\def\x{{\mathbf x}}
\def\z{{\mathbf z}}
\def\1{{\mathbf 1}}
\def\a{{\mathbf a}}
\def\c{{\mathbf c}}
\def\v{{\mathbf v}}
\def\X{{\mathbf X}}
\def\Z{{\mathbf Z}}
\def\hatx{{\bf \hat x}}
\def\hatf{{ \hat f}}
\def\hatw{{\bf \hat w}}
\def\hati{\hat{\imath}}
\def\tildej{\tilde{\jmath}}
\def\tildel{\tilde{l}}
\def\hatX{{\bf \hat X}}
\def\hatb{{\hat b}}
\def\ae{\text{~~a.e.}}
\def\as{\text{~~a.s.}}
\def\hatalpha{\mbox{\boldmath${\hat \alpha}$}}
\def\tildealpha{\mbox{\boldmath${\tilde \alpha}$}}
\def\tildebeta{\mbox{\boldmath${\tilde \beta}$}}
\def\hatDelta{\mbox{\boldmath${\hat \Delta}$}}
\def\Lambdab{{\boldsymbol\Lambda}}
\def\lambdab{{\boldsymbol\lambda}}
\def\nub{{\boldsymbol\nu}}
\def\Gammab{{\boldsymbol\Gamma}}
\def\Sigmab{{\boldsymbol\Sigma}}
\def\Deltab{{\boldsymbol\Delta}}
\def\betab{{\boldsymbol\beta}}
\def\epsilonb{{\boldsymbol\varepsilon}}
\def\alphab{{\boldsymbol\alpha}}
\def\alphabtilde{{\boldsymbol\tilde{\alpha}}}
\def\hatlambdab{{\boldsymbol{\hat\lambda}}}
\def\Gammab{{\boldsymbol\Gamma}}
\def\tildeGammab{{\bf \tilde{\boldsymbol\Gamma}}}
\def\y{{\mathbf y}}
\def\w{{\mathbf w}}
\def\D{{\mathbf D}}
\def\Y{{\mathbf Y}}
\def\M{{\mathbf M}}
\def\B{{\mathbf B}}
\def\Q{{\mathbf Q}}
\def\CC{{\mathbf C}}
\def\TT{{\mathcal T}}
\def\W{{\mathbf W}}
\def\d{{\mathbf d}}
\def\E{{\mathbb E}}
\def\O{{O}}
\def\N{{\mathcal N}}
\def\o{{\mathcal o}}
\def\s{{\mathbf s}}
\def\C{{\mathcal C}}
\def\CCC{{C}}
\def\L{{\mathcal L}}
\def\P{{\mathcal P}}
\def\PP{{\mathbf P}}
\def\PPP{{\mathbb P}}
\def\R{{\mathbf R}}
\def\S{{\mathcal S}}
\def\SS{{\mathbf SS}}
\def\H{{\mathcal H}}
\def\d{{\mathbf d}}
\def\e{{\mathbf e}}
\def\b{{\mathbf b}}
\def\u{{\mathbf u}}
\def\m{{\mathbf m}}
\def\hatD{{\bf \hat D}}
\def\tildeD{{\bf \tilde D}}
\def\tildeA{{\bf \tilde A}}
\def\tildeB{{\bf \tilde B}}
\def\tildeX{{\bf \tilde X}}
\def\tildex{{\bf \tilde x}}
\def\tildef{{ \tilde f}}
\def\hatE{{\bf \hat E}}
\def\hate{{\bf \hat e}}
\def\hatd{{\bf \hat d}}
\def\hatf{{ \hat f}}
\def\tilded{{\bf \tilde d}}
\def\Real{{\mathbb R}}
\def\r{{\mathbf r}}
\def\U{{\mathbf U}}
\def\u{{\mathbf u}}
\def\A{{\mathbf A}}
\def\G{{\mathbb G}}
\def\GG{{\mathcal G}}
\def\AA{{\mathcal A}}
\def\FF{{\mathcal F}}
\def\V{{\mathbf V}}
\def\K{{\mathbf K}}
\def\KK{{\mathcal K}}
\def\T{{\mathbf T}}
\def\I{{\mathbf I}}
\def\one{{\mathbb 1}}
\def\argmin{\operatornamewithlimits{arg\,min}}
\def\liminf{\operatornamewithlimits{lim\,inf}}
\def\argmax{\operatornamewithlimits{arg\,max}}
\def\Var{\operatorname{Var}}
\def\diag{\operatorname{diag}}
\def\trace{\operatorname{Tr}}
\def\range{\operatorname{range}}
\def\vec{\operatorname{vec}}
\def\sign{\operatorname{sign}}
\def\FL{\operatorname{FL}}
\def\corr{\operatorname{corr}}
\def\cov{\operatorname{cov}}
\def\st{~~\text{s.t.}~~}
\def\defin{\stackrel{\vartriangle}{=}}
%\def\TODO{ {\color{red} TODO }}
\newcommand{\TODO}[1]{{\textbf{\color{red} TODO: #1}}}
\newcommand{\InSet}[1]{[ 1;#1 ]}
\newcommand{\IntSet}[1]{[ 1;#1 ]}

\newcommand{\NormFro}[1]{\|#1\|_{\text{F}}}
\newcommand{\NormFrob}[1]{\|#1\|_{\text{F}}}

\def\kappab{{\boldsymbol\kappa}}
\def\gammab{{\boldsymbol\gamma}}
\newcommand{\citet}{\cite}
\newcommand{\citep}{\cite}
%\renewenvironment{displaymath}{\begin{equation}}{\end{equation}}
%\renewenvironment{multline*}{\begin{multline}}{\end{multline}}

%\renewcommand{\cite}{\citep}
% \newcommand{\citet}{\emcite}
% \renewenvironment{displaymath}{\begin{equation}}{\end{equation}}
%\renewenvironment{multline*}{\begin{multline}}{\end{multline}}

% \newtheorem{example}{Example} 
% \newtheorem{theorem}{Theorem}
% \newtheorem{lemma}{Lemma} 
% \newtheorem{proposition}{Proposition} 
% \newtheorem{remark}[theorem]{Remark}
% \newtheorem{corollary}[theorem]{Corollary}
% \newtheorem{definition}[theorem]{Definition}
% \newtheorem{conjecture}[theorem]{Conjecture}
% \newtheorem{axiom}[theorem]{Axiom}
% 
% \newcommand{\BlackBox}{\rule{1.5ex}{1.5ex}}  % end of proof
% \newenvironment{proof}{\par\noindent{\bf Proof sktech:}}{\hfill\BlackBox\\[2mm]}
% \newcommand{\algsetup}{ }
% \newcommand{\RETURN}{ \STATE {\textbf{Return}} }
% 
