$$ \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\argmin}{arg\,min} \newcommand{\nnz}[1]{\mbox{nnz}(#1)} \newcommand{\dotprod}[2]{\langle #1, #2 \rangle} \DeclareMathOperator*{\Pr}{\mathbf{Pr}} \DeclareMathOperator*{\E}{\mathbf{E}} \DeclareMathOperator*{\Ex}{\mathbf{E}} \DeclareMathOperator*{\Var}{\mathbf{Var}} \DeclareMathOperator*{\Cov}{\mathbf{Cov}} \DeclareMathOperator*{\stddev}{\mathbf{stddev}} \DeclareMathOperator{\littlesum} \DeclareMathOperator{\littleprod} \DeclareMathOperator*{\avg}{avg} % environments \declaretheorem[numberwithin=section]{theorem} \declaretheorem[sibling=theorem]{lemma} \declaretheorem[sibling=theorem]{claim} \declaretheorem[sibling=theorem]{proposition} \declaretheorem[sibling=theorem]{fact} \declaretheorem[sibling=theorem]{corollary} \declaretheorem[sibling=theorem]{conjecture} \declaretheorem[sibling=theorem]{question} \declaretheorem[sibling=theorem]{answer} \declaretheorem[sibling=theorem]{solution} \declaretheorem[sibling=theorem]{hypothesis} \declaretheorem[sibling=theorem]{exercise} \theoremstyle{definition} \declaretheorem[sibling=theorem]{definition} \declaretheorem[sibling=theorem]{remark} \declaretheorem[sibling=theorem]{notation} \declaretheorem[sibling=theorem]{observation} \declaretheorem[sibling=theorem]{example} \newcommand{\ignore}[1]{} % probability type operators \let\Pr\relax \DeclareMathOperator*{\Pr}{\mathbf{Pr}} \DeclareMathOperator*{\E}{\mathbf{E}} \DeclareMathOperator*{\Ex}{\mathbf{E}} \DeclareMathOperator*{\Var}{\mathbf{Var}} \DeclareMathOperator*{\Cov}{\mathbf{Cov}} \DeclareMathOperator*{\stddev}{\mathbf{stddev}} \DeclareMathOperator{\littlesum} \DeclareMathOperator{\littleprod} \DeclareMathOperator*{\avg}{avg} % math terms \DeclareMathOperator{\poly}{poly} \DeclareMathOperator{\polylog}{polylog} \DeclareMathOperator{\size}{size} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\vol}{vol} \DeclareMathOperator{\spn}{span} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\codim}{codim} \DeclareMathOperator{\diag}{diag} % number systems \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\F}{\mathbb F} \newcommand{\Q}{\mathbb Q} % complexity classes \newcommand{\PTIME}{\mathsf{P}} \newcommand{\LOGSPACE}{\mathsf{L}} \newcommand{\ZPP}{\mathsf{ZPP}} \newcommand{\RP}{\mathsf{RP}} \newcommand{\BPP}{\mathsf{BPP}} \newcommand{\P}{\mathsf{P}} \newcommand{\NP}{\mathsf{NP}} \newcommand{\TC}{\mathsf{TC}} \newcommand{\AC}{\mathsf{AC}} \newcommand{\SC}{\mathsf{SC}} \newcommand{\SZK}{\mathsf{SZK}} \newcommand{\AM}{\mathsf{AM}} \newcommand{\IP}{\mathsf{IP}} \newcommand{\PSPACE}{\mathsf{PSPACE}} \newcommand{\EXP}{\mathsf{EXP}} \newcommand{\MIP}{\mathsf{MIP}} \newcommand{\NEXP}{\mathsf{NEXP}} \newcommand{\BQP}{\mathsf{BQP}} \newcommand{\distP}{\mathsf{dist\textbf{P}}} \newcommand{\distNP}{\mathsf{dist\textbf{NP}}} % short forms \newcommand{\eps}{\epsilon} \newcommand{\lam}{\lambda} \newcommand{\dleta}{\delta} \newcommand{\simga}{\sigma} \newcommand{\vphi}{\varphi} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\wt}[1]{\widetilde{#1}} \newcommand{\wh}[1]{\widehat{#1}} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\ot}{\otimes} \newcommand{\zo}{\{0,1\}} \newcommand{\co}{:} %\newcommand{\co}{\colon} \newcommand{\bdry}{\partial} \newcommand{\grad}{\nabla} \newcommand{\transp}{^\intercal} \newcommand{\inv}{^{-1}} \newcommand{\symmdiff}{\triangle} \newcommand{\symdiff}{\symmdiff} \newcommand{\half}{\tfrac{1}{2}} \newcommand{\bbone}{\mathbbm 1} \newcommand{\Id}{\bbone} \newcommand{\SAT}{\mathsf{SAT}} % calligraphic letters \newcommand{\calA}{\mathcal{A}} \newcommand{\calB}{\mathcal{B}} \newcommand{\calC}{\mathcal{C}} \newcommand{\calD}{\mathcal{D}} \newcommand{\calE}{\mathcal{E}} \newcommand{\calF}{\mathcal{F}} \newcommand{\calG}{\mathcal{G}} \newcommand{\calH}{\mathcal{H}} \newcommand{\calI}{\mathcal{I}} \newcommand{\calJ}{\mathcal{J}} \newcommand{\calK}{\mathcal{K}} \newcommand{\calL}{\mathcal{L}} \newcommand{\calM}{\mathcal{M}} \newcommand{\calN}{\mathcal{N}} \newcommand{\calO}{\mathcal{O}} \newcommand{\calP}{\mathcal{P}} \newcommand{\calQ}{\mathcal{Q}} \newcommand{\calR}{\mathcal{R}} \newcommand{\calS}{\mathcal{S}} \newcommand{\calT}{\mathcal{T}} \newcommand{\calU}{\mathcal{U}} \newcommand{\calV}{\mathcal{V}} \newcommand{\calW}{\mathcal{W}} \newcommand{\calX}{\mathcal{X}} \newcommand{\calY}{\mathcal{Y}} \newcommand{\calZ}{\mathcal{Z}} % bold \newcommand{\bone}{\boldsymbol{1}} \newcommand{\bbeta}{\boldsymbol{\beta}} \newcommand{\bdelta}{\boldsymbol{\delta}} \newcommand{\bepsilon}{\boldsymbol{\epsilon}} \newcommand{\blambda}{\boldsymbol{\lambda}} \newcommand{\bomega}{\boldsymbol{\omega}} \newcommand{\bpi}{\boldsymbol{\pi}} \newcommand{\bphi}{\boldsymbol{\phi}} \newcommand{\bvphi}{\boldsymbol{\varphi}} \newcommand{\bpsi}{\boldsymbol{\psi}} \newcommand{\bsigma}{\boldsymbol{\sigma}} \newcommand{\btheta}{\boldsymbol{\theta}} \newcommand{\btau}{\boldsymbol{\tau}} \newcommand{\ba}{\boldsymbol{a}} \newcommand{\bb}{\boldsymbol{b}} \newcommand{\bc}{\boldsymbol{c}} \newcommand{\bd}{\boldsymbol{d}} \newcommand{\be}{\boldsymbol{e}} \newcommand{\boldf}{\boldsymbol{f}} \newcommand{\bg}{\boldsymbol{g}} \newcommand{\bh}{\boldsymbol{h}} \newcommand{\bi}{\boldsymbol{i}} \newcommand{\bj}{\boldsymbol{j}} \newcommand{\bk}{\boldsymbol{k}} \newcommand{\bell}{\boldsymbol{\ell}} \newcommand{\bm}{\boldsymbol{m}} \newcommand{\bn}{\boldsymbol{n}} \newcommand{\bo}{\boldsymbol{o}} \newcommand{\bp}{\boldsymbol{p}} \newcommand{\bq}{\boldsymbol{q}} \newcommand{\br}{\boldsymbol{r}} \newcommand{\bs}{\boldsymbol{s}} \newcommand{\bt}{\boldsymbol{t}} \newcommand{\bu}{\boldsymbol{u}} \newcommand{\bv}{\boldsymbol{v}} \newcommand{\bw}{\boldsymbol{w}} \newcommand{\bx}{\boldsymbol{x}} \newcommand{\by}{\boldsymbol{y}} \newcommand{\bz}{\boldsymbol{z}} \newcommand{\bA}{\boldsymbol{A}} \newcommand{\bB}{\boldsymbol{B}} \newcommand{\bC}{\boldsymbol{C}} \newcommand{\bD}{\boldsymbol{D}} \newcommand{\bE}{\boldsymbol{E}} \newcommand{\bF}{\boldsymbol{F}} \newcommand{\bG}{\boldsymbol{G}} \newcommand{\bH}{\boldsymbol{H}} \newcommand{\bI}{\boldsymbol{I}} \newcommand{\bJ}{\boldsymbol{J}} \newcommand{\bK}{\boldsymbol{K}} \newcommand{\bL}{\boldsymbol{L}} \newcommand{\bM}{\boldsymbol{M}} \newcommand{\bN}{\boldsymbol{N}} \newcommand{\bP}{\boldsymbol{P}} \newcommand{\bQ}{\boldsymbol{Q}} \newcommand{\bR}{\boldsymbol{R}} \newcommand{\bS}{\boldsymbol{S}} \newcommand{\bT}{\boldsymbol{T}} \newcommand{\bU}{\boldsymbol{U}} \newcommand{\bV}{\boldsymbol{V}} \newcommand{\bW}{\boldsymbol{W}} \newcommand{\bX}{\boldsymbol{X}} \newcommand{\bY}{\boldsymbol{Y}} \newcommand{\bZ}{\boldsymbol{Z}} % others \newcommand{\false}} \newcommand{\true}} % bold calligraphic \newcommand{\bcalG}{\boldsymbol{\calG}} \newcommand{\calbG}{\bcalG} \newcommand{\bcalX}{\boldsymbol{\calX}} \newcommand{\calbX}{\bcalX} \newcommand{\bcalY}{\boldsymbol{\calY}} \newcommand{\calbY}{\bcalY} \newcommand{\bcalZ}{\boldsymbol{\calZ}} \newcommand{\calbZ}{\bcalZ} % left-right wrappers \DeclarePairedDelimiter\parens{\lparen}{\rparen} \DeclarePairedDelimiter\abs{\lvert}{\rvert} \DeclarePairedDelimiter\norm{\lVert}{\rVert} \DeclarePairedDelimiter\floor{\lfloor}{\rfloor} \DeclarePairedDelimiter\ceil{\lceil}{\rceil} \DeclarePairedDelimiter\braces{\lbrace}{\rbrace} \DeclarePairedDelimiter\bracks{\lbrack}{\rbrack} \DeclarePairedDelimiter\angles{\langle}{\rangle} % something for figures \newcommand{\myfig}[4]{\begin{figure}[H] \centering \includegraphics[width=#1\textwidth]{#2} \caption{#3} \label{#4} \end{figure}} \definecolor{codegreen}{rgb}{0,0.6,0} \definecolor{codegray}{rgb}{0.5,0.5,0.5} \definecolor{codepurple}{rgb}{0.58,0,0.82} \definecolor{backcolour}{rgb}{0.95,0.95,0.92} \lstdefinestyle{mystyle}{ backgroundcolor=\color{backcolour}, commentstyle=\color{codegreen}, keywordstyle=\color{magenta}, numberstyle=\tiny\color{codegray}, stringstyle=\color{codepurple}, basicstyle=\ttfamily\footnotesize, breakatwhitespace=false, breaklines=true, captionpos=b, keepspaces=true, numbers=left, numbersep=5pt, showspaces=false, showstringspaces=false, showtabs=false, tabsize=2 } \lstset{style=mystyle} $$