\newcommand{\mutual}{\ensuremath{{I}}}
\newcommand{\entropy}{\ensuremath{{H}}}
\newtheorem{lemma}{Lemma}
\newtheorem{fact}{Fact}
\newtheorem{example}{Example}
\newtheorem{prop}{Proposition}
\newtheorem{theorem}{Theorem}
%\newcommand*{\defeq}{\stackrel{\text{def}}{=}}
\newcommand*{\defeq}{\equiv}
\newcommand{\var}{\mathop{\mathrm{Var}}}
\newcommand{\condexp}[2]{\mathop{\mathbb{E}}\left[#1|#2\right]}
\newcommand{\expt}[1]{\mathop{\mathbb{E}}\left[#1\right]}
\newcommand{\norm}[1]{\lVert#1\rVert}
\newcommand{\tr}[1]{#1^*}
\newcommand{\ip}[2]{\langle #1, #2 \rangle}
\newcommand{\mse}{\mathop{\mathrm{MSE}}}
\DeclareMathOperator{\trace}{tr}
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\newcommand{\reals}{\ensuremath{\mathbbm{R}}}
\newcommand{\stratis}[1]{\textcolor{red}{Stratis: #1}}