore, $d_{H'}^-(v)\\geq k=\\Delta\/2$ for $v\\in U$.\nNotice that for a vertex $v\\in U$, $N_{H'}^{-}(v)\\subseteq U$ by the definition. Hence, $d_{G[U]}^-(v)\\geq k=\\Delta\/2$ for $v\\in\nU$. Because the sum of in-degrees equals the sum of out-degrees, for  every vertex $v\\in U$, we have that\n$d_{G[U]}^-(v)=d_{G[U]}^+(v)=k=\\Delta\/2$. Then $C=G[U]$ is a component of $G$ such that for every $v\\in V(C)$,\n$d_C^-(v)=d_C^+(v)=k$, but such components are excluded by the preprocessing; a contradiction.\n\nObserve now that if $d_{H'}^-(v)<d_{H'}^+(v)$, then $d_{H'}^-(v)<k$ and thus $v\\in A$. Hence, by adding at most $\\Delta b$\n(maybe multiple) arcs from $V(H')\\setminus A$ to $A$, joining the vertices $v\\in V(H')$ of degrees $d_{H'}^-(v)>d_{H'}^+(v)$ with\nvertices of degrees  $d_{H'}^-(v)<d_{H'}^+(v)$, we  can  transform $H'$ into a disjoint union of  directed Eulerian graphs.\nSince $V(H')=R_{H'}^+(A)$, each of these directed Eulerian graphs contains at least one vertex of $A$. Thus  the set of arcs\nof $H'$ can be covered by at most $\\Delta b$ arc-disjoint directed walks,   each walk starting from a vertex of $A$ and never\ncoming back to $A$. Because $d_{H'}^-(v)\\geq k$ for $v\\in V(H')\\setminus A$, we have that $|E(G')|\\geq k(|V(H')|-b)> \\Delta\nkbp$. Then there is a walk $W$ with at least $kp+1$ arcs.  Let $a\\in A$ be the first vertex of $W$ and let $t$ be the last\nvertex of the walk. The walk $W$ visits $a$ only once, $t$  and all other vertices of $W$ are visited at most $k$ times. We\nconclude  that $W$ has at least $p$ vertices.\n\nLet $R=R_{H'-A}^-(t)$ and let $A'=\\{a\\in A\\ |\\ N_{H'}^+(a)\\cap R\\neq\\emptyset\\}$. Consider $H=G[R\\cup A']$. Since\n$V(W)\\subseteq V(H)$, $|V(H)|\\geq p$.  For any $v\\in V(H)\\setminus A$, the in-neighbors of $v$ in $H'$ are in $H$ by the\nconstruction and, therefore, $d_H^-(v)\\geq k$. It remains to observe that to select at most $b$ anchors, we take $A'\\subseteq\nV(H)$.\n\\end{proof}\n\nUsing  Claim~A, we proceed with our algorithm. We try to find a solution such that $H$ has at most $q=(\\Delta\np+1)b$ vertices by applying  Lemma~\\ref{lem:bounded}. It takes time $O(2^{O(\\Delta^2 bp)}\\cdot n\\log n)$. If we obtain a\nsolution, then we return it and stop. Otherwise, we conclude that every core  contains at least $(\\Delta p+1)b+1$\nvertices. By Claim~A, we can search for a solution   $H$ with a  non-anchor vertex $t$ which is reachable from\nall other vertices of $H$ by directed paths  avoiding $A$. Notice that since $t$ is a non-anchor vertex, we have that\n$d_G^-(t)\\geq k$. We try  at most $n$ possibilities for all possible choices of  $t$, and solve our problem for each choice.\nClearly, if we get a YES-answer for one of the choices, we return it and stop. Otherwise, if we fail,  we return a NO-answer.\n\nFrom now we assume that we already selected $t$. We denote by $G'$ the graph obtained from $G$ by adding  an artificial source\nvertex $s$ joined by arcs with all the vertices $v\\in V(G)$ with $d_G^-(v)<k$. Observe that $(s,t)\\notin E(G')$.\n\n\\medskip\n\nSuppose that $(A,H)$ is a solution with the set of anchors $A$ such that $t\\in V(H)\\setminus A$ is reachable in $H$ from any\nvertex of $H$ by a path with all inner vertices in $V(H)\\setminus A$. Denote by $\\delta_{G'}(H)$ the set $\\{v\\in V(H)\\ |\\\nN_{G'}^-(v)\\setminus V(H)\\neq \\emptyset\\}$, i.e., $\\delta_{G'}(H)$ contains vertices that have in-neighbors outside $H$. We\nneed a chain of claims about the structure of $H$ in $G'$.\n\n\n\\medskip\n\\noindent\n{\\bf Claim~B.} {\\it\n$|\\delta_{G'}(H)\\setminus A|\\leq \\Delta b$.\n}\n\n\n\\begin{proof}[Proof of Claim~B]\nLet $X=\\{v\\in V(H)\\ |\\ d_H^-(v)\\geq k\\text{ and }d_H^+(v)<k\\}$, $Y=\\{v\\in V(H)\\ |\\ d_H^-(v)=d_H^+(v)=k\\}$ and $Z=\\{v\\in V(H)\\\n|\\ d_H^-(v)<k\\}$. Clearly, $$\\sum_{v\\in X}(d_H^-(v)-d_H^+(v))+ \\sum_{v\\in Y}(d_H^-(v)-d_H^+(v))=\\sum_{v\\in\nZ}(d_H^+(v)-d_H^-(v))$$ Observe that $d_H^-(v)-d_H^+(v)\\geq 1$ for $v\\in X$,  $d_H^-(v)-d_H^+(v)=0$ for $v\\in Y$ and\n$d_H^+(v)-d_H^-(v)\\leq \\Delta$ for $v\\in Z$. Hence, $|X|\\leq \\Delta|Z|$. If $d_H^-(v)<k$ for $v\\in V(H)$, then $v\\in A$. It\nfollows that $Z\\subseteq A$ and $|Z|\\leq b$. We have $|X|\\leq \\Delta b$. Consider a vertex $v\\in \\delta_{G'}(H)\\setminus A$.\nIt has at least one in-neighbor outside $H$ in $G$ and $d_H^-(v)\\geq k$. Then $d_H^+(v)<k$ and  $v\\in X$. We conclude that\n$\\delta_{G'}(H)\\setminus A\\subseteq X$ and $|\\delta_{G'}(H)\\setminus A|\\leq \\Delta b$.\n\\end{proof}\n\n\n\\medskip\n\\noindent\n{\\bf Claim~C.} {\\it\nThere is  an $s-t$ separator $S$ in $G'$ of size at most $(\\Delta(k-1)+1)b$ such that\n $V(H)\\setminus A\\subseteq R_{G'-S}^-(t)$.\n}\n\n\\begin{proof}[Proof of Claim~C]\nLet $S=\\Big(\\delta_{G'}(H)\\cap A\\Big)\\cup\\Big(\\bigcup_{v\\in \\delta_{G'}(H)\\setminus A}(N_G^-(v)\\setminus V(H)\\Big)$, i.e., the\nset containing all anchors that are in $\\delta_{G'}$, and for each non-anchor vertex of $\\delta_{G'}$ containing all its\nin-neighbors outside of $H$. Consider a directed $(s,t)$-path $P$ in $G'$. Let $v$ be the first vertex in $P$ that is in\n$V(H)$ and let $u$ be its predecessor in $P$. If $v\\in A$, then $v\\in S$. If $v\\notin A$, then $u\\neq s$ as $H$ has no\nnon-anchor vertices with in-degree at most $k-1$ in $G$. Then $u\\in S$. We conclude that each $(s,t)$-path contains a vertex\nof $S$, i.e., this set is an $s-t$ separator.\n\nObserve that $V(H)\\setminus A\\subseteq R_{G'-S}^-(t)$ by the definition of $S$ and the fact that $t$ can be reached from any\nvertex of $H$ in this graph by a path with all inner vertices in $V(H)\\setminus A$.\n\nIt remains to show that $|S|\\leq (\\Delta(k-1)+1)b$.  By Claim~B, $|\\delta_{G'}(H)\\setminus A|\\leq \\Delta b$. A\nvertex $v\\in \\delta_{G'}(H)\\setminus A$ has at least one out-neighbor in $H$ because $t$ is reachable from $v$. Then $v$ has\nat most $k-1$ in-neighbors outside $H$. Hence $|S|\\leq |A|+(k-1)(\\delta_{G'}(H)\\setminus A)\\leq (\\Delta(k-1)+1)b$.\n\\end{proof}\n\nNow we can prove the following claim about important $s-t$ separators in $G'$.\n\n\\medskip\n\\noindent\n{\\bf Claim~D.} {\\it\nThere is an important $s-t$ separator $S^*$ of size at most $(\\Delta(k-1)+1)b$ in $G'$ such that $V(H)\\subseteq\nR_{G'-S^*}^-(t)\\cup S^*$.\n}\n\n\n\n\\begin{proof}[Proof of Claim~D]\nBy Claim~C, there is an $s-t$ separator $S'$ in $G'$ of size at most $(\\Delta(k-1)+1)b$ such that $V(H)\\setminus\nA\\subseteq R_{G'-S'}^-(t)$. Notice that $S'$ not necessary a minimal separator, but there is a minimal $s-t$ separator\n$S\\subseteq S'$. Clearly, $|S|\\leq (\\Delta(k-1)+1)b$.\n\nWe show that  $V(H)\\subseteq R_{G'-S}^-(t)\\cup S$.  \nBecause $R_{G'-S'}^-(t)\\subseteq  R_{G'-S}^-(t)$, we have that $V(H)\\setminus A\\subseteq R_{G'-S}^-(t)$. Also if an anchor $a$\nis in  $R_{G'-S'}^-(t)$, then $a\\in R_{G'-S}^-(t)$. Let $a\\in A\\cap S'$. If $a\\in A\\cap S$, then $a\\in R_{G'-S}^-(t)\\cup S$.\nIf  $a\\notin S$, then by Claim~C, $a$ has an out-neighbor $v\\in R_{G'-S'}^-(t)$ and in this case we have $a\\in\nR_{G'-S}^-(t)$.\n\nIt remains to observe that there is an important $s-t$ separator $S^*$ such that $|S^*|\\leq |S|\\leq (\\Delta(k-1)+1)b$ and\n$R_{G'-S}^-(t)\\subseteq   R_{G'-S^*}^-(t)$. Therefore, $V(H)\\subseteq R_{G'-S}^-(t)\\cup S\\subseteq R_{G'-S^*}^-(t)\\cup S^*$.\n\\end{proof}\n\nThe next step of our algorithm is to check all important $s-t$ separators in $G'$ of size at most  $(\\Delta(k-1)+1)$. By\nLemma~\\ref{lem:imp-sep}, there are at most $4^{(\\Delta(k-1)+1)b}$ important $s-t$ separators and they can be listed in time\n$2^{O(\\Delta^2b)}\\cdot n^c$. For each important $s-t$ separator $S^*$, we consider the set of vertices $U= R_{G'-S^*}^-(t)\\cup\nS^*$ and decide whether there is a solution such that $V(H)\\subseteq U$. If we have a solution for some $S^*$, then we return\na YES-answer and stop. Otherwise, if we fail  to find such a solution for all important separators, we use\nClaim~D  to deduce that there is no solution.\n\n\\medskip\nFrom now on, we assume that an important $s-t$ separator $S^*$ is given and that  $U= R_{G'-S^*}^-(t)\\cup S^*$.\nIn what follows, we describe a procedure of finding  a solution with  $V(H)\\subseteq U$.\n\n\n\\medskip\nDenote by $D$ the set $\\{v\\in U\\ |\\ d_G^-(v)>0\\}$. We need the following observation.\n\n\n\n\\medskip\n\\noindent\n{\\bf Claim~E.} {\\it\nSet $D$ contains at most $(\\Delta+1) (\\Delta(k-1)+1)b$ vertices.\n}\n\n\n\\begin{proof}[Proof of Claim~E]\nLet $Q=G[U]$. Let $X=\\{v\\in V(Q)\\ |\\ d_Q^-(v)\\geq k\\text{ and }d_Q^+(v)<k\\}$, $Y=\\{v\\in V(Q)\\ |\\ d_Q^-(v)=d_Q^+(v)=k\\}$ and\n$Z=\\{v\\in V(Q)\\ |\\ d_Q^-(v)<k\\}$. Clearly,\n$$\\sum_{v\\in X}(d_Q^-(v)-d_Q^+(v))+ \\sum_{v\\in Y}(d_Q^-(v)-d_Q^+(v))=\\sum_{v\\in Z}(d_Q^+(v)-d_Q^-(v))$$\nObserve that $d_Q^-(v)-d_Q^+(v)\\geq 1$ for $v\\in X$,  $d_Q^-(v)-d_Q^+(v)= 0$ for $v\\in Y$ and $d_Q^+(v)-d_Q^-(v)\\leq \\Delta$\nfor $v\\in Z$. Hence, $|X|\\leq \\Delta|Z|$.\n\nRecall that $G'$ is obtained from $G$ by joining $s$ with all vertices of in-degree at most $k-1$. Since $S^*$ is an $s-t$\nseparator, if for $v\\in U$, $d_Q^-(v)<k$, then $v\\in S^*$. Hence, $Z\\subseteq S^*$ and $|Z|\\leq |S^*|\\leq (\\Delta(k-1)+1)b$.\nIf for for $v\\in U$, $d_G^-(v)>k$, then $v\\in X\\cup Z$. We conclude that $|D|\\leq|X|+|Z|\\leq (\\Delta+1)|Z|\\leq (\\Delta+1)\n(\\Delta(k-1)+1)b$.\n\\end{proof}\n\nRecall   that  set $\\delta_{G'}(H)$ contains vertices of $H$ that have in-neighbors outside of $H$. If $v\\in\n\\delta_{G'}(H)\\setminus A$, then it has at least $k$ in-neighbors in $H$ and at least one in-neighbor outside $H$. Notice that\n$s\\notin N_{G'}^-(v)$ because $d_G^-(v)\\geq d_H^-(v)\\geq k$. Hence, $d_G^-(v)>k$. Because $V(H)\\subseteq U$,\n$\\delta_{G'}(H)\\setminus A\\subseteq D$. By Claim~C, $|\\delta_{G'}(H)\\setminus A|\\leq \\Delta b$, and by\nClaim~E, $|D|\\leq (\\Delta+1) (\\Delta(k-1)+1)b$.\nWe consider  all at most\n$  2^{(\\Delta+1) (\\Delta(k-1)+1)b}$ possibilities to select  $\\delta_{G'}(H)\\setminus A$. For each choice of\n$\\delta_{G'}(H)\\setminus A$, we guess the arcs that join the vertices that are outside $H$ with the vertices of\n$\\delta_{G'}(H)\\setminus A$ and delete them. Denote the graph obtained from $G$ by $F$. Recall that from each vertex $v$ of\n$\\delta_{G'}(H)\\setminus A$, there is a directed path to $t$ that avoids $A$. Hence, $v$ has at least one out-neighbor in $H$\nand at most $\\Delta-1$ in-neighbors in $G$. Also $v$ has at least $k$ in-neighbors in $H$, and we delete at most $d_G^-(v)-k$\narcs. Therefore, for $v$ we choose at most $k-1$ arcs out of at most $\\Delta-1$ arcs. We can upper bound the number of\npossibilities for $v$ by $2^{\\Delta-1}$, and the total number of possibilities for $\\delta_{G'}(H)\\setminus A$ is\n$2^{(\\Delta-1)\\Delta b}$.\n\nObserve that $(A,H)$ is a solution for the new instance of \\textsc{Dir-AKC}\\xspace, where $G$ is replaced by $F$ for a correct guess of the\ndeleted arcs. Also each solution for the new instance provides a solution for the graph $G$, because if we put deleted arcs\nback, then we can only increase in-degrees. Hence, we can check for each possible choice of the set of deleted arcs, whether\nthe new instance has a solution. If for some choice we obtain a solution, then we return a YES-answer. Otherwise, if we fail\nfor all choices, then we return a NO-answer. Further we assume that $F$ is given.\n\n\nDenote by $F'$ the graph obtained from $F$ by the addition of a vertex $s$ joined by arcs with all the vertices $N_{G'}^+(s)$.\nNow $\\delta_{F'}(H)=\\{v\\in V(H)\\ |\\ N_{F'}^-(v)\\setminus V(H)\\neq \\emptyset\\}$. By the choice of $F$,\n$\\delta_{F'}(H)=\\delta_{G'}(H)\\cap A$ and, therefore, $|\\delta_{F'}(H)|\\leq b$. Also $\\delta_{F'}(H)$ is an $s-t$ separator in\n$F'$ by Claim~C.\n\nNow we can prove the following.\n\n\\medskip\n\\noindent\n{\\bf Claim~F.} {\\it\nThere is an important $s-t$ separator $\\hat{S}$ of size at most $b$ in $F'$ such that $(\\hat{S},G[R_{F'-\\hat{S}}^-(t)\\cup\n\\hat{S}])$ is a solution for the instance of the \\textsc{Dir-AKC}\\xspace problem for the graph $G$.\n}\n\n\n\\begin{proof}[Proof of Claim~F]\nLet $U=R_{F'-\\hat{S}}^-(t)\\cup \\hat{S}$. It was already observed that $\\delta_{G'}^*(H)$ is an $s-t$ separator in $F'$ of size\nat most $b$. Then there is a minimal $s-t$ separator $S\\subseteq \\delta_{G'}^*(H)$. Clearly, $|S|\\leq b$.\n\nAs before in the proof of Claim~D, we show that  $V(H)\\subseteq R_{F'-S}^-(t)\\cup S$. Because for any vertex\n$v$ of $H$, there is a directed $(v,t)$ path with all inner vertices in $V(H)\\setminus A$, $V(H)\\setminus A\\subseteq\nR_{F'-\\delta_{F'}(H)}^-(t)$. Because $R_{F'-\\delta_{F'}(H)}^-(t)\\subseteq  R_{F'-S}^-(t)$ we have $V(H)\\setminus A\\subseteq\nR_{F'-S}^-(t)$. Also if  $a\\in A$ is in  $R_{F'-\\delta_{F'}(H)}^-(t)$, then $a\\in R_{F'-S}^-(t)$. Let $a\\in A\\cap\n\\delta_{F'}(H)$. Trivially, if $a\\in A\\cap S$, then $a\\in R_{F'-S}^-(t)\\cup S$. If  $a\\notin S$, then $a$ has an out-neighbor\n$v\\in R_{F'-\\delta_{F'}(H)}^-(t)$ and $a\\in R_{F'-S}^-(t)$. Then there is an important $s-t$ separator $\\hat{S}$ such that\n$|\\hat{S}|\\leq |S|\\leq b$ and $R_{F'-S}^-(t)\\subseteq R_{F'-\\hat{S}}^-(t)$. Therefore, $V(H)\\subseteq R_{F'-S}^-(t)\\cup\nS\\subseteq R_{F'-S^*}^-(t)\\cup S^*$, and $|U|\\geq p$.\n\nIt remains to observe that $s$ is adjacent to all vertices of $G$ with in-degrees at most $k-1$ and $S^*$ is an $s-t$\nseparator. It immediately follows that for any  vertex $v\\in R_{F'-S^*}^-(t)$, $d_{F(U)}^-(v)\\geq k$. Then\n$(\\hat{S},G[R_{F'-\\hat{S}}^-(t)\\cup \\hat{S}])$ is a solution.\n\\end{proof}\n\nThe final step of our algorithm is to enumerate all important $s-t$ separators $\\hat{S}$ of size at most $b$ in $F'$, which\nnumber by Lemma~\\ref{lem:imp-sep} is at most $4^b$, and for each $\\hat{S}$, check whether $(\\hat{S},G[R_{F'-\\hat{S}}^-(t)\\cup\n\\hat{S}])$ is a solution. Recall that all these separators can be listed in time $2^{O(b)}\\cdot n^c$. We return a YES-answer\nif we obtain a solution for some important separator, and a NO-answer otherwise.\n\nTo complete the proof, let us observe that each step of the algorithm runs either in polynomial or \\ensuremath{\\operatorClassFPT}~ time.\nParticularly, the preprocessing is done in time $O(\\Delta n)$. Then we check the existence of a solution of a bounded size in\ntime $2^{O(\\Delta^2 bp)}\\cdot n\\log n$. Further we consider at most $n$ possibilities to choose $t$. For each $t$, we consider\nat most $4^{(\\Delta(k-1)+1)b}$ important $s-t$ separators $S^*$. Recall, that they can be listed in time\n$2^{O(\\Delta^2b)}\\cdot n^c$ for some constant $c$. Then for each $S^*$, we have  at most  $2^{(\\Delta+1)\n(\\Delta(k-1)+1)b+(\\Delta-1)}$ possibilities to construct $F$, and it can be done in time $2^{O(\\Delta^3b)}+O(\\Delta n)$.\nFinally, there are at most $4^b$ important $s-t$ separators $\\hat{S}$ and they can be listed in time $2^{O(b)}\\cdot n$ for\nsome $c$. We conclude that the total running time is $2^{O(\\Delta^3b+\\Delta^2bp)}\\cdot n^c$ for some constant $c$.\n\\end{proof}\n\nCombining Lemmas~\\ref{lem:bound-deg-anchors} and \\ref{lem:bound-deg-saved}, we obtain the following theorem.\n\n\\begin{theorem}\\label{thm:bound-deg}\nLet $\\Delta$ be a positive integer. If $k\\geq \\frac{\\Delta}{2}$, then the \\textsc{Dir-AKC}\\xspace problem can be solved in time $2^{O(\\Delta^3\nb+ \\Delta^2 b p)}\\cdot n^{O(1)}$ for $n$-vertex directed graphs of maximum degree at most $\\Delta$.\n\\end{theorem}\n\nTheorems~\\ref{thm:fpt-d-1} and \\ref{thm:bound-deg} give the next corollary.\n\n\\begin{corollary}\\label{cor:fpt-delta}\nThe \\textsc{Dir-AKC}\\xspace problem can be solved in time $2^{O(bp)}\\cdot n^{O(1)}$  for $n$-vertex directed graphs of maximum degree at most\n$4$.\n\\end{corollary}\n\n\n\\section{Conclusions}\\label{sec:concl}\nWe proved that \\textsc{Dir-AKC}\\xspace is \\ensuremath{\\operatorClassNP}-complete even for planar DAGs of maximum degree at most $k+2$. It was also shown  that \\textsc{Dir-AKC}\\xspace\nis \\ensuremath{\\operatorClassFPT}~ when parameterized by $p+\\Delta$ for directed graphs of maximum degree at most\n$\\Delta$ whenever $k\\geq\\Delta\/2$. It is natural to ask whether the problem is \\ensuremath{\\operatorClassFPT}~ for other values $k$. This question\nis interesting even for the special case $\\Delta=5$ and $k=2$.\n\nFor the special case of  directed acyclic graphs (DAGs) we understand  the complexity of the problem much better.\nTheorem~\\ref{thm:w-saved} showed that \\textsc{Dir-AKC}\\xspace on DAGs is \\classW{1}-hard parameterized by $p$ for every fixed $k\\geq 2$, when\nthe degree of the graph is not bounded. We now show the following theorem  that gives\n\\classW{2}-hardness of \\textsc{Dir-AKC}\\xspace when parameterized  by the number of anchors $b$ (recall that we can always assume that $b\\leq\np$).\n\n\\begin{theorem}\\label{thm:w-hardness-dags-bounded-degree}\nFor any $\\Delta\\geq 3$ and any positive $k<\\frac{\\Delta}{2}$,  \\textsc{Dir-AKC}\\xspace  is \\classW{2}-hard (even on DAGs) when parameterized by\nthe number of anchors $b$ on graphs of maximum degree at most $\\Delta$.\n\\end{theorem}\n\n\n\\begin{proof}\nFirst, we prove the claim for $k=1$ and $\\Delta=3$. We reduce from the {\\sc $b$-Set Cover} problem which is known to be\n\\classW2-hard~\\cite{downey-fellows-book}:\n\n\\begin{center}\n\\noindent\\framebox{\\begin{minipage}{4.50in}{\\textsc{$b$-Set Cover} }\\\\\n\\emph{Input }: A collection $X=\\{X_1,\\ldots,X_r\\}$ of subsets of a finite $n$-element set $U$ and a positive integer $b$. \\\\\n\\emph{Parameter}: $b$\\\\\n\\emph{Question}: Are there at most $b$ subsets $X_{i_1},\\ldots,X_{i_{b}}$ such that these sets cover $U$, i.e., $U=\n    \\bigcup_{j=1}^{b}X_{i_j}$?\n\\end{minipage}}\n\\end{center}\n\n\\begin{figure}[ht]\n\\centering\\scalebox{0.7}{\\input{fig2.pdf_t}}\n\\caption{Construction of $G$ for $U=\\{u_1,u_2,u_3\\}$ and $X_1=\\{u_1,u_2\\},X_2=\\{u_2,u_3\\} $.\n\\label{fig:W2h-1}}\n\\end{figure}\n\n\nLet $U=\\{u_1,\\ldots,u_n\\}$. We construct the directed graph $G$ as follows (see Fig.~\\ref{fig:W2h-1}).\n\\begin{itemize}\n\\item For $i\\in\\{1,\\dots,r\\}$, assume that $X_i=\\{u_{j_1},\\ldots,u_{j_s}\\}$ and\n\\begin{itemize}\n\\item construct a vertex $v_i$ and $s$ vertices $x_{ij_1},\\ldots,x_{ij_s}$;\n\\item construct arcs $(v_i,x_{ij_1}),(x_{ij_1},x_{ij_2}),\\ldots,(x_{ij_{s-1}},x_{ij_s})$.\n\\end{itemize}\n\\item For $j\\in\\{1,\\ldots,n\\}$, assume that $u_j$ is included in the sets $X_{i_1},\\ldots,X_{i_t}$ and\n\\begin{itemize}\n\\item construct a vertex $w_j$ and $t$ vertices $y_{ji_1},\\ldots,y_{ji_t}$;\n\\item construct arcs $(y_{ji_1},y_{ji_2}),\\ldots,(y_{ji_{t-1}},y_{ji_t})$;\n\\item join $y_{ji_t}$ with $w_j$ by a directed path $P_j$ of length $\\ell=2rn+r$.\n\\end{itemize}\n\\item For $i\\in\\{1,\\dots,r\\}$ and $j\\in\\{1,\\ldots,n\\}$, if $u_j\\in X_i$, then construct an arc $(x_{ij},y_{ji})$.\n\\end{itemize}\nIt is straightforward to see that $G$ is a directed acyclic graph of maximum degree at most 3. We set $p=n \\ell$. We claim\nthat $U$ can be covered by at most $b$ sets if and only if there is a set of at most $b$ vertices $A$ such that there exists\nan induced subgraph $H$ of $G$ with at least $p$ vertices, $A\\subseteq V(H)$ and for any $v\\in V(H)\\setminus A$, $d_H^-(v)\\geq\n1$.\n\nNotice that $v_1,\\ldots,v_r$ are the sources of $G$, $w_1,\\ldots,w_n$ are the sinks, and $V(G)=\\bigcup_{i=1}^rR_{G}^+(v_i)$.\nObserve also that $w_j$ can be reached from $v_i$ if and only if $u_j\\in X_i$.\n\nSuppose that $U$ can be covered by at most $b$ sets say $X_{i_1},\\ldots,X_{i_{b}}$. Let $A=\\{v_{i_1},\\ldots,v_{i_{b}}\\}$ and\n$H=G[R_{G}^+(A)]$. It is straightforward to see that for any vertex $z\\in V[H]$, $d_{H}^-(z)\\geq 1$. Because $U$ is covered,\nall vertices $w_1,\\ldots,w_n$ are in $H$ and, therefore, $V(P_1)\\cup\\ldots\\cup V(P_n)\\subseteq V(H)$. It remains to observe\nthat $|V(P_1)\\cup\\ldots\\cup V(P_n)|=n(\\ell+1)\\geq p$ and we conclude that $(A,H)$ is a solution of our instance of \\textsc{Dir-AKC}\\xspace.\n\nAssume now that $(A,H)$ is a solution of the \\textsc{Dir-AKC}\\xspace problem. Without loss of generality we can assume that that each $a\\in A$\nis a source of $G$. Otherwise, $a\\in R_G^+(v_i)$ for some source $v_i$, and we can replace $a$ by $v_i$ in $A$ (or delete it\nif $v_i\\in A$ already). Let $\\{i\\ |\\ 1\\leq i\\leq n,~v_i\\in A\\}=\\{i_1,\\ldots,i_{b}\\}$. We show that $X_1,\\ldots,X_{i_{b}}$\ncover $U$. To obtain a contradiction, assume that there is an element $u_j\\in U$ such that $u_j\\notin X_{i_1}\\cup \\ldots\\cup\nX_{i_{b}}$. Then the vertex $w_j$ is not reachable from $A$. Hence, the vertices of $P_j$ are not reachable from $A$. It\nfollows that $V(P_j)\\cap V(H)=\\emptyset$. We have that $|V(H)|\\leq |V(G)|-|V(P_j)|$. Because $|X_i|\\leq n$ for\n$i\\in\\{1,\\ldots,r\\}$ and each $u_h$ is included in at most $r$ sets for $h\\in\\{1,\\ldots,n\\}$, $|V(G)|\\leq\nr(n+1)+n(r+\\ell)=2rn+r+n \\ell=2rn+r+p$. Therefore, $|V(H)|\\leq p+(2rn+r-(\\ell+1))<p$ because $P_j$ has $\\ell+1$ vertices; a\ncontradiction.\n\nNow we prove \\classW{2}-hardness for $k\\geq 2$ and $\\Delta>2k$. We reduce from an instance of the \\textsc{Dir-AKC}\\xspace problem with $k=1$\nand $\\Delta=3$. Consider an instance of this problem with a directed acyclic graph $G$ and positive integers $b, p$. Assume\nthat $b\\leq p\\leq |V(G)|$ and $|V(G)|\\geq 3$. We construct the graph $G'$ as follows (see Fig.~\\ref{fig:W2h-2}).\n\n\\begin{figure}[ht]\n\\centering\\scalebox{0.7}{\\input{fig3.pdf_t}}\n\\caption{Construction of $G'$ for $k=4$.\n\\label{fig:W2h-2}}\n\\end{figure}\n\n\\begin{itemize}\n\\item Construct a copy of $G$ and denote its vertices by $v_1,\\ldots,v_n$.\n\\item For each $i\\in\\{1,\\ldots,n\\}$, construct a set of $k$ vertices $D_i$ and join $k-1$ vertices of this set with $v_i$\n    by arcs.\n\\item For each $i\\in \\{2,\\ldots,n\\}$, join each vertex of $D_{i-1}$ with all vertices of $D_i$ by arcs.\n\\end{itemize}\nClearly, $G'$ is a directed acyclic graph. We let $b'=b+k$ and $p'=p+nk$. Let also $D=D_1\\cup\\ldots\\cup D_n$. Notice that for\neach $v\\in V(G)$, $d_{G'}(v)=d_G(v)+k-1\\leq k+2\\leq\\Delta$ as maximum degree of $G$ is 3. For $v\\in D$, $d_{G'}(v)\\leq\n2k+1\\leq\\Delta$. Hence maximum degree of $G'$ is at most $\\Delta$. We now claim that there is a set of at most $b$ vertices\n$A\\subseteq V(G)$ such that there exists an an induced subgraph $H$ of $G$ with at least $p$ vertices, $A\\subseteq V(H)$ and\nfor any $v\\in V(H')\\setminus A$,  $d_{H}^-(v)\\geq 1$ if and only if there is a set of at most $b'$ vertices $A'\\subseteq\nV(G')$ such that there exists an an induced subgraph $H'$ of $G'$ with at least $p'$ vertices, $A'\\subseteq V(H')$ and for any\n$v\\in V(H)\\setminus A$,  $d_{H'}^-(v)\\geq k$.\n\nSuppose that our original instance of \\textsc{Dir-AKC}\\xspace has a solution $(A,H)$. We let $A'=A\\cup D_1$ and $H'=G'[V(H)\\cup D]$. Then each\nvertex $v\\in D\\setminus A'$ has $k$ in-neighbors in $D$. It remains to observe that each vertex $v$ of $G'$ from\n$V(G)\\setminus A'$ has at least one in-neighbor in $V(G)$ and $k-1$ in-neighbors in $D$. Therefore, $d_{G'}^-(v)\\geq k$.\n\nAssume now that $(A',H')$ is a solution for the constructed instance of \\textsc{Dir-AKC}\\xspace with $|A'|\\leq b'$ and $|V(H)|\\geq p'$. If\n$|D\\cap A'|<k$, then we claim that $D\\cap V(H')\\subseteq A'$. To prove it, suppose that $(V(H')\\cap D)\\setminus\nA\\neq\\emptyset$ and consider the smallest index $i$ such that there is $v\\in (V(H')\\cap D_i)\\setminus A$. Clearly, $i\\geq 2$.\nThe vertex $v$ has in-neighbors only in $D_{i-1}$. By the choice of $i$, $D_{i-1}$ has at most $k-1$ vertices of $H'$, because\nthey can be only anchors and $|D\\cap A'|<k$. Then $d_{H'}^-(v)<k$, a contradiction.\n\nThen if $|D\\cap A'|<k$, $V(H')\\subseteq V(G)\\cup A'$ and $|V(H')|\\leq n+b+k\\leq n+p+k<p'$ as $n\\geq 3$ and $k\\geq 2$. This\ncontradicts our assumption about size of $H'$. Hence, at least $k$ anchors are in $D$ and $|A'\\setminus D|\\leq b$. Let\n$A=A'\\setminus D$ and $H=H'-D$. If $v\\in V(H)\\setminus A$, then $d_{H'}^-(v)\\geq k$ and $v$ has at most $k-1$ in-neighbors\nfrom $D$ in $H'$. Then $v$ has at least one in-neighbor in $V(H)$ and $d_H^-(v)\\geq 1$.\n\\end{proof}\n\n\nThe case of $k\\geq \\frac{\\Delta}{2}$ the complexity of parameterization by $b$ on DAGs is left open. However we can show that\n\\textsc{Dir-AKC}\\xspace is \\ensuremath{\\operatorClassFPT} on DAGs of maximum degree $\\Delta$, when parameterized by $\\Delta+p$.\n\n\\begin{theorem}\\label{thm:dags}\nFor any positive integers $p$ and $\\Delta$, \\textsc{Dir-AKC}\\xspace can be solved in time $2^{O(\\Delta p)}\\cdot n^2\\log n$ for\n$n$-vertex DAGs of maximum degree at most $\\Delta$.\n\\end{theorem}\n\n\\begin{proof}\nConsider an instance of \\textsc{Dir-AKC}\\xspace with an $n$-vertex directed acyclic graph $G$. Without loss of generality we can assume that\n$b\\leq p\\leq n$.\n\nWe apply Lemma~\\ref{lem:bounded} for $q=p$. In time $2^{O(\\Delta p)}\\cdot n\\log n$ we either obtain a solution or conclude\nthat for any solution $(A,H)$, $H$ has size at least $p+1$. If we obtain a solution, we return it. Suppose that we got a\nNO-answer. If $p=n$, then we return a NO-answer. Otherwise,  we select a sink $t\\in V(G)$ using the fact that any directed\nacyclic graph has at least one such vertex. Observe that we can assume that $t$ is not an anchor in any solution. Also if $t$\nis included in a solution $H$ of size at least $p+1$, then $H-t$ is a solution of size at least $p$, because $t$ is not joined\nby arcs with other vertices of $H$. Then we solve the instance $G-t$ of \\textsc{Dir-AKC}\\xspace recursively.\n\nAs each step is done in time $2^{O(\\Delta p)}\\cdot n\\log n$ and the number of steps is at most $n$, the claim follows.\n\\end{proof}\n\nLet us remark that this result can be easily extended for any class of directed acyclic graphs $\\mathcal{G}$ such that the\ncorresponding class of underlaying graphs $\\{G^*|G\\in \\mathcal{G}\\}$ has (locally) bounded expansion by making use of  the\nresults by Dvorak et al.~\\cite{DvorakKT10}.\nFinally, what happens when the input graph is planar? We know that the problem is \\ensuremath{\\operatorClassNP}-complete on planar graphs for fixed\n$k\\geq 1$ and maximum degree $k+2$. Is the problem \\ensuremath{\\operatorClassFPT}\\ on planar directed graphs when parameterized by the size of the\ncore $p$?\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nKnee osteoarthritis (OA) is a leading cause of disability in the United States, affecting over 10\\% percent of the adult population, and accounting for over 600,000 knee replacements costing in excess of \\$10 billion per year \\cite{Zhang2010, Murphy2012, Cram2012, Kremers2015}. Although commonly characterized by joint pain and functional disability, OA is a heterogeneous disease with multiple risk factors and pathways contributing to variable disease symptoms and progression. The pathogenesis of OA is complex, with biomechanical joint stressors, inflammatory mediators, and structural changes to bone, ligaments, joint synovium, and menisci leading to articular cartilage degradation and loss \\cite{Robinson2016, Lane2011}. No FDA-approved drugs are available which can forestall cartilage degeneration. Imaging biomarkers that are sensitive to structural changes are needed for early detection of OA in an effort to provide useful insights to clinical trials investigating disease modifying drugs. Magnetic resonance imaging (MRI) of the knee provides a complete 3D view of the knee anatomy unlike traditional 2D x-rays. Moreover, changes in cartilage morphology are detected earlier in MRIs than x-rays \\cite{Sharma2008}.  In a longitudinal study by Raynauld et al.\\ \\cite{Raynauld2005}, MRI was used to reliably measure structural changes in arthritic knees and detect cartilage volume loss as early as 12 months from baseline. Balamoody et al. \\cite{Balamoody2010} demonstrated that similar MRI protocols across different 3T scanners gave comparable morphologic results, further suggesting that MR imaging is a dependable tool for quantitative cartilage morphology analysis.\n\nFor analysis of the cartilage structures, accurate segmentation is a crucial first step. In clinical research there is a need for reproducible fully or highly automated segmentation as it offers consistent accuracy and speed over manual segmentation efforts. However, automated knee segmentation is challenging due to the thinning cartilage, appearance of osteophytes, bone marrow and cartilage lesions and surface fibrillation in MRI data. Several of these disease artifacts appear similar to the cartilage further increasing the segmentation complexity (Fig.~\\ref{fig:4d}a).\n\nAutomated knee segmentation approaches previously reported in the literature were surveyed based on the employed methodology. Active shape and appearance models (ASM\/AAM) were widely used in a variety of combinations such as deformable active shape models \\cite{Fripp2010, Fripp2007}, volumetric and surface based appearance models \\cite{Williams2010}, semi-automated initialization of active shape models \\cite{Lynch2001}, and minimum descriptor length based group-wise image registration followed by active appearance model  segmentation \\cite{Vincent2010}. In the presence of pathology, active shape\/active appearance models need to be flexible enough to properly represent many local changes of shape\/appearance, thus requiring large training-set sizes and yielding high-dimensional models. The associated model-training and computational complexity is already substantial in 3D and may become impractical if 4D longitudinal analysis is considered. Multi-atlas based methods were used in several algorithms such as \\cite{Shan2014} with a non-local patch based fusion, Dam et al.\\ \\cite{Dam2015}  extended their multi-atlas based rigid registration by k-nearest neighbor voxel based classification in \\cite{Folkesson2007}. Multi-atlas registration followed by locally weighted voting was reported in \\cite{Lee2014},  multi-atlas based registration followed by outlier detection to segment the tibio-femoral joint in \\cite{Tamez-Pena2012}. Since multi-atlas methods optimize a match between the overall template labeling as well as the individual labeled templates in the set, they may be sensitive to imperfect label templates in the presence of pathology. Both multi-atlas and ASM\/AAM approaches may also suffer from converging to a locally rather than globally optimal solutions. Markov random fields were used to construct and optimize local image patches for the region and boundary probabilities from local shape and appearance information in \\cite{Lee2011}. Graph cuts optimization was applied to the output of a hierarchical two-stage classifier, which was trained to identify the cartilage and  bone voxels \\cite{Wang2014}. As the underlying multi-label graph cuts jointly consider independent classifier outputs for cartilage,  bone, and background, label-conflict-resolution  may be challenging in regions with multiple labels. In Zhang et al.\\ \\cite{Zhang2013}, four different MR acquisitions of the same patients were used to leverage the benefits of different sequence contrasts to model the spatial contextual information using support vector machines in a discrete random field framework. While a conceptually interesting approach, the need for multiple MR acquisitions limits the method's applicability.\n\nUsing cost functions designed as gradient-based features with heuristically determined weighting combination based on human expert knowledge (hand-tuned cost functions, Section II) limits the segmentation performance especially in the pathologic cases. Machine learning based methods can be used to identify the boundary properties of the anatomy of interest. Such learning-based cost function design has shown promise previously \\cite{Brejl-TMI-00}. \n\nA major limitation of learning-based approaches is the time-consuming task of manually segmenting a large number of accurate examples used for training. Several post-processing interactive techniques were reported to ease the corrections, e.g., thin plate splines \\cite{ross2009lung}, live wire techniques \\cite{schenk2000efficient} and active shape models \\cite{schwarz2008interactive} with the major issue being that they correct for the surfaces directly to match the object boundaries and are sometimes infeasible to be computed in close-to-real time.\n\nWe present a fully automated LOGISMOS segmentation algorithm that guarantees global optimality with respect to the cost functions provided. A novel hierarchical random forest (RF) classifiers was designed to provide the learned costs. An interactive correction method called just-enough interaction (JEI) \\cite{Sun2013,Kashyap-IMIC2016} was used to substantially reduce the interaction time needed to prepare the training data in comparison with voxel-by-voxel editing approaches. In our method, two variations of the RF classifier were used --- a neighborhood approximation forests (NAF) followed by an RF classifier collected on a geometric graph. The novelty of this approach is that the NAF classifier gathers contextual and textural information from a global neighborhood of 3D image patches while the RF classifier collects local feature information along the geometric graph search columns. This combination enables training on contextual and textural features  both locally and globally thus improving segmentation accuracy.\n \nOA as a disease progresses slowly in the early stages with the rate of cartilage loss accelerating with advancing disease \\cite{Eckstein2014a}. Every patient in the OAI was enrolled in the longitudinal study with follow-up MRI scans done at regular clinical visits. The availability of multi-time-point information can be leveraged to provide additional information proving beneficial in patients with progressively worsening OA. Using the spatial and temporal contextual information from the neighboring time-points of the same patient helps reducing the inter-time-point variability ensuring that the cartilage losses are within physiologically feasible ranges. The LOGISMOS framework was extended to 4D to incorporate this information and simultaneously segment the multiple follow-up scans of the same patient. To the best of our knowledge, this is the first attempt at a 4D knee MRI segmentation algorithm incorporating the additional available information in a LOGISMOS framework.\n\nThe work reported here improves several aspects of the LOGISMOS algorithm for cartilage segmentation previously presented in \\cite{Yin2010}. Our novel hierarchical random forest classifier outperforms the previous method, in which a single RF classifier did not account for the regionally-specific appearance of the surrounding menisci, muscle, bone and other anatomies. As a result, certain intensity profiles appearing in normal-cartilage regions of the knee also occurred in other local regions in a pathological case. This ambiguity resulted in the imperfect training of the classifier and contributed to segmentation inaccuracy.  Further, the LOGISMOS algorithm was extended to handle simultaneous multi-3D (4D) segmentation.\n\nThe remainder of the paper is organized as follows: Section \\ref{LOGISMOS} introduces the LOGISMOS algorithm framework and the 3D graph construction. In Section \\ref{cost_function} our proposed hierarchical RF classifier design is described in detail followed by the 4D LOGISMOS extension in Section \\ref{4D}. The experimental design, validation, and discussion follow in Sections \\ref{expMethods}, \\ref{results} and \\ref{conclusions}.\n\n\\subsection{LOGISMOS Segmentation}\n\\label{LOGISMOS}\n\nLOGISMOS algorithm \\cite{Yin2010} solves the simultaneous segmentation of multiple objects and surfaces in a graph-based framework with guarantee of global optimality. To ensure global optimality, the LOGISMOS algorithm has to satisfy two important criteria: a) The geometric graph must have non-intersecting column construction and b) the graph edge construction must satisfy the self-closure property (see \\cite{Wu2005, Li2006PAMI} for proof). The focus of this paper is reporting a new method for cost function design and extending the 3D LOGISMOS to 4D. The workflow originally proposed in \\cite{Yin2010} is maintained.\n\nThe algorithm is initialized by identifying the volume of interest (VOI) which uses an AdaBoost classifier \\cite{Freund1997} trained on manually identified VOI's. The purpose of VOI detection was to localize smaller regions to reduce computation time. Further the VOI bounds are used for affine fitting of the mean femur and tibia bone shape mesh $S_0$. A patient specific bone mesh $S$ is computed by a single surface single object LOGISMOS segmentation using $S_0$. This step is necessary since the final segmentation of the bone and the cartilage surfaces is defined by this prior.\n\nA geometric graph with non-intersecting columns representing each surface (bone and cartilage of femur and tibia) is built by mimicking the behavior of charged particles resulting in electric lines of force (ELF) based columns. The cost functions designed are assigned to the graph nodes of every column. The graph-based multiple surfaces and objects segmentation edge construction and optimization are described in \\cite{Yin2010}. \n\n\\section{Cost function design}\n\\label{cost_function}\nLOGISMOS guarantees global optimality of its solution with respect to the cost functions provided. \n\nSimple hand-tuned cost functions were first designed to capture the bone to cartilage interface, which appeared as a strong dark to bright edge when traversing the image along the graph column from inside of the bone surface outwards to the bright cartilage. Directional 1D derivative operators gave costs based on the edges encountered along the search columns. For the cartilage, a weighted combination of the first and second derivative operators was used. This helps prevent interpreting cartilage inhomogeneities as edges. Although human expert designed cost functions are very effective at capturing the desired features there are limitations. 1) Choosing the correct weighting combination is challenging. 2) Same costs may not work for all parts of the anatomy. 3) The same anatomical objects (e.g., bone, cartilage) appear differently in pathological cases due to the loss of structure and\/or the presence of lesions. \n\nRF classifiers \\cite{Breiman2001} were used to provide better and location-specific cost functions. RF classifiers use the concept of bagging, where for each decision tree a random subset of features is chosen thereby reducing inter-decision-tree correlation,  which improves accuracy. A single RF classifier was used in \\cite{Yin2010} where the limitation was that all the information used was localized along the graph columns. Because of this locality, features were unable to capture information of the neighborhood that appeared larger than a few nodes along the column. There are several anatomical features that appear locally like a cartilage boundary but when examined in a global neighborhood may be combined with pathology (e.g., synovial fluid, Fig.~\\ref{fig:4d}). Further, the single RF classifier did not account for the regionally-specific appearance of the surrounding menisci, muscle, bone and other anatomies. As a  result, certain intensity profiles appearing in a specific normal-cartilage region of the knee also occurred in another local region in a pathological case. This discrepancy resulted in the improper training of the classifier and contributed to segmentation inaccuracy.\n\nIn this work, a combination of global and local contextual features was explored. Two RF based classifiers working in hierarchy were used to learn image appearance properties in the cartilage regions (Fig.\\ \\ref{fig:workflow_train}). The first stage used a NAF \\footnote{Based on code available at: \\url{http:\/\/www.nmr.mgh.harvard.edu\/~enderk\/software.html}} \\cite{konukoglu2013} trained on example image patches. The output probability maps of the NAF were used with other image-based features for training each of the regionally specific second-stage RF classifiers. The second RF classifier collected features along the geometric graph columns. For the second RF classifier, the local regions of the knee were spatially clustered using $k$-means clustering to account for variable local anatomy appearances surrounding the cartilage (Fig.~\\ref{fig:clusterRF}). Each of the spatially clustered regions was trained using a different RF classifier in the second stage. Disjoint training sets were used to help build a more realistic RF model based on actual NAF performance on unseen images.\n\n\\subsection{Neighborhood Approximation Forests}\nNAF uses a random forest framework consisting of a several binary decision trees where each tree independently learns to predict the closest neighborhood. NAF uses the training set to learn which clusters of image patches have the most similar neighborhood structures based on a similarity criterion. The definition of similarity is application specific with the advantage of using a wide variety of distance-based metrics without modifying the core underlying approach. Upon training, NAF approximates the neighbors of an unseen test image based on this similarity criterion. \n\nFor each tree in the forest, the training phase starts with the root node and continues to add new branches. Each node branch is trained to learn a set of binary tests which progressively partition the image into subsets with respect to the user-defined distance metric. The node split is optimized such that the user-defined distance function yields the most compact partitioning. \nFor every tree in the forest, a subset of the entire feature set is chosen for training thereby improving generalization and producing an independent prediction. \n\nIn this work, the NAF was trained on image patches where the pairwise distance function $\\rho(I,J)$ captured the similarity of the image patches according to their segmentation labels. The distance function between training image patches were defined as $\\rho(I,J)~=~\\parallel seg(I) - seg(J) \\parallel_{l_0}$ where $seg(.)$ is the segmentation label map for the corresponding image patch. The algorithm learned to group image patches that appeared similar to each other based on the segmentation similarity. \n\nThe test image was partitioned into several smaller patches. Each patch passed through a trained NAF classifier. The output probability of all patches were combined to produce the final probability map (Fig.\\ \\ref{fig:NAF}). This was used as one of the inputs for the second clustered RF classifier. \n\n\\begin{figure}[htb]\n\t\\centering\n\t{\\includegraphics*[height=0.5\\columnwidth,width=0.6\\columnwidth]{NAP_probablity_Ver2_9496443.png}}\n\t\\vspace{-.3cm}\n\t\\caption{The output probability map of the NAF for an image overlaid on the image volume. The color map indicates the probability output values for each 3D voxel belonging to the femur (top) and tibia cartilage. The brighter yellow denotes higher probability while the red  indicates regions of lower probability of the voxel being a cartilage.}\n\t\\label{fig:NAF}\n\t\t\\vspace{-.4cm}\n\\end{figure}\n\n\\subsection{Clustered Random Forest Classifier}\nThe second RF classifier was trained on features collected at every node along the search columns of the geometric graph. Bone mesh surfaces that were corrected using just-enough interaction (see Appendix \\ref{jei}) were used for geometric graph construction during training. Positive example labels corresponded to the nearest cartilage mesh intersection along each graph column. The different features collected at each graph node are shown in Table \\ref{featureLabel} with volumes of every feature computed individually. The value at each node point along the geometric ELF graph column was linearly interpolated from the closest point in the feature volume.\n\nTo handle the variability of cartilage image intensities and the regionally specific surrounding anatomy, a $k$-means clustering algorithm was applied on the coordinates of femur and tibia mean shape $S_0$ mesh points. The spatial parcellation of the pre-segmented mesh in 40 clusters each was empirically designed (total 80, Fig.~\\ref{fig:clusterRF}). Having too many clusters resulted in a reduced set of graph columns yielding insufficient sets of training examples. Having only a small number of clusters resulted in larger more inhomogeneous sets  while negatively affecting the quality of  classification. Training used a separate RF classifier for each of the clusters to learn variable cartilage appearance and consider surrounding anatomies. \n\nThe probability response to the features along the search nodes in the testing datasets provided the node costs for LOGISMOS segmentation. \n\n\\begin{figure}[htb]\n\\vspace{-.2cm}\n\t\\centering\n\t\\includegraphics*[height=1in]{ClusteredRF_FEMUR.png} \n  \\includegraphics*[height=1in]{ClusteredRF_TIBIA.png}\n  \\vspace{-.2cm}\n\t\\caption{Parcellation of the $S_0$ mesh using $k$-means clustering of the femur and tibia into 40 clusters each. Each colored region represents a different cluster that is trained using a separate RF classifier to account for the regionally-specific appearance of surrounding menisci, muscle bone and other anatomies.\n}\n\t\\label{fig:clusterRF}\n\t\\vspace{-.4cm}\n\\end{figure}\n\n\\begin{table}[htb]\n\t\\large\n\t\\caption{List of features for the second RF classifier.}\n\t\\label{featureLabel}\n\t\\vspace{-.6cm}\n\t\\begin{center}\n\t  \\resizebox{\\columnwidth}{!}{\n\t\t\\begin{tabular}{l|l}\n\t\t\t\\hline\n\t\t\t{\\bf Index} & {\\bf Description} \\\\ \\hline\n\t\t\t1--9           & 3 eigenvalues of Hessian matrices on \\\\ & intensity image at $\\sigma=0.5,1.0,2.0$ mm \\cite{frangi1998multiscale}\\\\\n\t\t\t10--15         & Gaussian gradient on intensity and NAF \\\\&probability volumes at $\\sigma=0.36,0.7,1.4$ mm \\cite{zhao2009congenital}\\\\\n\t\t\t16--18         & Intensity, Gaussian smoothed intensity,\\\\ &and NAF probability volumes \\\\\n\t\t\t19--20         & Laplacian derivative of intensity volume\\\\ &at $\\sigma=0.36,0.7$ mm \\cite{zhao2009congenital} \\\\\n\t\t\t21             & Gabor texture feature \\cite{daugman1985uncertainty} \\\\\n\t\t\t22--25         & Intensity statistics: mean, variance, skewness\\\\& and kurtosis of a 2 mm$^3$ region\\\\& centered around each graph node \\cite{antonie2001application}\\\\\n\t\t\t26--28         & Haar features (1.5mm kernel) along \\\\& horizontal, vertical \\& diagonal directions \\cite{viola2004robust} \\\\\n\t\t\t29--30         & 1D directional gradient along the search column \\\\& direction on NAF probability and intensity volume\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t  }\n\t\t\\vspace{-.6cm}\n  \\end{center}\n\\end{table}\n\n\\section{4D Longitudinal Segmentation of Knee MRI}\n\\label{4D}\nAlong with the proposed learning-based cost functions, it is beneficial to leverage the contextual information from multiple time-points to improve the overall segmentation accuracy. Sequential yearly follow-up MRIs of the same knee provide spatial and temporal contextual information which helps to reduce the inter-time-point variability while ensuring physiologically feasible losses of cartilage during osteoarthritis progression. The crucial first step in 4D segmentation is to register the pre-segmented mesh surfaces and the respective images across time-points to establish correspondences temporally between the similar regions of the knee in 4D. A rigid registration was used given that only slight changes in bone shape occur with disease progression. The same configuration is maintained across all time-points in terms of the geometric graph parameters and topology. Enforcing contextual information temporally is done by linking adjacent time-points using  inter-time-point edges in the underlying 4D LOGISMOS graph.\n\n\\subsection{Establishing Temporal Correspondences}\n\nAfter pre-segmentation of each of the time-points, iterative closest points (ICP) algorithm was used to register the pre-segmented mesh surfaces. With a large translational or rotational movement between the two-time-points, there is a tendency to mismatch the surfaces, i.e., femur matched to the tibia or vice versa if they were the closest in terms of the least squares optimization. To prevent the mismatch, a two step registration was employed. The first step used the femur mesh only for ICP registration. This transform matrix was applied on both the femur and tibia meshes. After the first transform ensured that femoral and tibial surfaces were reasonably aligned, the ICP registration was run again using both the femur and tibia points together as a unified point cloud to further refine the registration. This ensured that vertex to vertex correspondences and thereby column correspondences were established. Fig.\\ \\ref{fig:icp} illustrates the establishment of correspondence between two time-points. The same two-step transformation matrices were applied to the entire images. \n\n\\begin{figure}[htb]\n\t\\includegraphics*[width=0.49\\columnwidth]{BeforeICP_with_Borders.png} \n\t\\includegraphics*[width=0.49\\columnwidth,height=3.7cm]{AfterICP_with_Borders.png}\n\t\\vspace{-.3cm}\n\t\\caption{Establishing point correspondences between time-points after pre-segmentation. The meshes are registered using ICP to establish column correspondences. The same transformation is applied to the corresponding volumes. (a)~Two time-points before registration. The meshes are colored differently and the volume border is highlighted in yellow to indicate its positioning. (b)~Meshes registered after ICP.}\n\t\\vspace{-.3cm}\n\t\\label{fig:icp}\n\\end{figure}\n\n\\subsection{Incorporating Inter-Time Point Context}\nFor the 4D segmentation, the datasets at each time-point have their respective 3D multi-surface multi-object constraints enforced as described earlier in \\cite{Yin2010}. In order to enforce the contextual constraints across adjacent time-points, inter-time-point edges are constructed connecting nodes in every column with the corresponding columns on the registered surfaces. Using the prior information we can set the different minimum and maximum permissible limits for the bones and cartilages respectively so that cartilage thickness changes are within a physiologically feasible range. The edges introduced define inter-time-point min and max ($\\delta_{tmin}, \\delta_{tmax}$) where the inter-time-point maximum allowed change ($\\delta_{tmax}$) was set to 0.6 mm\/year based on available clinical literature \\cite{Eckstein2014a}. The minimum $\\delta_{tmin}$ was set to zero.\n\nThe inter-time-point edges $E^{t}$ between neighboring time-points $t_1$, $t_2$ are constructed for all nodes of the graph:\n\\begin{align*}\nE^{t} = \\{\\langle v_i(t_1,n_{cart},k), v_i(t_2,n_{cart},k-\\delta_{tmin})\\rangle\\}\\\\\n\\cup \\{\\langle v_i(t_1,n_{cart},k-\\delta_{tmax}), v_i(t_2,n_{cart},k)\\rangle\\}\\\\\n\\cup \\{\\langle v_i(t_1,n_{bone},k), v_i(t_2,n_{bone},k - \\delta_{tmin})\\rangle\\}\\\\\n\\cup  \\{\\langle v_i(t_1,n_{bone},k-\\delta_{tmax}), v_i(t_2,n_{bone},k)\\rangle\\},\n\\end{align*}\nwhere $v_i(t,n,k)$ represents the $k$th node on the $i$th column, the inter-time-point edges are constructed between nodes $v_i(t_1,n,k), v_i(t_2,n,k)$ of longitudinally corresponding columns $t_1$, $t_2$ for every $n \\in \\{cart,bone\\}$. After the graph edges and nodes have been constructed, the 4D LOGISMOS segmentation problem is solved by computing a minimum $\\emph{s-t}$ cut in a derived edge weighted digraph \\cite{Boykov2004}. \n\n\\section{Experimental Methods} \n\\label{expMethods}\nKnee MRI data used for evaluation originated from the Osteoarthritis Initiative (OAI) database available for public access\\footnote{\\url{https:\/\/oai.epi-ucsf.org\/datarelease\/}}. All MR acquisitions used double echo steady state (DESS) pulse sequence with in-plane resolution of 0.36$\\times$0.36mm and a slice spacing of 0.7mm, resulting in image volumes with 384 $\\times$ 384 $\\times$ 160 voxels. 88 datasets at baseline (BL) and 12-months follow-up (12M) scans (176 3D MRIs in total) were used for which manually defined independent standards are available via OAI. BL images of 34 patients were used for training the hierarchical RF classifiers. The 12M data of the same patients were excluded from the testing set given the similarity of appearance resulting in a testing set of 54 MRIs at BL and 12M respectively (total of 108 MRIs).\n\nFig.\\ \\ref{fig:workflow_train} shows the training workflow for the hierarchical RF classifiers. For bone surface segmentation, the initially employed gradient-based costs were very robust and remained unchanged. For learning the cartilage-surface costs, the 34 patients were divided into two training sets with 15 and 19 patients each that were used to train the NAF and the clustered RF classifier respectively. The OAI provided independent standards had the cartilage and bone boundaries only in the region of articular cartilage (Fig.\\ \\ref{fig:qual}a). However the LOGISMOS segmentation and further analysis required the entire bone structure. The datasets used for training the clustered RF classifier were first inspected and JEI edited (see Appendix \\ref{jei} for details). \n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[width=\\columnwidth]{Workflow_Training2.png}\n\t\\vspace{-.3cm}\n\t\\caption{Learning-based workflow for training the hierarchical RF classifiers.}\n\t\\label{fig:workflow_train}\n\\end{figure}\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\vspace{-.3cm}\n\t\\includegraphics[width=0.8\\columnwidth]{Workflow_Testing_Ver4.png}\n  \\vspace{-.3cm}\n\t\\caption{Testing workflow to compare the clustered random forest classifiers with the existing methods on a set of 108 patients.}\n\t\\vspace{-.3cm}\n\t\\label{fig:workflow_test}\n\\end{figure}\n\n\\begin{table}[htb]\n\t\\caption{Parameters used for graph construction.}\n\t\\centering\n\t\\label{graph-parameters}\n\t\\begin{tabular}{lccccc}\n\t\t\\hline\n\t\t& Inter-surface & Inter-object & Smoothness & Column size \\\\\n\t\t& max (mm)   & max (mm)  & (mm)    & (mm)     \\\\\n\t\t\\hline\n\t\tLearned cost  & 6            & 18          & 0.6          & 18.15 \\\\\n\t\tGradient cost & 4            & 12           & 0.4         & 12.2   \\\\\n\t\t\\hline\n\t\t& & & & \\\\\n\t\t\\multicolumn{5}{l}{*Minimum inter-surface, inter-object and inter-time-point separations are zero.}\n\t\\end{tabular}\n\\end{table}\n\nAll image volumes were first LOGISMOS-segmented using gradient costs. The geometric graphs had 8006 and 8002 graph columns for the femur and tibia objects, respectively. The graph parameters are listed in Table \\ref{graph-parameters} with the additional inter-time-point constraint set at 0.6 mm for the 4D LOGISMOS. \n\nThe NAF features were sampled over 15 datasets with 1521 samples per image patch. Due to the imbalance in the negative to positive labels ratio, a neighborhood around the cartilage was marked as negative examples. A set of 200 trees with 40,000 image patches were used as inputs to the training. The second RF classifier was trained on 19 JEI corrected datasets with every node along the ELF search column collecting 30 features (Table~\\ref{featureLabel}). A total of 80 RF classifiers (one per cluster) were trained with 800 trees per each forest. \n\nOA is characterized by thickness losses in subregions of the surface. Analyzing the entire cartilage structure may not be sensitive to these localized and regionally smaller changes. Therefore a consensus based nomenclature was proposed by clinicians identifying regions of the cartilage most likely affected by OA \\cite{Eckstein2006a}. The existing methodologies in the literature require manual initialization for the sub-plate analysis \\cite{Wirth2008,Wirth2010}. We developed a fully automated nomenclature-compliant sub-plate analysis algorithm. The algorithm subdivided the segmented surface by automated identification of the trochlear notch anatomy and implicit cutting plane geometry (Appendix \\ref{sub-plate}). All of our experimental results were subplate analyzed and reported for the entire femur and tibia cartilage regions, central load-bearing lateral femur\/tibia (cLF\/cLT), and central load-bearing medial femur\/tibia (cMF\/cMT).\n\nThe experimental results are reported as signed and unsigned errors. The signed errors indicate the measurement bias with the positive errors denoting underestimation and the negative errors denoting overestimation with respect to the independent standard. The unsigned errors indicate the variability of the measurement. The error measurements were collected in regions for which the independent standard was available. The first experiment was to validate the accuracy of the different cost functions in 3D LOGISMOS segmentation. The costs by the proposed hierarchical RF classifier were compared against the gradient costs. To evaluate the benefits of adding the NAF stage, a single RF classifier was trained on the same 19 patient datasets excluding the NAF features while maintaining the same graph parameters as the hierarchical classifier system. The testing workflow is shown in Fig.~\\ref{fig:workflow_test}.\nThe second experiment simultaneously 4D segmented the 54 knees using the proposed hierarchical costs and compared against the 108 individually 3D LOGISMOS segmented results using the same segmentation cost functions. It is designed to reveal the benefits of adding contextual information between the adjacent time-points. Given that OA is a slowly progressing disease, the expectation was that there would not be a significant appearance of artifacts or thickness losses over a one year study period. Fig.\\ \\ref{fig:3Dvs4D-flowchart} shows the flowchart highlighting the key differences between the 3D and 4D LOGISMOS pipelines. \n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[width=0.9\\columnwidth]{3D_vs_4D_flowchart.png}\n\t\\caption{Comparison of 3D and 4D LOGISMOS pipelines. The black and red arrows highlight the flow of 3D and 4D LOGISMOS algorithms respectively.}\n\t\\label{fig:3Dvs4D-flowchart}\n\\end{figure}\n\nTo demonstrate the benefits of 4D analysis over a longer study period, the same 54 patient datasets with BL, 12M, 24M, 36M and 48M follow-up scans were used resulting in a total of 237 3D MRIs analyzed for the third experiment (out of 270 due to subjects missing some follow-up visits). Since OAI-provided  independent standards were available only for BL and 12 M, a new independent standard was defined in house by an expert analyst using JEI approach (Appendix A).  To highlight the improvements, we compared the maximum cartilage thickness errors for 3D and 4D across all time-points and determined statistical significance of their differences.\n\n\n\\section{Results}\n\\label{results}\nTable \\ref{bone_error} shows surface positioning errors for the bone segmentation with the signed errors being sub-voxel accurate. Since high bone-segmentation accuracy was already achieved using the gradient-based costs, a learning based cost function design was not necessary. Although we saw presence of bone lesions with progression of OA, the gradient-based cost was sufficiently robust. Thus all the different methodologies in both 3D and 4D LOGISMOS used the same gradient-based cost functions for the bone.\n\n\\begin{table}[htb]\n\t\\centering\n\t\\caption{Bone segmentation errors (in mm) using the gradient-based cost functions.}\n\t\\label{bone_error}\n\t\\begin{tabular}{l|cc}\n\t\t\\hline\n\t\t& Signed & Unsigned \\\\\n\t\t\\hline\n\t\tFemur & -0.19$\\pm$0.07 & 0.40$\\pm$0.06 \\\\\n\t\tTibia & -0.13$\\pm$0.09 &  0.39$\\pm$0.09 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nCatilage surface positioning errors (compared against independent standard) using the hierarchical classifier, gradient cost, and single-stage RF classifier are reported. Each resulting surface from the above methods were sub-plate analyzed. Table \\ref{error1} shows a significant reduction in signed and unsigned errors for all the femoral sub-plates ($p\\ll0.001$). Significant reduction in unsigned errors over the segmentation using gradient costs was seen for all the tibial sub-plates while the signed errors showed a significant reduction of errors over the gradient-based costs on all sub-plates except the central medial tibia (cMT). Comparison with the single-stage RF classifier saw significant reduction of unsigned errors in the whole tibia and central medial tibia (cMT) plates. \n\nFig.\\ \\ref{fig:qual} qualitatively compares the segmentation accuracies between the gradient-based costs and the hierarchical classifier with respect to the independent standard. Both the femur and tibia are shown with their respective bone and cartilage segmentations showing good agreement between learning-based segmentation and the independent standard.  \n\n\\begin{table*}[htb]\n\t\\centering\n\t\\caption{Cartilage surface positioning errors (in mm) achieved by hierarchical classifier, gradient cost and single-stage RF classifier using 3D LOGISMOS algorithm. Paired $t$-test comparisons are reported for (NAF+RF vs. Gradient) and (NAF+RF vs. RF only)}\n\t\\label{error1}\n\t\\begin{tabular}{lr|rr|rr}\n\t\t\\hline\n\t\t\\bf n=108 & NAF+RF           & Gradient          & $p$-value & RF only & $p$-value       \\\\ \n\t\t\\hline                                           \n\t\tFemur signed   & \\bf -0.01$\\pm$0.18 &-0.31$\\pm$0.24     & $\\ll0.001$  & -0.10$\\pm$0.17& $\\ll0.001$  \\\\ \n\t\tFemur unsigned & \\bf 0.55$\\pm$0.11 & 0.69$\\pm$0.13     & $\\ll0.001$  &  0.56$\\pm$0.10& $\\ll0.001$  \\\\ \n\t\t\\hline     \n\t\tcMF signed   & \\bf -0.04$\\pm$0.29 &-0.38$\\pm$0.58     & $\\ll0.001$  & -0.11$\\pm$0.27& $\\ll0.001$  \\\\ \n\t\tcMF unsigned & \\bf 0.52$\\pm$0.16 & 0.78$\\pm$0.35     & $\\ll0.001$  &  0.55$\\pm$0.17& $\\ll0.001$  \\\\ \n\t\t\\hline\n\t\tcLF signed   & \\bf -0.26$\\pm$0.24 &-0.52$\\pm$0.35     & $\\ll0.001$  & -0.36$\\pm$0.20& $\\ll0.001$  \\\\ \n\t\tcLF unsigned & \\bf 0.42$\\pm$0.12 & 0.65$\\pm$0.20     & $\\ll0.001$  &  0.47$\\pm$0.11& $\\ll0.001$  \\\\ \n\t\t\\hline\n\t\tTibia signed   &    \\bf 0.06$\\pm$0.17 & -0.11$\\pm$0.35 & $\\ll0.001$      & 0.11$\\pm$0.22& $\\ll0.001$  \\\\\n\t\tTibia unsigned & \\bf 0.60$\\pm$0.14 &     0.79$\\pm$0.20 & $\\ll0.001$  & 0.62$\\pm$0.18& $\\ll0.001$  \\\\\n\t\t\\hline\n\t\tcMT signed   & -0.15$\\pm$0.31 &-0.25$\\pm$0.77     & $0.193$  & \\bf -0.09$\\pm$0.34& $0.003$ \\\\ \n\t\tcMT unsigned & \\bf 0.52$\\pm$0.20 & 0.92$\\pm$0.41     & $\\ll0.001$  &  0.58$\\pm$0.22& $\\ll0.001$ \\\\ \n\t\t\\hline\n\t\tcLT signed   & \\bf -0.03$\\pm$0.32 &0.36$\\pm$1.12     & $\\ll0.001$  & -0.01$\\pm$0.32& $0.16$ \\\\ \n\t\tcLT unsigned & \\bf 0.46$\\pm$0.17 & 0.79$\\pm$0.99     & $\\ll0.001$  &  0.47$\\pm$0.18& $0.31$ \\\\ \n\t\t\\hline \n\t\t\\multicolumn{6}{l}{*Bold indicates statistically significant reduction in positioning errors.}\n\t\\end{tabular}\n\\end{table*}\n\n\\begin{figure}[htb]\n  \\centering\n\t\\begin{tabular}{cc}\n\t\t\\multicolumn{2}{c}{\\includegraphics[width=0.4\\columnwidth]{IndependentStandard.png}}\\\\\n\t\t\\includegraphics[width=0.4\\columnwidth]{Gradient.png} &\n\t\t\\includegraphics[width=0.4\\columnwidth]{Learned.png} \\\\\n\t\\end{tabular}\n\t\\caption{Segmentation accuracy in a representative subject.\n\t(a)~Independent standard.\n\t(b)~Gradient-costs LOGISMOS segmentation.\n\t(c)~Learned-costs LOGISMOS segmentation. Region marked by the arrow shows clear improvement in the segmentation quality when using the learned costs.}\n\t\\label{fig:qual}\n\\end{figure}\n\nTable \\ref{error-4Dvs3D} compares signed and unsigned border positioning errors with respect to the independent standard for 3D and 4D LOGISMOS segmentation at BL and 12M follow-up respectively. As expected, the overall segmentation accuracy is almost the same between the two methods except for a few of the plate regions highlighted in bold showing statistically significant differences over 3D ($p<0.05$).\n\nTable VI shows the performance improvement obtained by the  4D LOGISMOS over 3D for a longer study period of five years (BL\u2013--48 M, 5 time-points). To demonstrate improvements in local accuracy, Table VI gives the unsigned  errors of cartilage thickness averaged for three quantiles within 90-100\\% of the largest errors  per patient combined for all available time-points for each sub-plate. These unsigned maximum thickness errors  were statistically significantly  smaller in each of cMF, cMT and cLT  for 4D LOGISMOS compared with separate 3D analyses.\n\nFig.\\ \\ref{fig:4d} qualitatively shows the improvement of 4D LOGISMOS over 3D. Note the lack of an obvious edge distinguishing the tibia and femur cartilage. Using the spatial and temporal contextual information from all the time-points,  the 4D method is able to correctly position the cartilage for the tibia and femur despite the lack of image information locally. \n\n\\begin{figure}[htb]\n  \\centering\n\t\\begin{tabular}{cc}\n\t\t\\multicolumn{2}{c}{\\includegraphics[width=0.4\\columnwidth]{9279291L_IS.png}}\\\\\n\t\t\\includegraphics[width=0.4\\columnwidth]{9279291L_3D.png} &\n\t\t\\includegraphics[width=0.4\\columnwidth]{9279291L_4D.png} \\\\\n\t\\end{tabular}\n\t\\caption{Qualitative improvement of 4D segmentation versus 3D for a patient with severe osteoarthritis. Parts of the region between the femur and tibia exhibit synovial fluid leakage, the appearance and texture of which mimic that of a cartilage. (a) Independent standard, (b) 3D segmentation, (c) 4D segmentation. }\n\t\\label{fig:4d}\n\\end{figure}\n\n\\begin{table*}[htb]\n  \\caption{Cartilage surface positioning errors (in mm) of 4D versus 3D LOGISMOS.}\n  \\label{error-4Dvs3D}\n  \\centering\n  \\begin{tabular}{l|rrr|rrr}\n  \\hline\n                  & \\multicolumn{3}{c|}{\\bf n=54 Baseline} & \\multicolumn{3}{c}{\\bf n=54 12 months} \\\\\n                  & \\multicolumn{1}{c}{4D} & \\multicolumn{1}{c}{3D} & $p$-value & \\multicolumn{1}{c}{4D} & \\multicolumn{1}{c}{3D} & $p$-value \\\\ \\hline\n  Femur signed    & \\bf  0.01$\\pm$0.18  & 0.01$\\pm$0.19     & 0.027      & -0.02$\\pm$0.17    &-0.04$\\pm$0.17 & 0.062 \\\\ \n  Femur unsigned  & \\bf  0.53$\\pm$0.11  & 0.54$\\pm$0.11     & $\\ll$0.001 &  0.55$\\pm$0.11    & 0.55$\\pm$0.11 & 0.601 \\\\ \\hline\n  cMF signed      & \\bf -0.01$\\pm$0.25  &-0.03$\\pm$0.27     & 0.015      & \\bf 0.02$\\pm$0.27 &-0.04$\\pm$0.30 & 0.016 \\\\\n  cMF unsigned    &  0.51$\\pm$0.16      & 0.52$\\pm$0.17     & 0.092      &  0.52$\\pm$0.15    & 0.53$\\pm$0.16 & 0.614 \\\\ \\hline\n  cLF signed      & -0.26$\\pm$0.22      &-0.25$\\pm$0.23     & 0.108      & -0.30$\\pm$0.21    &-0.30$\\pm$0.23 & 0.853 \\\\\n  cLF unsigned    &  0.42$\\pm$0.10      & 0.42$\\pm$0.10     & 0.790      &  0.43$\\pm$0.12    & 0.44$\\pm$0.13 & 0.318 \\\\ \\hline\n  Tibia signed    &  0.08$\\pm$0.17      &\\bf 0.07$\\pm$0.16  & 0.039      &  0.07$\\pm$0.19    & 0.06$\\pm$0.18 & 0.464 \\\\\n  Tibia unsigned  &  0.59$\\pm$0.14      & 0.60$\\pm$0.14     & 0.140      &  0.60$\\pm$0.16    & 0.60$\\pm$0.14 & 0.310 \\\\ \\hline\n  cMT signed      & -0.13$\\pm$0.28      &-0.14$\\pm$0.29     & 0.125      & -0.14$\\pm$0.29    &-0.15$\\pm$0.31 & 0.619 \\\\\n  cMT unsigned    & \\bf 0.50$\\pm$0.18   & 0.51$\\pm$0.20     & 0.039      &  0.52$\\pm$0.17    & 0.53$\\pm$0.17 & 0.341 \\\\ \\hline\n  cLT signed      &  0.00$\\pm$0.30      & 0.00$\\pm$0.31     & 0.338      & -0.06$\\pm$0.31    &-0.06$\\pm$0.30 & 0.629 \\\\\n  cLT unsigned    & \\bf 0.45$\\pm$0.17   & 0.46$\\pm$0.18     & $\\ll$0.001 &  0.45$\\pm$0.16    & 0.44$\\pm$0.16 & 0.256 \\\\ \n  \\hline\n\t\\multicolumn{7}{l}{*Bold indicates statistically significant reduction in positioning errors.}\n  \\end{tabular}\n\\end{table*}\n\n\\begin{table*}[htb]\n\t\\caption{Longitudinal max thickness error values of 4D versus 3D LOGISMOS.}\n\t\\label{max_thicknessError-4Dvs3D}\n\t\\centering\n\t\\begin{tabular}{l|rrr|rrr|rrr}\n\t\t\\hline\n    & \\multicolumn{3}{c|}{\\bf 2\\% Max Average Errors (98-100\\%)} & \\multicolumn{3}{c|}{\\bf 5\\% Max Average Errors (95-100\\%)} & \\multicolumn{3}{c}{\\bf 10\\% Max Average Errors (90-100\\%)} \\\\\n\t\t& \\multicolumn{1}{c}{4D} & \\multicolumn{1}{c}{3D} & $p$-value & \\multicolumn{1}{c}{4D} & \\multicolumn{1}{c}{3D} & $p$-value & \\multicolumn{1}{c}{4D} & \\multicolumn{1}{c}{3D} & $p$-value \\\\ \\hline\n    cMF unsigned    &  \\bf 0.76$\\pm$0.60 & 1.11$\\pm$0.63     & $\\ll$0.001 & \\bf 0.57$\\pm$0.43    & 0.87$\\pm$0.41 & $\\ll$0.001  &  \\bf 0.43$\\pm$0.32    & 0.68$\\pm$0.29 & $\\ll$0.001 \\\\ \\hline\n    cLF unsigned    &  1.46$\\pm$2.36     & 1.67$\\pm$1.82     & 0.610      &  1.07$\\pm$1.73    & 1.16$\\pm$1.02    & 0.74        &  0.74$\\pm$0.95    & 0.84$\\pm$0.55 & 0.528 \\\\ \\hline\n    cMT unsigned    & \\bf 1.28$\\pm$0.47  & 1.78$\\pm$1.36     & 0.008      &  \\bf 1.09$\\pm$0.42    & 1.48$\\pm$1.00 & 0.004      &  \\bf 0.92$\\pm$0.38    & 1.19$\\pm$0.63 & $\\ll$0.001 \\\\ \\hline\n    cLT unsigned    & \\bf 1.29$\\pm$0.47  & 1.40$\\pm$0.49     & $\\ll$0.001 &  \\bf 1.09$\\pm$0.40    & 1.17$\\pm$0.41 & $\\ll$0.001 &  \\bf 0.90$\\pm$0.33    & 0.98$\\pm$0.34 & $\\ll$0.001 \\\\ \n    \\hline\n\t\t\\multicolumn{7}{l}{*Bold indicates statistically significant reduction in positioning errors.}\n\t\\end{tabular}\n\\end{table*}\n\n\\section{Discussion and Conclusion}\n\\label{conclusions}\n\nA novel method for learning segmentation cost functions using  hierarchical RF classifiers  was reported and its performance was demonstrated in 3D and 4D LOGISMOS segmentation of knee MR images. The preparation of the training data was done using our just-enough interaction (JEI) approach, which provided fast and accurate post-processing correction. To minimize the impact of the underlying automated LOGISMOS segmentation on the independent standard used for training and evaluation, a great level of flexibility was embedded in the graph construction so that the expert analyst could freely modify the surfaces as needed and thus create unbiased corrections of the knee surfaces.\n\nUsing the single RF classifier improved the segmentation accuracy over the initial gradient-based costs. In addition, using a hierarchy of NAF and RF stages showed further statistically significant improvement in segmentation performance. The previous LOGISMOS-based method \\cite{Yin2010} used a single-stage RF with the graph constructed in the image-voxel space and a subset of current features contributing to the cost function (Table~\\ref{featureLabel}) and it placed third on the publicly available SKI10 challenge \\cite{heimann2010segmentation}. The method reported here shows a significant  improvement. Further comparison with the various other methods would inevitably be imperfect due to the different testing datasets utilized. In the OAI cohort, our method achieved high accuracy in comparison with the publicly available independent standard. Training of the hierarchical RF classifier and the subsequent 3D\/4D LOGISMOS analyses ran on a Linux system with CentOS-7.3, 128 GB RAM. Once trained, the average segmentation times were 34 and 22~minutes per time-point for individual 3D and simultaneous 4D LOGISMOS, respectively.\n\nLeveraging the benefits of the available multi-time-point MRI datasets for the OA patients, we designed a 4D extension of the LOGISMOS segmentation framework. The added temporal inter-time-point contextual constraints helped constrain the changes of cartilage thickness to  what was physiologically feasible. This benefit of using information from the previous time-points along with the new cost function design helped notably improve the segmentation accuracy. Further, the advantages of the 4D LOGISMOS were demonstrated on an OAI longitudinal patient dataset with 5 imaging sessions from BL to 48 month. Our sub-plate detection algorithm was used to study the accuracy of segmentation on the load-bearing regions of the knee and provided good performance insight. The robustness of the automated sub-plate division was demonstrated showing consistent measurements between different surfaces used to initialize the automated extraction of the sub-plate load-bearing regions. \n\nThe combination of LOGISMOS and learning-based methods has been  very effective. Other learning-based techniques such as deep learning may further improve the performance once more ground-truth data become available to determine further benefits of such approaches. 4D LOGISMOS also offers a platform for future clinically-oriented work, such as investigating the rate of cartilage degradation and its impact on OA progression, and identifying cartilage thickness that correlates to disease severity and\/or predicts the need for joint replacement.\nAssessing reproducibility of the reported method is left for future work and the OAI scan-rescan dataset is well-suited for this task. Patellar segmentation was not considered in the work reported here due to the limited validation material available for patellar segmentation development. Although patellar cartilage was not segmented, our early LOGISMOS approach \\cite{Yin2010} was designed to segment all three bones and cartilages of the knee joint simultaneously.\n\n\\appendices\n\n\\section{Just Enough Interaction} \n\\label{jei}\n\nIn this work, JEI was used to prepare data for training the proposed hierarchical RF classifier (Fig.~\\ref{fig:workflow_train}) and to assess the benefits of 4D LOGISMOS over a longitudinal study period. JEI is a novel interaction technique that uses the graph based LOGISMOS framework. The user inspects the automated segmentation results and provides segmentation correction points that interact directly with the underlying graph framework. This strategy differs from traditional voxel-by-voxel editing by only requiring limited (i.e., just-enough) interactions to correct the automated LOGISMOS segmentation. The JEI interaction method still guarantees global optimality even with the cost-changes resulting from the JEI corrections.\n\nFollowing automated LOGISMOS segmentation, the JEI editing was performed on the resulting surface as needed. A custom graphical user interface (GUI) was designed onto which the image volume, residual graph and the ELF geometric graph were loaded. The JEI workflow is shown in Fig. \\ref{fig:jei_workflow}. The details of the JEI architecture and GUI are given in \\cite{Kashyap-IMIC2016}.\n \n\\begin{figure}[htb]\n \t{\\includegraphics[width=\\columnwidth]{JEI_Workflow.png}}\n \t\\caption{A graphical depiction of the JEI workflow showing the interactive correction steps. A single slice that is identified with improper segmentation of the tibia cartilage (yellow) is shown. The user provides the correcting set of boundary points (cyan) which identifies the 3D neighborhood of graph columns on which the costs are modified locally. The max-flow was recomputed in 3D resulting in the corrected surfaces within milliseconds. The resulting correction from JEI on a single 2D slice can be appreciated in the corresponding 3D model representation of the surface.}\n \t\\label{fig:jei_workflow}\n\\end{figure}\n \n\n\\section{Automated Extraction of Load-Bearing Sub-Regions}\n\\label{sub-plate}\nAnalyzing the whole cartilage structure may not be sensitive to cartilage losses that occur locally. Cartilage sub-regions have been identified as areas which bear the maximum stresses during motion with a consensus based nomenclature by the clinicians on the sub-regions of the femur and tibia that need to be analyzed separately \\cite{Eckstein2006a}. Several techniques \\cite{Wirth2008,Wirth2010} for identifying the sub-plates exist all of which require human interaction to provide an initialization for the sub-region analysis. We have developed a fully automated sub-plate detection algorithm that uses LOGISMOS surface meshes to define individual sub-plates.\n\n\\subsection{Trochlear Notch Identification}\n\nThe first step in the sub-plate extraction is the detection of the trochlear notch. The notch is at the base of a groove along which the patella (knee cap) slides over the femur (Fig.\\ \\ref{fig:notch_eg}). The main anatomic feature exploited is anterior to posterior (AP) curvature of the groove of the femoral bone (Fig.\\ \\ref{fig:notch_eg}b). The notch is at the base of the curvature before the bone ridge rises sharply. An implicit cutting plane was used to isolate the analysis region represented by $\\hat{n}.p$ where $ \\hat{n}$ is the normal direction from the plane and $p$ is a point on the plane.  When isolating the groove regions using simple implicit plane cutting, a family of contours were drawn along this surface. The sharp rise of the bone structure along the contour near the base of the ridge gives a large change in contour positioning value traversing in the AP direction. To increase robustness we find the positions of largest change on all of the closely positioned groove contour lines and average them to find the desired trochlear notch (Fig.\\ \\ref{fig:notch}). \n\n\\begin{figure}[htb]\n\t\t{\\includegraphics*[width=0.49\\columnwidth,height=4.1cm]{Trochlear_Groove.png}}\n\t\t{\\includegraphics*[width=0.40\\columnwidth]{notch_Identification.png}}\\\\\t  \n\t\\caption{Trochlear notch.  (a)~LOGISMOS-segmented bone mesh with the trochlear groove region highlighted. \n\t(b)~Bone mesh highlighting the trochlear groove (overlayed in yellow) in the anterior to posterior direction. The blue arrow indicates the trochlear notch on the groove at the base of the groove curvature before the bone ridge structure rises sharply.\n\t}\n\t\\label{fig:notch_eg}\n\\end{figure}\n\n\\begin{figure}[htb]\n\t\\includegraphics[width=0.46\\columnwidth]{Trochlear_Notch.png}\n\t\\includegraphics[width=0.46\\columnwidth]{Central_Femur.png}\n\t\\caption{ Trochlear notch.  (a)~Use of AP curvature of the groove on the femoral bone. After isolating the groove region, the trochlear notch is identified by taking an average gradient on a family of closely positioned contour lines along this surface. (b)~The load-bearing regions of the femur identified by isolating 60 \\% of the distance from the trochlear notch to  posterior most in the AP direction on each condyle respectively.}\n\t\\label{fig:notch}\n\\end{figure}\n\\subsection{Sub-Plate Detection using Implicit Cutting Plane Geometry}\n\nUsing implicit cutting planes at the trochlear notch separates the posterior region from the anterior and further splits them into the medial and lateral condyle. For each femoral condyle, the load-bearing regions is defined as 60\\% of the distance in the AP direction from the trochlear notch to the posterior-most point of the respective condyles. Isolating them using cutting planes normal to the AP plane at the 60\\% region isolates the load-bearing regions of the central medial and lateral femur (Fig.\\ \\ref{fig:notch}b).\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[height=0.31\\columnwidth]{Tibia_separation.png}\n\t\\includegraphics[height=0.31\\columnwidth]{Tibia_Subplate.png}\n\t\\caption{(a)~Isolated medial and lateral tibia, (b)~All the tibial sub-plates.}\n\t\\label{fig:tibia}\n\\end{figure}\n\nThe tibia is divided into the medial and lateral compartments using cutting planes positioned at the trochlear notch with its normal perpendicular to the AP plane (Fig.\\ \\ref{fig:tibia}). After subdividing the plates into the medial and lateral regions, we isolate the central tibia and the peripheral sub-regions. The central 20\\% elliptical area of the medial and the lateral plate is computed around the center of mass of each respective plate. The radius of the major and the minor axis is computed as a ratio of bounds of the respective medial\/lateral compartments. The major axis radius is computed from the compartment bounds along the AP direction and similarly, the minor axis radius is computed from the ratio of the compartment bounds perpendicular to the AP direction. Furthermore, the remaining regions are isolated using a 45$\\degree$ and 135$\\degree$ cutting planes around the center of mass of their respective compartments to give all four peripheral sub-regions as shown in Fig.\\ \\ref{fig:tibia}. The final list of all the extracted sub-plates from the automated algorithm is visually shown in Fig.\\ \\ref{fig:all_plates}. \n\n\\begin{figure}[htb]\n\t\\includegraphics[width=\\columnwidth]{Sub-platesCropped.png}\n\t\\includegraphics[width=\\columnwidth,height=0.6\\columnwidth]{Screenshot_2.png}\n\t\\caption{ Automated sub-plate division of the cartilage. The 60\\% central lateral (cLF) and central medial femur (cMF) are shown. Each Medial (MT) and lateral tibia (LT) regions are subdivided as: central (cLT\/cMT), interior (iLT\/iMT), exterior (eLT\/eMT), anterior (aLT\/aMT) and posterior (pLT\/pMT) regions respectively.}\n\t\\label{fig:all_plates}\n\\end{figure}\n\n\\subsection{Robustness of Sub-plate Detection}\nAutomated identification of sub-plate regions uses multi-surface knee joint segmentation as input. Consequently, the sensitivity of such surface segmentation on sub-plate region identification and thus on cartilage thickness measurement must be established. Load-bearing sub-plates were derived from two surface segmentation results (4D LOGISMOS, JEI-defined surfaces) and used to determine sub-plate-specific cartilage thickness from the manually segmented independent standard.\nIn 108 patients, the two sets of sub-plate-specific cartilage thickness measurements were compared using regression analysis, which showed high robustness (low sensitivity) of the sub-plate detection algorithm on the identification of the load-bearing regions (using manually segmented independent standard surface) with respect to the underlying knee-joint segmentation. The sub-plate-specific correlation  of the two thickness measurements ranged from $R^2 = 0.98$ to $0.99$.\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\\bibliographystyle{IEEEtranbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}