%%%%% newcommands from Alizadeh %%%\newcommand{\zup}{\overline{z}} %%%\newcommand{\zdown}{\underline{z}} %%%\newcommand{\phiup}{\overline{\phi}} %%%\newcommand{\phidown}{\underline{\phi}} %%%\newcommand{\Xhalf}{X^{1/2}} %%%\newcommand{\Xnhalf}{X^{-1/2}} %%%%\newcommand{\VEC}{{\bf vec}} %%%%\newcommand{\MAT}{{\rm Mat}} %%% %%%%\newcommand{\rk}{\operatornamewithlimits{rk}} %changed by Henry below %%%\newcommand{\rk}{\rm rank} %%%\newcommand{\jrk}[1]{\rk_{#1}} %%%%\newcommand{\rank}{\operatorname{rank}} %%%%\newcommand{\Arw}{\operatorname{Arw}} % changed to below by Henry %%%\newcommand{\Arw}{{\cal Arw}} %%%\newcommand{\Ocomm}{C_{\J}} %%%%\newcommand{\Comm}{\mathfrak{C}} % commented out by Henry %%%\newcommand{\bigO}{\mathcal{O}} %%%\newcommand{\Cn}{\mathcal{C}_n} %%%\newcommand{\defeq}{\stackrel{\mathrm{def}}{=}} %%%\newcommand{\pd}{\succ} %%%\nc{\VV}{\mbox{\bf vec\hspace{.03cm}}} %%%\newcommand{\xbar}{\bar{x}} %%%\newcommand{\ybar}{\bar{y}} %%%\newcommand{\zbar}{\bar{z}} %%%\newcommand{\bvv}[1]{\bar{\mathbf{#1}}} %%%\newcommand{\Th}[1]{$#1^\mathrm{th}$} %%%\newcommand{\ubar}{\bar{u}} %%%\newcommand{\tlup}[1]{\widetilde{#1}} %%%\newcommand{\tldn}[1]{\undertilde{#1}} %%%%\newcommand{\Int}{\operatorname{Int}} % commented out by Henry to ?? %%%%%%%%%%%%%%%%%%%%%%%???????? %%%%\newcommand{\sgn}{\operatornamewithlimits{sgn}} %%%\newcommand{\vzero}{\mathbf{0}} %%%\newcommand{\vone}{\mathbf{1}} %%%\newcommand{\va}{\mathbf{a}} %%%\newcommand{\vb}{\mathbf{b}} %%%\newcommand{\vc}{\mathbf{c}} %%%\newcommand{\vd}{\mathbf{d}} %%%\newcommand{\ve}{\mathbf{e}} %%%\newcommand{\vf}{\mathbf{f}} %%%\newcommand{\vg}{\mathbf{g}} %%%\newcommand{\vp}{\mathbf{p}} %%%\newcommand{\vq}{\mathbf{q}} %%%\newcommand{\vt}{\mathbf{t}} %%%\newcommand{\vu}{\mathbf{u}} %%%%\newcommand{\vv}{\mathbf{v}} %%%\renewcommand{\vv}{\mathbf{v}} %%%\newcommand{\vw}{\mathbf{w}} %%%\newcommand{\vx}{\mathbf{x}} %%%\newcommand{\vy}{\mathbf{y}} %%%\newcommand{\vz}{\mathbf{z}} %%%\newcommand{\xt}{\tlup{\vx}} %%%\newcommand{\wt}{\tlup{\vw}} %%%\newcommand{\At}{\tldn{A}} %%%\newcommand{\at}{\tldn{\va_i}} %%%\newcommand{\Xt}{\tilde{X}} %%%\newcommand{\ct}{\tldn{\vc}} %%%\renewcommand{\et}{\tldn{\ve}} %%%\newcommand{\st}{\tldn{\vs}} %%%\newcommand{\St}{\tldn{S}} %%%\newcommand{\dXt}{\widetilde{\dX}} %%%\newcommand{\dSt}{\tldn{\dS}} %%%\newcommand{\dxt}{\tlup{\dx}} %%%\newcommand{\dst}{\tldn{\ds}} %%%\newcommand{\onehalf}{{1\!/\!2}} %%%\newcommand{\nonehalf}{{-\!1\!/\!2}} %%%\newcommand{\xtsq}{\xt^{\onehalf}} %%%\newcommand{\xtisq}{\xt^{\nonehalf}} %%%\newcommand{\stsq}{\st^{\onehalf}} %%%\newcommand{\stisq}{\st^{\nonehalf}} %%%\newcommand{\Xtsq}{\Xt^{\onehalf}} %%%\newcommand{\Xtisq}{\Xt^{\nonehalf}} %%%\newcommand{\Stsq}{\St^{\onehalf}} %%%\newcommand{\Stisq}{\tldn{S}^{\nonehalf}} %%%\newcommand{\Xsq}{X^{\onehalf}} %%%\newcommand{\Xisq}{X^{\nonehalf}} %%%\newcommand{\Ssq}{S^{\onehalf}} %%%\newcommand{\Sisq}{S^{\nonehalf}} %%%\newcommand{\xsq}{\vx^{\onehalf}} %%%\newcommand{\xisq}{\vx^{\nonehalf}} %%%\newcommand{\ssq}{\vs^{\onehalf}} %%%\newcommand{\sisq}{\vs^{\nonehalf}} %%%\newcommand{\ah}{\underaccent{\hat}{\va_i}} %%%\newcommand{\Ah}{\underaccent{\hat}{A}} %%%\newcommand{\sh}{\hat{\vs}} %%%\newcommand{\xh}{\hat{\vx}} %%%\newcommand{\yh}{\hat{\vy}} %%%\newcommand{\hx}{\hat{x}_j} %%%\newcommand{\norminf}[1]{\left\|{#1}\right\|_{\infty}} %%%\newcommand{\normtwo}[1]{\left\|{#1}\right\|_2} %%%\newcommand{\normF}[1]{\left\|{#1}\right\|_{\mathrm{F}}} %%%\newcommand{\normP}[1]{\left\|{#1}\right\|_{\mathrm{p}}} %%%\newcommand{\jnormF}[2]{\left\| {#1}\right\|_{\mathrm{F/{#2}}}} %%%\newcommand{\scp}[2]{\langle{#1},{#2}\rangle} %%%\newcommand{\jscp}[3]{\langle{#1},{#2}\rangle_{#3}} %%%\newcommand{\jdet}[1]{\det_{#1}} %%%\newcommand{\jtr}[1]{\tr_{#1}} %%%\newcommand{\Bn}{\mathcal{B}_n} %%%\newcommand{\Gp}{\mathcal{G}_+} %%%\newcommand{\Qp}[1]{\mathbf{Q}_{#1}} %%%\newcommand{\jprod}{\circ} %%%%\newcommand{\J}{\mathcal{J}} %%%%\newcommand{\J}{\mathfrak{J}} % commented out by Henry %%%\newcommand{\Lo}{\mathcal{L}} %%%\newcommand{\ProjA}{\mathcal{P}} %%%\newcommand{\Q}{\mathcal{Q}} %%%\newcommand{\Oc}{\mathcal{O}} %%%\newcommand{\Sy}{\mathcal{S}} %%%\newcommand{\bbC}{\mathbb{C}} %%%\newcommand{\bbH}{\mathbb{H}} %%%\newcommand{\bbM}{\mathbb{M}} %%%\newcommand{\bbO}{\mathbb{O}} %%%\newcommand{\bbQ}{\mathbb{Q}} %%%%\newcommand{\mathA}{\mathfrak{A}} % changed to following by Henry %%%\newcommand{\mathA}{\A} %%%\newcommand{\mathC}{\A} %%%\newcommand{\mathH}{\A} %%%\newcommand{\mathM}{\A} %%%\newcommand{\mathO}{\A} %%%\newcommand{\mathQ}{\A} %%%\newcommand{\mathS}{\A} %%%%\newcommand{\mathC}{\mathfrak{C}} % changed till math S below %%%%\newcommand{\mathH}{\mathfrak{H}} %%%%\newcommand{\mathM}{\mathfrak{M}} %%%%\newcommand{\mathO}{\mathfrak{O}} %%%%\newcommand{\mathQ}{\mathfrak{Q}} %%%%\newcommand{\mathS}{\mathfrak{S}} %%%\newcommand{\ST}{{\mbox{s.t.}}} %%%\newcommand{\NF}[1]{\N_{F/{#1}}} %%%\newcommand{\Quart}{\mbox{\boldmath H}} %%%%\newcommand{\psd}{\succcurlyeq} % changed by Henry %%%\newcommand{\psd}{\succeq} %%%\newcommand{\pdef}{\succ} %%%%\newcommand{\tr}{\operatorname{tr}} %%%%\renewcommand{\tr}{\operatorname{tr}} % commented out by Henry %%%%\newcommand{\Det}{\operatorname{Det}} %%%\renewcommand{\Det}{\operatorname{Det}} %%%%\newcommand{\Tr}{\operatorname{Tr}} %%%%\renewcommand{\Tr}{\operatorname{Tr}} % changed by Henry to above %%%% i.e. commented out %%%%\newcommand{\A}{\mathcal{A}} %%%\renewcommand{\A}{\mathcal{A}} %%%%\newcommand{\B}{\mathcal{B}} %%%\renewcommand{\B}{\mathcal{B}} %%%%\newcommand{\C}{\mathcal{C}} %%%\renewcommand{\C}{\mathcal{C}} %%%%\newcommand{\E}{\mathcal{E}} %%%\renewcommand{\E}{\mathcal{E}} %%%%\newcommand{\F}{\mathcal{F}} %%%\renewcommand{\F}{\mathcal{F}} %%%%\newcommand{\G}{\mathcal{G}} %%%\renewcommand{\G}{\mathcal{G}} %%%\renewcommand{\H}{\mathcal{H}} %%%%\newcommand{\N}{\mathcal{N}} %%%\renewcommand{\N}{\mathcal{N}} %%%%\newcommand{\K}{\mathcal{K}} %%%\renewcommand{\K}{\mathcal{K}} %%%%\newcommand{\M}{\mathcal{M}} %%%\newcommand{\J}{\mathcal{J}} %%%\renewcommand{\M}{\mathcal{M}} %%%%\newcommand{\T}{\mathcal{T}} %%%\renewcommand{\T}{\mathcal{T}} %%%%\newcommand{\vs}{\mathbf{s}} %%%\renewcommand{\vs}{\mathbf{s}} %%%%\newcommand{\dx}{\Delta \vx} %%%\renewcommand{\dx}{\Delta \vx} %%%%\newcommand{\dy}{\Delta \vy} %%%\renewcommand{\dy}{\Delta \vy} %%%%\newcommand{\ds}{\Delta \vs} %%%\renewcommand{\ds}{\Delta \vs} %%%%\newcommand{\ch}{\hat{\vc}} %%%\renewcommand{\ch}{\underaccent{\hat}{\vc}} %%%%% end from Alizadeh