
% \newtheorem{theorem}{Theorem}[section]
% \newtheorem{definition}[theorem]{Definition}
% \newtheorem{lemma}[theorem]{Lemma}
% \newtheorem{remark}[theorem]{Remark}
% \newtheorem{corollary}[theorem]{Corollary}
% \newtheorem{proposition}[theorem]{Proposition}
% \newtheorem{observation}[theorem]{Observation}
% \newtheorem{assumption}[theorem]{Assumption}
% \newtheorem{example}[theorem]{Example}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\newtheorem{corollary}{Corollary}
\newtheorem{proposition}{Proposition}
\newtheorem{observation}{Observation}
\newtheorem{assumption}{Assumption}
\newtheorem{example}{Example}
%==================================

\newcommand{\viet}[1]{{\color{red}(Viet says: #1)}}
\newcommand{\hieu}[1]{{\color{brown}(Hieu says: #1)}}
\newcommand{\note}[1]{{\color{blue}(Note: #1)}}
%======================================
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{equation*}\begin{aligned}}
\newcommand{\eea}{\end{aligned}\end{equation*}}
\newcommand{\ds}{\displaystyle}
\newcommand{\independent}{\protect\mathpalette{\protect\independenT}{\perp}}
\newcommand{\R}{\mathbb{R}}

\newcommand{\Max}{\max\limits_}
\newcommand{\Min}{\min\limits_}
\newcommand{\Sup}{\sup\limits_}
\newcommand{\Inf}{\inf\limits_}
\newcommand{\Tr}[1]{\mathrm{Tr} \big[ #1 \big]}

\newcommand{\wh}{\widehat}
\newcommand{\mc}{\mathcal}
\newcommand{\mbb}{\mathbb}
\newcommand{\inner}[2]{\big \langle #1, #2 \big \rangle }

\newcommand{\PP}{\mbb P}
\newcommand{\Pnom}{\wh{\mbb P}}
\newcommand{\QQ}{\mbb Q}
\newcommand{\D}{\mathds D}
\newcommand{\dd}{\mathrm{d}}

\newcommand{\cov}{\Sigma}
\newcommand{\covsa}{\wh \cov}
\newcommand{\FR}{\mathds{F}}

\DeclareMathOperator{\Trace}{Tr}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\Diag}{Diag}
\DeclareMathOperator{\sign}{sign}

\DeclareMathOperator{\st}{s.t.}
\DeclareMathOperator{\nullspace}{null}
\DeclareMathOperator{\vect}{vec}
\newcommand{\KL}{\mathrm{KL}}
\newcommand{\W}{\mathds{W}}
\DeclareMathOperator{\Argmin}{Argmin}
\DeclareMathOperator{\Minval}{Min}
\DeclareMathOperator{\grad}{grad}
\DeclareMathOperator{\Hess}{Hess}

\newcommand{\defeq}{%
  \mathrel{\vbox{\offinterlineskip\ialign{%
    \hfil##\hfil\cr
    $\scriptscriptstyle\triangle$\cr
    %\noalign{\kern0ex}
    $=$\cr
}}}}

% Peyman's comments
\newcommand{\Sym}{\mathbb{S}}
\newcommand{\PSD}{\mathbb{S}_{+}} % the set of positive semi-definite matrices
\newcommand{\PD}{\mathbb{S}_{++}} % the set of positive definite matrices
\newcommand{\Let}{\triangleq}
\newcommand{\opt}{^\star}
\newcommand{\eps}{\varepsilon}
\newcommand{\ra}{\rightarrow}
\newcommand{\M}{\mc M}
\newcommand{\BB}{\mbb B}
\newcommand{\B}{\mc B}
\newcommand{\X}{X}
\newcommand{\Y}{Y}
\newcommand{\Wass}{\mathds{W}}
\newcommand{\V}{\mathds{W}_S}
\newcommand{\Q}{\mbb{Q}}
\newcommand{\EE}{\mathds{E}}
% \DeclareMathOperator{\deg}{deg}


\newcommand{\half}{\frac{1}{2}}
\newcommand{\dualvar}{\gamma}


% \newcommand{\ie}{{\em i.e.}}
\newcommand{\da}{\downarrow}
\newcommand{\J}{\mc J}
\newcommand{\Dataset}{\wh{\mc D}}

\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\NA}{\mathrm{NA}}