

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{claim}{Claim}
\newtheorem{prop}{Proposition}
\newtheorem{conj}{Conjecture}
\newtheorem{cor}{Corollary}
\newtheorem{assumptions}{Assumption}

\theoremstyle{definition}
\newtheorem{quest}{Question}
\newtheorem{defn}{Definition}
\newtheorem{scheme}{Scheme}
\newtheorem{rem}{Remark}


\DeclareMathOperator*{\arginf}{\arg\!\inf}
\DeclareMathOperator*{\argsup}{\arg\!\sup}
\DeclareMathOperator*{\argmin}{\arg\!\min}
\DeclareMathOperator*{\argmax}{\arg\!\max}
\DeclareMathOperator{\atantwo}{atan2}

\newcommand{\cA}{\mathcal{A}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}

\newcommand{\bu}{\bm{u}}
\newcommand{\ba}{\bm{a}}
\newcommand{\bb}{\bm{b}}
\newcommand{\bc}{\bm{c}}
\newcommand{\bp}{\bm{p}}
\newcommand{\be}{\bm{e}}
\newcommand{\bx}{\bm{x}}
\newcommand{\by}{\bm{y}}
\newcommand{\bv}{\bm{v}}
\newcommand{\bz}{\bm{z}}
\newcommand{\bw}{\bm{w}}
\newcommand{\bh}{\bm{h}}
\newcommand{\bg}{\bm{g}}
\newcommand{\bH}{\bm{H}}
\newcommand{\bG}{\bm{G}}
\newcommand{\bI}{\bm{I}}
\newcommand{\bM}{\bm{M}}
\newcommand{\bW}{\bm{W}}
\newcommand{\bA}{\bm{A}}
\newcommand{\bR}{\bm{R}}
\newcommand{\bP}{\bm{P}}
\newcommand{\bQ}{\bm{Q}}
\newcommand{\bD}{\bm{D}}
\newcommand{\bU}{\bm{U}}
\newcommand{\bV}{\bm{V}}
\newcommand{\bX}{\bm{X}}
\newcommand{\bY}{\bm{Y}}
\newcommand{\bZ}{\bm{Z}}

\newcommand{\btheta}{\pmb{\theta}}
\newcommand{\blambda}{\pmb{\lambda}}
\newcommand{\bmu}{\pmb{\mu}}
\newcommand{\bsigma}{\pmb{\sigma}}
\newcommand{\bSigma}{\pmb{\Sigma}}
\newcommand{\bLambda}{\pmb{\Lambda}}

\newcommand{\rB}{\textnormal{B}}
\newcommand{\rU}{\textnormal{U}}
\newcommand{\rV}{\textnormal{V}}
\newcommand{\rW}{\textnormal{W}}
\newcommand{\rX}{\textnormal{X}}
\newcommand{\rY}{\textnormal{Y}}
\newcommand{\rZ}{\textnormal{Z}}
\newcommand{\rA}{\textnormal{A}}
\newcommand{\rS}{\textnormal{S}}
\newcommand{\rL}{\textnormal{L}}
\newcommand{\rJ}{\textnormal{J}}
\newcommand{\rR}{\textnormal{R}}
\newcommand{\rN}{\textnormal{N}}

\newcommand{\tr}{\mathsf{tr}}
\newcommand{\diag}{\mathsf{diag}}
\newcommand{\rank}{\mbox{rank}}
\newcommand{\ones}{\bm{1}}

\newcommand{\binstr}{\{0,1\}} 
\newcommand{\supp}{\mbox{supp}}
\renewcommand{\tilde}{\widetilde}

\newcommand{\Reals}{\mathbb{R}}
\newcommand{\defined}{\triangleq}
\newcommand{\etal}{\textit{et al.}~}
\newcommand{\sto}{\mbox{\normalfont s.t.}}
\newcommand{\suchthat}{\,\mid\,}
\newcommand{\indicator}[1]{\mathbb{I}_{#1}}
\newcommand{\bigO}{O}
\newcommand{\dif}{\textrm{d}}


\newcommand{\ExpVal}[2]{\mathbb{E}\left[ #2 \right]}
\newcommand{\EE}[1]{\ExpVal{}{#1}}
\newcommand{\EEE}[2]{\mathbb{E}_{#1}\big[ #2 \big]}
\newcommand{\mmse}{\mathsf{mmse}}
\newcommand{\mmae}{\mathsf{mmae}}
\newcommand{\median}{\mathsf{median}}
\newcommand{\Ent}[1]{\mathsf{Entropy}\left(#1\right)}
\newcommand{\Entt}[2]{\mathsf{Entropy}_{#1}\left[#2\right]}
\newcommand{\Varr}[2]{\mathsf{Var}_{#1}\left(#2\right)}
\newcommand{\Covv}[2]{\mathsf{Cov}_{#1}\left[#2\right]}
\newcommand{\Var}[1]{\mathsf{Var}\left(#1\right)}
\newcommand{\Unc}[1]{\mathsf{Unc}\left(#1\right)}
\newcommand{\Uncc}[2]{\mathsf{Unc}_{#1}\left(#2\right)}
\newcommand{\Sim}[1]{\mathsf{Sim}\left(#1\right)}
\newcommand{\Perf}[1]{\mathsf{Perf}_t\left(#1\right)}
\newcommand{\myPerf}[1]{\mathsf{Perf}_t \left(g_{h,#1} \circ h, ~ \bx^*\right)}  %
\newcommand{\pH}{\mathcal{P}_{\mathcal{H}}}

\newcommand{\indep}{\rotatebox[origin=c]{90}{$\models$}}
\newcommand{\dep}{\rotatebox[origin=c]{90}{$\not\models$}}

\newcommand{\KL}{\textnormal{D}_{\scalebox{.6}{\textnormal KL}}}
\newcommand{\TV}{\textnormal{D}_{\scalebox{.6}{\textnormal TV}}}
\newcommand{\chisquare}{\textnormal{D}_{\chi^2}}
\newcommand{\wass}{\mathbb{W}}

\newcommand{\floor}[1]{\lfloor #1 \rfloor}
\newcommand{\ceil}[1]{\lceil #1 \rceil}

\usepackage{color}

\newcommand{\gitrepo}[0]{[{\color{red}{URL REDACTED FOR BLIND REVIEW}}]}



\newcommand{\hao}{}
\newcommand{\yj}{}
\newcommand{\rev}{}  %
\newcommand{\uai}{\textcolor{magenta}}  %

\newcommand{\inp}{\mathbf{x}}
\newcommand{\rep}{\mathbf{z}}
\newcommand{\rdim}{d}
\newcommand{\amk}{\mathsf{Dist}_{k}}
\newcommand{\amone}{\mathsf{Dist}_{1}}
\newcommand{\aggf}{\texttt{AGGREGATE}}

\newcommand{\kendall}{Kendall's $\tau$ coefficient }


