%\newtheorem{theorem}{Theorem}[section]
\newtheorem{question}[theorem]{Question}
\newtheorem{conj}{Conjecture}
\newtheorem{cond}[theorem]{Condition}
%\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corrollary}{Corrollary}
\newtheorem{informal theorem}[theorem]{Theorem (informal statement)}
\newtheorem{condition}[theorem]{Condition}
%\newtheorem{proposition}[theorem]{Proposition}
%\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{obs}[theorem]{Observation}
%\newtheorem{remark}[theorem]{Remark}
\newtheorem{infproposition}[theorem]{Proposition (Informal)}
\newtheorem{observation}[theorem]{Observation}
%\newtheorem{assumption}[theorem]{Assumption}
%\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}{Example}
\newcommand{\eqdef}{\coloneqq}

\newcommand{\bA}{\mathbf{A}}
\newcommand{\bB}{\mathbf{B}}
\newcommand{\bC}{\mathbf{C}}
\newcommand{\bD}{\mathbf{D}}
\newcommand{\bE}{\mathbf{E}}
\newcommand{\bF}{\mathbf{F}}
\newcommand{\bG}{\mathbf{G}}
\newcommand{\bH}{\mathbf{H}}
\newcommand{\bI}{\mathbf{I}}
\newcommand{\bJ}{\mathbf{J}}
\newcommand{\bK}{\mathbf{K}}
\newcommand{\bL}{\mathbf{L}}
\newcommand{\bM}{\mathbf{M}}
\newcommand{\bN}{\mathbf{N}}
\newcommand{\bO}{\mathbf{O}}
\newcommand{\bP}{\mathbf{P}}
\newcommand{\bQ}{\mathbf{Q}}
\newcommand{\bR}{\mathbf{R}}
\newcommand{\bS}{\mathbf{S}}
\newcommand{\bT}{\mathbf{T}}
\newcommand{\bU}{\mathbf{U}}
\newcommand{\bV}{\mathbf{V}}
\newcommand{\bW}{\mathbf{W}}
\newcommand{\bX}{\mathbf{X}}
\newcommand{\bY}{\mathbf{Y}}
\newcommand{\bZ}{\mathbf{Z}}

\newcommand{\ba}{\mathbf{a}}
\newcommand{\bb}{\mathbf{b}}
\newcommand{\bc}{\mathbf{c}}
\newcommand{\bd}{\mathbf{d}}
\newcommand{\be}{\mathbf{e}}
% \newcommand{\bf}{\mathbf{f}}
\newcommand{\bg}{\mathbf{g}}
\newcommand{\bh}{\mathbf{h}}
\newcommand{\bi}{\mathbf{i}}
\newcommand{\bj}{\mathbf{j}}
\newcommand{\bk}{\mathbf{k}}
\newcommand{\bl}{\mathbf{l}}
\newcommand{\bm}{\mathbf{m}}
\newcommand{\bn}{\mathbf{n}}
\newcommand{\bo}{\mathbf{o}}
\newcommand{\bp}{\mathbf{p}}
\newcommand{\bq}{\mathbf{q}}
\newcommand{\br}{\mathbf{r}}
\newcommand{\bs}{\mathbf{s}}
\newcommand{\bt}{\mathbf{t}}
\newcommand{\bu}{\mathbf{u}}
\newcommand{\bv}{\mathbf{v}}
\newcommand{\bw}{\mathbf{w}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\by}{\mathbf{y}}
\newcommand{\bz}{\mathbf{z}}


\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}

\newcommand{\ca}{\mathcal{a}}
\newcommand{\cb}{\mathcal{b}}
\newcommand{\cc}{\mathcal{c}}
\newcommand{\cd}{\mathcal{d}}
\newcommand{\ce}{\mathcal{e}}
\newcommand{\cf}{\mathcal{f}}
\newcommand{\cg}{\mathcal{g}}
\newcommand{\ch}{\mathcal{h}}
\newcommand{\ci}{\mathcal{i}}
\newcommand{\cj}{\mathcal{j}}
\newcommand{\ck}{\mathcal{k}}
\newcommand{\cl}{\mathcal{l}}
\newcommand{\cm}{\mathcal{m}}
\newcommand{\cn}{\mathcal{n}}
\newcommand{\co}{\mathcal{o}}
\newcommand{\cp}{\mathcal{p}}
\newcommand{\cq}{\mathcal{q}}
% \newcommand{\cr}{\mathcal{r}}
\newcommand{\cs}{\mathcal{s}}
\newcommand{\ct}{\mathcal{t}}
\newcommand{\cu}{\mathcal{u}}
\newcommand{\cv}{\mathcal{v}}
\newcommand{\cw}{\mathcal{w}}
\newcommand{\cx}{\mathcal{x}}
\newcommand{\cy}{\mathcal{y}}
\newcommand{\cz}{\mathcal{z}}


\newcommand{\sA}{\mathbb{A}}
\newcommand{\sB}{\mathbb{B}}
\newcommand{\sC}{\mathbb{C}}
\newcommand{\sD}{\mathbb{D}}
\newcommand{\sE}{\mathbb{E}}
\newcommand{\sF}{\mathbb{F}}
\newcommand{\sG}{\mathbb{G}}
\newcommand{\sH}{\mathbb{H}}
\newcommand{\sI}{\mathbb{I}}
\newcommand{\sJ}{\mathbb{J}}
\newcommand{\sK}{\mathbb{K}}
\newcommand{\sL}{\mathbb{L}}
\newcommand{\sM}{\mathbb{M}}
\newcommand{\sN}{\mathbb{N}}
\newcommand{\sO}{\mathbb{O}}
\newcommand{\sP}{\mathbb{P}}
\newcommand{\sQ}{\mathbb{Q}}
\newcommand{\sR}{\mathbb{R}}
\newcommand{\sS}{\mathbb{S}}
\newcommand{\sT}{\mathbb{T}}
\newcommand{\sU}{\mathbb{U}}
\newcommand{\sV}{\mathbb{V}}
\newcommand{\sW}{\mathbb{W}}
\newcommand{\sX}{\mathbb{X}}
\newcommand{\sY}{\mathbb{Y}}
\newcommand{\sZ}{\mathbb{Z}}



\newcommand{\density}{\gamma}
%\newcommand{\W}{ {\cal W} }
\newcommand{\wperp}{(\vec w^\ast)^{\perp_\vec w}}
\newcommand{\lp}{\left}
\newcommand{\rp}{\right}
\newcommand\norm[1]{\left\| #1 \right\|}
\newcommand\snorm[2]{\left\| #2 \right\|_{#1}}
%\renewcommand\vec[1]{\mathbf{#1}}
\DeclareMathOperator*{\pr}{\mathbf{Pr}}
\DeclareMathOperator*{\E}{\mathbf{E}}
\newcommand{\proj}{\mathrm{proj}}
\newcommand{\me}{\mathrm{e}}
\newcommand{\mrm}{\mathrm}
\def\d{\mathrm{d}}
\newcommand{\normal}{\mathcal{N}}

\DeclareMathOperator*{\argmin}{argmin}
\DeclareMathOperator*{\argmax}{argmax}
\DeclareMathOperator*{\influ}{Inf}
\DeclareMathOperator*{\he}{He}


\newcommand{\sample}[2]{#1^{(#2)}}
\newcommand{\tr}{\mathrm{tr}}
\newcommand{\e}{\mathbf{e}}
\newcommand{\Sp}{\mathbb{S}}
%\newcommand{\bz}{\mathbf{z}}
\newcommand{\sym}{\mathrm{sym}}


\newcommand{\err}{\mathrm{err}}
\newcommand{\LTF}{\mathcal{H}}
\newcommand{\B}{\mathbb{B}} 
\renewcommand{\H}{\mathcal H }
\newcommand{\X}{\mathcal X }



\newcommand{\p}{\sP}
\newcommand{\q}{\sQ}
\newcommand{\h}{\sH}
\newcommand{\R}{\sR}
\newcommand{\s}{\sS}
\newcommand{\Z}{\sZ}
\newcommand{\N}{\sN}
\newcommand{\C}{\sC}

\newcommand{\eps}{\epsilon}
\newcommand{\dtv}{d_{\mathrm TV}}
\newcommand{\poly}{\mathrm{poly}}
\newcommand{\dom}{\mathrm{dom}}
\newcommand{\polylog}{\mathrm{polylog}}
\newcommand{\var}{\mathbf{Var}}
\newcommand{\cov}{\mathbf{Cov}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\nul}{\mathrm{null}}



\newcommand{\sgn}{\mathrm{sign}}
\newcommand{\sign}{\mathrm{sign}}
\newcommand{\calN}{{\cal N}}
\newcommand{\calL}{{\cal L}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\Ex}{\mathop{{\bf E}\/}}
\newcommand{\OPT}{\mathrm{opt}}
%\newcommand{\D}{D}
\newcommand{\LR}{\mathrm{LeakyRelu}}
\newcommand{\Ind}{\mathds{1}}
\newcommand{\1}{\Ind}
\newcommand{\matr[1]}{\boldsymbol{#1}}
\newcommand{\littleint}{\mathop{\textstyle \int}}
\newcommand{\littlesum}{\mathop{\textstyle \sum}}
\newcommand{\littleprod}{\mathop{\textstyle \prod}}
\newcommand{\wt}{\widetilde}
\newcommand{\wh}{\widehat}


\newcommand{\ltwo}[1]{\left\lVert#1\right\rVert_2}
\newcommand{\dotp}[2]{ #1 \cdot #2 }
\newcommand{\miscl}{\err_{0-1}^{\D}}
\newcommand{\wstar}{\bw^{\ast}}
\newcommand{\x}{\mathbf{x}}
\newcommand{\w}{\vec w}
\DeclareMathOperator\erf{erf}
\newcommand{\normald}[1]{\mathcal{N}_{#1}}

\newcommand{\ith}{^{(i)}}
\newcommand{\tth}{^{(t)}}
\newcommand{\Exx}{\E_{\x\sim \D_\x}}
\newcommand{\Exn}{\E_{\x\sim \normal}}
\newcommand{\Ey}{\E_{(\x,y)\sim \D}}
\newcommand{\vpar}[2]{\vec #1^{\|_{\vec #2}}}
\newcommand{\vperp}[2]{\vec #1^{\perp{\vec #2}}}
\newcommand{\bounda}{\nu}
\newcommand{\boundb}{\zeta}
\newcommand{\boundc}{\rho}
\newcommand{\boundd}{\subvector}
\newcommand{\booundd}{\subvector}
\newcommand{\esp}{\eps}
\newcommand{\lamdba}{\lambda}

\newcommand{\mderiv}{D_{\rho}}

\newcommand{\OU}{Ornstein–Uhlenbeck\xspace}
\newcommand{\lip}{Lipschitz\xspace}
\newcommand{\subvector}{h}
\DeclareMathOperator{\floor}{floor}
\newcommand{\opt}{\mathrm{opt}}
\newcommand{\Cagn}{C_{\mathrm{A}}}
\newcommand{\iid}{{i.i.d.}\ }
\newcommand{\abs}[1]{\lp| #1 \rp|}
\newcommand{\A}{\mathcal{A}}
\newcommand{\spa}{\mathrm{span}}

%\newcommand{\inote}[1]{\color{magenta} I: #1 \color{black}}
\newcommand{\mnote}[1]{\color{magenta} M: #1 \color{black}}
\newcommand{\cnote}[1]{\color{magenta} C: #1 \color{black}}
\newcommand{\vnote}[1]{\color{magenta} V: #1 \color{black}}
\newcommand{\rnote}[1]{\color{magenta} R: #1 \color{black}}
\newcommand{\jnote}[1]{\color{magenta} J: #1 \color{black}}
\newcommand{\relu}{\mathrm{Relu}}
\newcommand{\card}[1]{|#1|}

\DeclareMathOperator{\Var}{Var}
\renewcommand\Pr{\pr}

\newcommand{\CS}{Cauchy-Schwarz\xspace}
\newcommand{\holder}{H\"older}

\newcommand{\softmax}{\text{softmax}}
\newcommand{\Attn}{\mathrm{Attn}}
\newcommand{\SSM}{\mathrm{SSM}}
\newcommand{\MB}{\mathrm{Mamba}}
\renewcommand{\^}[1]{^{(#1)}}
\newcommand{\One}{\mathbbm{1}}

\newcommand{\AT}{\mathrm{AT}}
\newcommand{\TF}{\mathrm{TF}}
\newcommand{\MLP}{\mathrm{MLP}}
%\newcommand{\SSM}{\mathrm{SSM}}

\newcommand{\W}{\mathbf{W}}

\renewcommand{\poly}{\text{poly}}