\newtheorem{lemma}{Lemma} \newtheorem{theorem}{Theorem} \newtheorem{proof}{Proof} \newtheorem{defn}{Definition} \newtheorem{algo}{Algorithm} \newtheorem{proc}{Procedure} \newtheorem{rem}{Remark} \newtheorem{observation}{Observation} % general symbols \newcommand{\RF}{\mathbb{R}} \newcommand{\FF}{\mathbb{F}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\nn}{\nonumber} \newcommand{\paren}[1]{\left( #1 \right)} \newcommand{\sqprn}[1]{\left[ #1 \right]} \newcommand{\tlprn}[1]{\left\{ #1 \right\}} \newcommand{\abs}[1]{\left| #1 \right|} \newcommand{\tr}[2]{\left<\left< #1 , #2 \right>\right>} \newcommand{\trs}[2]{\left< #1 , #2 \right>} \newcommand{\iprod}[2]{\left\langle #1 , #2 \right\rangle} \newcommand{\del}{\partial} \newcommand{\dd}[2]{\frac{d#1}{d#2}} \newcommand{\ddel}[2]{\frac{\del#1}{\del#2}} \newcommand{\ddell}[2]{\frac{\del^2#1}{{\del#2}^2}} \newcommand{\ddelll}[3]{\frac{\del^2#1}{{\del#2}{\del#3}}} \newcommand{\beq}{\begin{eqnarray*}} \newcommand{\eeq}{\end{eqnarray*}} \newcommand{\beqn}{\begin{eqnarray}} \newcommand{\eeqn}{\end{eqnarray}} \newcommand{\ben}{\begin{enumerate}} \newcommand{\een}{\end{enumerate}} \newcommand{\bit}{\begin{itemize}} \newcommand{\eit}{\end{itemize}} \newcommand{\hide}[1]{} \newcommand{\oo}[1]{\frac{1}{#1}} \newcommand{\eps}{\epsilon} \newcommand{\evalat}[2]{\left.#1\right|_{#2}} \newcommand{\ceil}[1]{\ensuremath{\left\lceil#1\right\rceil}} \newcommand{\floor}[1]{\ensuremath{\left\lfloor#1\right\rfloor}} \newcommand{\ontop}[2]{\overset{{\scriptsize #1}}{#2}} \newcommand{\Perp}{\perp\!\!\!\perp} \newcommand{\limsu}{\mathop{\overline{\mathrm{lim}}}} \newcommand{\limin}{\mathop{\underline{\mathrm{lim}}}} \newcommand{\argmin}{\mathop{\mathrm{argmin}}} \newcommand{\argmax}{\mathop{\mathrm{argmax}}} \newcommand{\map}{\mathop{\mathrm{map}}} \newcommand{\powrs}[1]{\langle\!\langle #1 \rangle\!\rangle} \newcommand{\polys}[1]{\langle #1 \rangle} \newcommand{\contf}{\mathscr{C}} %\newcommand{\trn}{^{{}^\intercal}} %operator transpose \newcommand{\trn}{^\intercal} %operator transpose %\newcommand{\trn}{^{{}^{\!\intercal}}} %operator transpose \newcommand{\inv}{^{-1}} %inverse \newcommand{\unit}{\boldsymbol{1}} \newcommand{\zero}{\boldsymbol{0}} \newcommand{\defeq}{\stackrel{\mathrm{def}}{=}} \newcommand{\pleq}{~+\!=} \newcommand{\chr}{\mathrm{\bs{1}\hspace{-3pt}I}} % characteristic function % function names \newcommand{\zeros}{\mathrm{zeros}} \newcommand{\rank}{\mathrm{rank}} \newcommand{\TR}{\mathrm{tr}} \newcommand{\vol}{\mathrm{vol}} \newcommand{\sgn}{\mathrm{sgn}} \newcommand{\acos}{\mathrm{acos}} \newcommand{\intr}{\mathrm{int}\,} \newcommand{\extr}{\mathrm{ext}\,} \newcommand{\dom}{\mathrm{dom}\,} \newcommand{\conv}{\mathrm{conv}\hspace{1pt}} \newcommand{\cone}{\mathrm{cone}\hspace{1pt}} %probability & statistics symbols \newcommand{\convd}{\overset{{\scriptsize d}}{\to}} \newcommand{\convp}{\overset{{\scriptsize p}}{\to}} \newcommand{\unif}{\mathrm{Uniform}} \newcommand{\bern}{\mathrm{Bernoulli}} \newcommand{\negbin}{\mathrm{NegBinom}} \newcommand{\Binom}{\mathrm{Bin}} \newcommand{\pois}{\mathrm{Poisson}} %\newcommand{\ndistr}{\mathrm{Normal}} \newcommand{\ndistr}{\mathrm{n}} \newcommand{\Bdistr}{\mathrm{Beta}} \newcommand{\Gdistr}{\mathrm{Gamma}} % f(x) = \frac{1}{\beta^\alpha\Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta} \newcommand{\iGdistr}{\mathrm{InvGamma}} % f(x) = \frac{1}{\beta^\alpha\Gamma(\alpha)}x^{-\alpha-1}e^{-1/(\beta x)} \newcommand{\MSE}{\mathrm{MSE}} \newcommand{\MAE}{\mathrm{MAE}} \newcommand{\ARE}{\mathrm{ARE}} \newcommand{\se}{\mathrm{se}} \newcommand{\eg}{\mathrm{e.g.}} \newcommand{\st}{\mathrm{s.t.}} \newcommand{\ie}{\mathrm{i.e.}} \newcommand{\calA}{\mathcal{A}} \newcommand{\calC}{\mathcal{C}} \newcommand{\calD}{\mathcal{D}} \newcommand{\calE}{\mathcal{E}} \newcommand{\calF}{\mathcal{F}} \newcommand{\calH}{\mathcal{H}} \newcommand{\calI}{\mathcal{I}} \newcommand{\calK}{\mathcal{K}} \newcommand{\calL}{\mathcal{L}} \newcommand{\calM}{\mathcal{M}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calP}{\mathcal{P}} \newcommand{\calS}{\mathcal{S}} \newcommand{\calT}{\mathcal{T}} \newcommand{\calV}{\mathcal{V}} \newcommand{\calX}{\mathcal{X}} \newcommand{\calY}{\mathcal{Y}} \newcommand{\calZ}{\mathcal{Z}} \newcommand{\tha}{\theta} %\newcommand{\mexp}{\mathbb{E}} %\newcommand{\mexp}[1]{\ensuremath{\text{{\bf E}$\left[#1\right]$}}} \newcommand{\mexp}{\text{{\bf E}}} \newcommand{\var}{\mathrm{Var}} \newcommand{\cov}{\mathrm{Cov}} \newcommand{\corr}{\mathrm{Cor}} \newcommand{\bX}{\bs{X}} \newcommand{\bx}{\bs{x}} \newcommand{\bY}{\bs{Y}} \newcommand{\by}{\bs{y}} \newcommand{\bZ}{\bs{Z}} \newcommand{\bz}{\bs{z}} \newcommand{\Xbar}{\overline{X}} \newcommand{\ev}{\mathrm{Ev}} %evidence \newcommand{\pr}{\boldsymbol{\mathrm{P}}} %\newcommand{\prob}{\mathrm{Pr}} % embeddings \newcommand{\embeds}[1]{\;\lhook\joinrel\xrightarrow{#1}\;} %misc commands \newcommand{\ds}{\displaystyle} \newcommand{\ts}{\textstyle} \newcommand{\bs}{\boldsymbol} \renewcommand{\th}{\ensuremath{^{\mathrm{th}}}~}