%% mlmath.sty %% Copyright 2020 by Zheng Ma, Zhiqin Xu, Tao Luo and Yaoyu Zhang % % This work may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either version 1.3c % of this license or (at your option) any later version. % The latest version of this license is in % http://www.latex-project.org/lppl.txt % and version 1.3c or later is part of all distributions of LaTeX % version 2005/12/01 or later. % % This work has the LPPL maintenance status `maintained'. % % The Current Maintainer of this work is Zheng Ma. % % This work consists of the main source files mlmath.sty, mlmath.tex, mlmath.md, % and the derived documentation mlmath.pdf, README.md as well as the resource % files (they are svg file) of README.md under the folder svgs/. \ProvidesPackage{mlmath}[2020/06/15 v1.0.0 zyboys] \RequirePackage{amsmath} \RequirePackage{amsthm} \RequirePackage{amssymb} \RequirePackage{bm} \RequirePackage{bbm} \newcommand{\mat}[1]{\bm{#1}} \renewcommand{\vec}[1]{\bm{#1}} \newcommand{\D}{\mathrm{d}} \newcommand{\I}{\mathrm{i}} \newcommand{\E}{\mathrm{e}} \newcommand{\eps}{\varepsilon} \newcommand{\sgn}{\mathrm{sgn}} \newcommand{\cc}{\mathrm{c.c.}} \newcommand{\T}{\intercal} \newcommand{\Exp}{\mathbb{E}} \newcommand{\Prob}{\mathbb{P}} \newcommand*\diff{\mathop{}\!\D} \newcommand*\Diff[1]{\mathop{}\!\mathrm{d^#1}} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\Abs}[1]{\left\lvert#1\right\rvert} \newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\Norm}[1]{\left\lVert#1\right\rVert} \newcommand{\ceil}[1]{\lceil#1\rceil} \newcommand{\floor}[1]{\lfloor#1\rfloor} \newcommand{\lsim}{\lesssim} \newcommand{\gsim}{\gtrsim} % Random variables \newcommand{\ralpha}{\textnormal{\alpha}} \newcommand{\rbeta}{\textnormal{\beta}} \newcommand{\rgamma}{\textnormal{\gamma}} \newcommand{\rdelta}{\textnormal{\delta}} \newcommand{\repsilon}{\textnormal{\epsilon}} \newcommand{\reps}{\textnormal{\varepsilon}} \newcommand{\rzeta}{\textnormal{\zeta}} \newcommand{\reta}{\textnormal{\eta}} \newcommand{\rtheta}{\textnormal{\theta}} \newcommand{\rvartheta}{\textnormal{\vartheta}} \newcommand{\riota}{\textnormal{\iota}} \newcommand{\rkappa}{\textnormal{\kappa}} \newcommand{\rlambda}{\textnormal{\lambda}} \newcommand{\rmu}{\textnormal{\mu}} \newcommand{\rnu}{\textnormal{\nu}} \newcommand{\rxi}{\textnormal{\xi}} \newcommand{\rpi}{\textnormal{\pi}} \newcommand{\rrho}{\textnormal{\rho}} \newcommand{\rsigma}{\textnormal{\sigma}} \newcommand{\rtau}{\textnormal{\tau}} \newcommand{\rupsilon}{\textnormal{\upsilon}} \newcommand{\rphi}{\textnormal{\phi}} \newcommand{\rvarphi}{\textnormal{\varphi}} \newcommand{\rchi}{\textnormal{\chi}} \newcommand{\rpsi}{\textnormal{\psi}} \newcommand{\romega}{\textnormal{\omega}} \newcommand{\rGamma}{\textnormal{\Gamma}} \newcommand{\rDelta}{\textnormal{\Delta}} \newcommand{\rTheta}{\textnormal{\Theta}} \newcommand{\rLambda}{\textnormal{\Lambda}} \newcommand{\rXi}{\textnormal{\Xi}} \newcommand{\rPi}{\textnormal{\Pi}} \newcommand{\rSigma}{\textnormal{\Sigma}} \newcommand{\rUpsilon}{\textnormal{\Upsilon}} \newcommand{\rPhi}{\textnormal{\Phi}} \newcommand{\rPsi}{\textnormal{\Psi}} \newcommand{\rOmega}{\textnormal{\Omega}} \newcommand{\ra}{\textnormal{a}} \newcommand{\rb}{\textnormal{b}} \newcommand{\rc}{\textnormal{c}} \newcommand{\rd}{\textnormal{d}} \newcommand{\re}{\textnormal{e}} \newcommand{\rf}{\textnormal{f}} \newcommand{\rg}{\textnormal{g}} \newcommand{\rh}{\textnormal{h}} \newcommand{\ri}{\textnormal{i}} \newcommand{\rj}{\textnormal{j}} \newcommand{\rk}{\textnormal{k}} \newcommand{\rl}{\textnormal{l}} % rm is already a command, just don't name any random variables m \newcommand{\rn}{\textnormal{n}} \newcommand{\ro}{\textnormal{o}} \newcommand{\rp}{\textnormal{p}} \newcommand{\rqq}{\textnormal{q}} \newcommand{\rr}{\textnormal{r}} \newcommand{\rs}{\textnormal{s}} \newcommand{\rt}{\textnormal{t}} \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{\rB}{\textnormal{B}} \newcommand{\rC}{\textnormal{C}} \newcommand{\rD}{\textnormal{D}} \newcommand{\rE}{\textnormal{E}} \newcommand{\rF}{\textnormal{F}} \newcommand{\rG}{\textnormal{G}} \newcommand{\rH}{\textnormal{H}} \newcommand{\rI}{\textnormal{I}} \newcommand{\rJ}{\textnormal{J}} \newcommand{\rK}{\textnormal{K}} \newcommand{\rL}{\textnormal{L}} \newcommand{\rM}{\textnormal{M}} \newcommand{\rN}{\textnormal{N}} \newcommand{\rO}{\textnormal{O}} \newcommand{\rP}{\textnormal{P}} \newcommand{\rQ}{\textnormal{Q}} \newcommand{\rR}{\textnormal{R}} \newcommand{\rS}{\textnormal{S}} \newcommand{\rT}{\textnormal{T}} \newcommand{\rU}{\textnormal{U}} \newcommand{\rV}{\textnormal{V}} \newcommand{\rW}{\textnormal{W}} \newcommand{\rX}{\textnormal{X}} \newcommand{\rY}{\textnormal{Y}} \newcommand{\rZ}{\textnormal{Z}} % Random vectors \newcommand{\rvalpha}{\mathbf{\alpha}} \newcommand{\rvbeta}{\mathbf{\beta}} \newcommand{\rvgamma}{\mathbf{\gamma}} \newcommand{\rvdelta}{\mathbf{\delta}} \newcommand{\rvepsilon}{\mathbf{\epsilon}} \newcommand{\rveps}{\mathbf{\varepsilon}} \newcommand{\rvzeta}{\mathbf{\zeta}} \newcommand{\rveta}{\mathbf{\eta}} \newcommand{\rvtheta}{\mathbf{\theta}} \newcommand{\rvvartheta}{\mathbf{\vartheta}} \newcommand{\rviota}{\mathbf{\iota}} \newcommand{\rvkappa}{\mathbf{\kappa}} \newcommand{\rvlambda}{\mathbf{\lambda}} \newcommand{\rvmu}{\mathbf{\mu}} \newcommand{\rvnu}{\mathbf{\nu}} \newcommand{\rvxi}{\mathbf{\xi}} \newcommand{\rvpi}{\mathbf{\pi}} \newcommand{\rvrho}{\mathbf{\rho}} \newcommand{\rvsigma}{\mathbf{\sigma}} \newcommand{\rvtau}{\mathbf{\tau}} \newcommand{\rvupsilon}{\mathbf{\upsilon}} \newcommand{\rvphi}{\mathbf{\phi}} \newcommand{\rvvarphi}{\mathbf{\varphi}} \newcommand{\rvchi}{\mathbf{\chi}} \newcommand{\rvpsi}{\mathbf{\psi}} \newcommand{\rvomega}{\mathbf{\omega}} \newcommand{\rva}{\mathbf{a}} \newcommand{\rvb}{\mathbf{b}} \newcommand{\rvc}{\mathbf{c}} \newcommand{\rvd}{\mathbf{d}} \newcommand{\rve}{\mathbf{e}} \newcommand{\rvf}{\mathbf{f}} \newcommand{\rvg}{\mathbf{g}} \newcommand{\rvh}{\mathbf{h}} \newcommand{\rvi}{\mathbf{i}} \newcommand{\rvj}{\mathbf{j}} \newcommand{\rvk}{\mathbf{k}} \newcommand{\rvl}{\mathbf{l}} \newcommand{\rvm}{\mathbf{m}} \newcommand{\rvo}{\mathbf{o}} \newcommand{\rvp}{\mathbf{p}} \newcommand{\rvq}{\mathbf{q}} \newcommand{\rvr}{\mathbf{r}} \newcommand{\rvs}{\mathbf{s}} \newcommand{\rvt}{\mathbf{t}} \newcommand{\rvu}{\mathbf{u}} \newcommand{\rvv}{\mathbf{v}} \newcommand{\rvw}{\mathbf{w}} \newcommand{\rvx}{\mathbf{x}} \newcommand{\rvy}{\mathbf{y}} \newcommand{\rvz}{\mathbf{z}} % Random matrices \newcommand{\rmGamma}{\mathbf{\Gamma}} \newcommand{\rmDelta}{\mathbf{\Delta}} \newcommand{\rmTheta}{\mathbf{\Theta}} \newcommand{\rmLambda}{\mathbf{\Lambda}} \newcommand{\rmXi}{\mathbf{\Xi}} \newcommand{\rmPi}{\mathbf{\Pi}} \newcommand{\rmSigma}{\mathbf{\Sigma}} \newcommand{\rmUpsilon}{\mathbf{\Upsilon}} \newcommand{\rmPhi}{\mathbf{\Phi}} \newcommand{\rmPsi}{\mathbf{\Psi}} \newcommand{\rmOmega}{\mathbf{\Omega}} \newcommand{\rmA}{\mathbf{A}} \newcommand{\rmB}{\mathbf{B}} \newcommand{\rmC}{\mathbf{C}} \newcommand{\rmD}{\mathbf{D}} \newcommand{\rmE}{\mathbf{E}} \newcommand{\rmF}{\mathbf{F}} \newcommand{\rmG}{\mathbf{G}} \newcommand{\rmH}{\mathbf{H}} \newcommand{\rmI}{\mathbf{I}} \newcommand{\rmJ}{\mathbf{J}} \newcommand{\rmK}{\mathbf{K}} \newcommand{\rmL}{\mathbf{L}} \newcommand{\rmM}{\mathbf{M}} \newcommand{\rmN}{\mathbf{N}} \newcommand{\rmO}{\mathbf{O}} \newcommand{\rmP}{\mathbf{P}} \newcommand{\rmQ}{\mathbf{Q}} \newcommand{\rmR}{\mathbf{R}} \newcommand{\rmS}{\mathbf{S}} \newcommand{\rmT}{\mathbf{T}} \newcommand{\rmU}{\mathbf{U}} \newcommand{\rmV}{\mathbf{V}} \newcommand{\rmW}{\mathbf{W}} \newcommand{\rmX}{\mathbf{X}} \newcommand{\rmY}{\mathbf{Y}} \newcommand{\rmZ}{\mathbf{Z}} % Vectors \newcommand{\vzero}{\bm{0}} \newcommand{\vone}{\bm{1}} \newcommand{\valpha}{\bm{\alpha}} \newcommand{\vbeta}{\bm{\beta}} \newcommand{\vgamma}{\bm{\gamma}} \newcommand{\vdelta}{\bm{\delta}} \newcommand{\vepsilon}{\bm{\epsilon}} \newcommand{\veps}{\bm{\varepsilon}} \newcommand{\vzeta}{\bm{\zeta}} \newcommand{\veta}{\bm{\eta}} \newcommand{\vtheta}{\bm{\theta}} \newcommand{\vvartheta}{\bm{\vartheta}} \newcommand{\viota}{\bm{\iota}} \newcommand{\vkappa}{\bm{\kappa}} \newcommand{\vlambda}{\bm{\lambda}} \newcommand{\vmu}{\bm{\mu}} \newcommand{\vnu}{\bm{\nu}} \newcommand{\vxi}{\bm{\xi}} \newcommand{\vpi}{\bm{\pi}} \newcommand{\vrho}{\bm{\rho}} \newcommand{\vsigma}{\bm{\sigma}} \newcommand{\vtau}{\bm{\tau}} \newcommand{\vupsilon}{\bm{\upsilon}} \newcommand{\vphi}{\bm{\phi}} \newcommand{\vvarphi}{\bm{\varphi}} \newcommand{\vchi}{\bm{\chi}} \newcommand{\vpsi}{\bm{\psi}} \newcommand{\vomega}{\bm{\omega}} \newcommand{\vGamma}{\bm{\Gamma}} \newcommand{\vDelta}{\bm{\Delta}} \newcommand{\vTheta}{\bm{\Theta}} \newcommand{\vLambda}{\bm{\Lambda}} \newcommand{\vXi}{\bm{\Xi}} \newcommand{\vPi}{\bm{\Pi}} \newcommand{\vSigma}{\bm{\Sigma}} \newcommand{\vUpsilon}{\bm{\Upsilon}} \newcommand{\vPhi}{\bm{\Phi}} \newcommand{\vPsi}{\bm{\Psi}} \newcommand{\vOmega}{\bm{\Omega}} \newcommand{\va}{\bm{a}} \newcommand{\vb}{\bm{b}} \newcommand{\vc}{\bm{c}} \newcommand{\vd}{\bm{d}} \newcommand{\ve}{\bm{e}} \newcommand{\vf}{\bm{f}} \newcommand{\vg}{\bm{g}} \newcommand{\vh}{\bm{h}} \newcommand{\vi}{\bm{i}} \newcommand{\vj}{\bm{j}} \newcommand{\vk}{\bm{k}} \newcommand{\vl}{\bm{l}} \newcommand{\vm}{\bm{m}} \newcommand{\vn}{\bm{n}} \newcommand{\vo}{\bm{o}} \newcommand{\vp}{\bm{p}} \newcommand{\vq}{\bm{q}} \newcommand{\vr}{\bm{r}} \newcommand{\vs}{\bm{s}} \newcommand{\vt}{\bm{t}} \newcommand{\vu}{\bm{u}} \newcommand{\vv}{\bm{v}} \newcommand{\vw}{\bm{w}} \newcommand{\vx}{\bm{x}} \newcommand{\vy}{\bm{y}} \newcommand{\vz}{\bm{z}} \newcommand{\vA}{\bm{A}} \newcommand{\vB}{\bm{B}} \newcommand{\vC}{\bm{C}} \newcommand{\vD}{\bm{D}} \newcommand{\vE}{\bm{E}} \newcommand{\vF}{\bm{F}} \newcommand{\vG}{\bm{G}} \newcommand{\vH}{\bm{H}} \newcommand{\vI}{\bm{I}} \newcommand{\vJ}{\bm{J}} \newcommand{\vK}{\bm{K}} \newcommand{\vL}{\bm{L}} \newcommand{\vM}{\bm{M}} \newcommand{\vN}{\bm{N}} \newcommand{\vO}{\bm{O}} \newcommand{\vP}{\bm{P}} \newcommand{\vQ}{\bm{Q}} \newcommand{\vR}{\bm{R}} \newcommand{\vS}{\bm{S}} \newcommand{\vT}{\bm{T}} \newcommand{\vU}{\bm{U}} \newcommand{\vV}{\bm{V}} \newcommand{\vW}{\bm{W}} \newcommand{\vX}{\bm{X}} \newcommand{\vY}{\bm{Y}} \newcommand{\vZ}{\bm{Z}} % Matrices \newcommand{\mzero}{\bm{0}} \newcommand{\malpha}{\bm{\alpha}} \newcommand{\mbeta}{\bm{\beta}} \newcommand{\mgamma}{\bm{\gamma}} \newcommand{\mdelta}{\bm{\delta}} \newcommand{\mepsilon}{\bm{\epsilon}} \newcommand{\meps}{\bm{\varepsilon}} \newcommand{\mzeta}{\bm{\zeta}} \newcommand{\meta}{\bm{\eta}} \newcommand{\mtheta}{\bm{\theta}} \newcommand{\mvartheta}{\bm{\vartheta}} \newcommand{\miota}{\bm{\iota}} \newcommand{\mkappa}{\bm{\kappa}} \newcommand{\mlambda}{\bm{\lambda}} \newcommand{\mmu}{\bm{\mu}} \newcommand{\mnu}{\bm{\nu}} \newcommand{\mxi}{\bm{\xi}} \newcommand{\mpi}{\bm{\pi}} \newcommand{\mrho}{\bm{\rho}} \newcommand{\msigma}{\bm{\sigma}} \newcommand{\mtau}{\bm{\tau}} \newcommand{\mupsilon}{\bm{\upsilon}} \newcommand{\mphi}{\bm{\phi}} \newcommand{\mvarphi}{\bm{\varphi}} \newcommand{\mchi}{\bm{\chi}} \newcommand{\mpsi}{\bm{\psi}} \newcommand{\momega}{\bm{\omega}} \newcommand{\mGamma}{\bm{\Gamma}} \newcommand{\mDelta}{\bm{\Delta}} \newcommand{\mTheta}{\bm{\Theta}} \newcommand{\mLambda}{\bm{\Lambda}} \newcommand{\mXi}{\bm{\Xi}} \newcommand{\mPi}{\bm{\Pi}} \newcommand{\mSigma}{\bm{\Sigma}} \newcommand{\mUpsilon}{\bm{\Upsilon}} \newcommand{\mPhi}{\bm{\Phi}} \newcommand{\mPsi}{\bm{\Psi}} \newcommand{\mOmega}{\bm{\Omega}} \newcommand{\mA}{\bm{A}} \newcommand{\mB}{\bm{B}} \newcommand{\mC}{\bm{C}} \newcommand{\mD}{\bm{D}} \newcommand{\mE}{\bm{E}} \newcommand{\mF}{\bm{F}} \newcommand{\mG}{\bm{G}} \newcommand{\mH}{\bm{H}} \newcommand{\mI}{\bm{I}} \newcommand{\mJ}{\bm{J}} \newcommand{\mK}{\bm{K}} \newcommand{\mL}{\bm{L}} \newcommand{\mM}{\bm{M}} \newcommand{\mN}{\bm{N}} \newcommand{\mO}{\bm{O}} \newcommand{\mP}{\bm{P}} \newcommand{\mQ}{\bm{Q}} \newcommand{\mR}{\bm{R}} \newcommand{\mS}{\bm{S}} \newcommand{\mT}{\bm{T}} \newcommand{\mU}{\bm{U}} \newcommand{\mV}{\bm{V}} \newcommand{\mW}{\bm{W}} \newcommand{\mX}{\bm{X}} \newcommand{\mY}{\bm{Y}} \newcommand{\mZ}{\bm{Z}} % Tensor \DeclareMathAlphabet{\mathsfit}{\encodingdefault}{\sfdefault}{m}{sl} \SetMathAlphabet{\mathsfit}{bold}{\encodingdefault}{\sfdefault}{bx}{n} \newcommand{\tens}[1]{\bm{\mathsfit{#1}}} \newcommand{\talpha}{\tens{\alpha}} \newcommand{\tbeta}{\tens{\beta}} \newcommand{\tgamma}{\tens{\gamma}} \newcommand{\tdelta}{\tens{\delta}} \newcommand{\tepsilon}{\tens{\epsilon}} \newcommand{\teps}{\tens{\varepsilon}} \newcommand{\tzeta}{\tens{\zeta}} \newcommand{\teta}{\tens{\eta}} \newcommand{\ttheta}{\tens{\theta}} \newcommand{\tvartheta}{\tens{\vartheta}} \newcommand{\tiota}{\tens{\iota}} \newcommand{\tkappa}{\tens{\kappa}} \newcommand{\tlambda}{\tens{\lambda}} \newcommand{\tmu}{\tens{\mu}} \newcommand{\tnu}{\tens{\nu}} \newcommand{\txi}{\tens{\xi}} \newcommand{\tpi}{\tens{\pi}} \newcommand{\trho}{\tens{\rho}} \newcommand{\tsigma}{\tens{\sigma}} \newcommand{\ttau}{\tens{\tau}} \newcommand{\tupsilon}{\tens{\upsilon}} \newcommand{\tphi}{\tens{\phi}} \newcommand{\tvarphi}{\tens{\varphi}} \newcommand{\tchi}{\tens{\chi}} \newcommand{\tpsi}{\tens{\psi}} \newcommand{\tomega}{\tens{\omega}} \newcommand{\tGamma}{\tens{\Gamma}} \newcommand{\tDelta}{\tens{\Delta}} \newcommand{\tTheta}{\tens{\Theta}} \newcommand{\tLambda}{\tens{\Lambda}} \newcommand{\tXi}{\tens{\Xi}} \newcommand{\tPi}{\tens{\Pi}} \newcommand{\tSigma}{\tens{\Sigma}} \newcommand{\tUpsilon}{\tens{\Upsilon}} \newcommand{\tPhi}{\tens{\Phi}} \newcommand{\tPsi}{\tens{\Psi}} \newcommand{\tOmega}{\tens{\Omega}} \newcommand{\tA}{\tens{A}} \newcommand{\tB}{\tens{B}} \newcommand{\tC}{\tens{C}} \newcommand{\tD}{\tens{D}} \newcommand{\tE}{\tens{E}} \newcommand{\tF}{\tens{F}} \newcommand{\tG}{\tens{G}} \newcommand{\tH}{\tens{H}} \newcommand{\tI}{\tens{I}} \newcommand{\tJ}{\tens{J}} \newcommand{\tK}{\tens{K}} \newcommand{\tL}{\tens{L}} \newcommand{\tM}{\tens{M}} \newcommand{\tN}{\tens{N}} \newcommand{\tO}{\tens{O}} \newcommand{\tP}{\tens{P}} \newcommand{\tQ}{\tens{Q}} \newcommand{\tR}{\tens{R}} \newcommand{\tS}{\tens{S}} \newcommand{\tT}{\tens{T}} \newcommand{\tU}{\tens{U}} \newcommand{\tV}{\tens{V}} \newcommand{\tW}{\tens{W}} \newcommand{\tX}{\tens{X}} \newcommand{\tY}{\tens{Y}} \newcommand{\tZ}{\tens{Z}} % entries of a tensor % Same font as tensor, without \bm wrapper \newcommand{\etens}[1]{\mathsfit{#1}} \newcommand{\etalpha}{\etens{\alpha}} \newcommand{\etbeta}{\etens{\beta}} \newcommand{\etgamma}{\etens{\gamma}} \newcommand{\etdelta}{\etens{\delta}} \newcommand{\etepsilon}{\etens{\epsilon}} \newcommand{\eteps}{\etens{\varepsilon}} \newcommand{\etzeta}{\etens{\zeta}} \newcommand{\eteta}{\etens{\eta}} \newcommand{\ettheta}{\etens{\theta}} \newcommand{\etvartheta}{\etens{\vartheta}} \newcommand{\etiota}{\etens{\iota}} \newcommand{\etkappa}{\etens{\kappa}} \newcommand{\etlambda}{\etens{\lambda}} \newcommand{\etmu}{\etens{\mu}} \newcommand{\etnu}{\etens{\nu}} \newcommand{\etxi}{\etens{\xi}} \newcommand{\etpi}{\etens{\pi}} \newcommand{\etrho}{\etens{\rho}} \newcommand{\etsigma}{\etens{\sigma}} \newcommand{\ettau}{\etens{\tau}} \newcommand{\etupsilon}{\etens{\upsilon}} \newcommand{\etphi}{\etens{\phi}} \newcommand{\etvarphi}{\etens{\varphi}} \newcommand{\etchi}{\etens{\chi}} \newcommand{\etpsi}{\etens{\psi}} \newcommand{\etomega}{\etens{\omega}} \newcommand{\etGamma}{\etens{\Gamma}} \newcommand{\etDelta}{\etens{\Delta}} \newcommand{\etTheta}{\etens{\Theta}} \newcommand{\etLambda}{\etens{\Lambda}} \newcommand{\etXi}{\etens{\Xi}} \newcommand{\etPi}{\etens{\Pi}} \newcommand{\etSigma}{\etens{\Sigma}} \newcommand{\etUpsilon}{\etens{\Upsilon}} \newcommand{\etPhi}{\etens{\Phi}} \newcommand{\etPsi}{\etens{\Psi}} \newcommand{\etOmega}{\etens{\Omega}} \newcommand{\etA}{\etens{A}} \newcommand{\etB}{\etens{B}} \newcommand{\etC}{\etens{C}} \newcommand{\etD}{\etens{D}} \newcommand{\etE}{\etens{E}} \newcommand{\etF}{\etens{F}} \newcommand{\etG}{\etens{G}} \newcommand{\etH}{\etens{H}} \newcommand{\etI}{\etens{I}} \newcommand{\etJ}{\etens{J}} \newcommand{\etK}{\etens{K}} \newcommand{\etL}{\etens{L}} \newcommand{\etM}{\etens{M}} \newcommand{\etN}{\etens{N}} \newcommand{\etO}{\etens{O}} \newcommand{\etP}{\etens{P}} \newcommand{\etQ}{\etens{Q}} \newcommand{\etR}{\etens{R}} \newcommand{\etS}{\etens{S}} \newcommand{\etT}{\etens{T}} \newcommand{\etU}{\etens{U}} \newcommand{\etV}{\etens{V}} \newcommand{\etW}{\etens{W}} \newcommand{\etX}{\etens{X}} \newcommand{\etY}{\etens{Y}} \newcommand{\etZ}{\etens{Z}} % Sets \newcommand{\sone}{\mathbbm{1}} \newcommand{\sA}{\mathbb{A}} \newcommand{\sB}{\mathbb{B}} \newcommand{\sC}{\mathbb{C}} \newcommand{\sD}{\mathbb{D}} % Don't use a set called E, because this would be the same as our symbol for expectation. \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}} % Don't use a set called P, because this would be the same as our symbol for probability. \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}} % Function class, spaces, operator, etc. \newcommand{\fA}{\mathcal{A}} \newcommand{\fB}{\mathcal{B}} \newcommand{\fC}{\mathcal{C}} \newcommand{\fD}{\mathcal{D}} \newcommand{\fE}{\mathcal{E}} \newcommand{\fF}{\mathcal{F}} \newcommand{\fG}{\mathcal{G}} \newcommand{\fH}{\mathcal{H}} \newcommand{\fI}{\mathcal{I}} \newcommand{\fJ}{\mathcal{J}} \newcommand{\fK}{\mathcal{K}} \newcommand{\fL}{\mathcal{L}} \newcommand{\fM}{\mathcal{M}} \newcommand{\fN}{\mathcal{N}} \newcommand{\fO}{\mathcal{O}} \newcommand{\fP}{\mathcal{P}} \newcommand{\fQ}{\mathcal{Q}} \newcommand{\fR}{\mathcal{R}} \newcommand{\fS}{\mathcal{S}} \newcommand{\fT}{\mathcal{T}} \newcommand{\fU}{\mathcal{U}} \newcommand{\fV}{\mathcal{V}} \newcommand{\fW}{\mathcal{W}} \newcommand{\fX}{\mathcal{X}} \newcommand{\fY}{\mathcal{Y}} \newcommand{\fZ}{\mathcal{Z}} %neural networks \newcommand{\LS}{L_S} \newcommand{\LD}{L_\fD} \newcommand{\RS}{R_S} \newcommand{\Rn}{R_n} \newcommand{\RD}{R_\fD} \newcommand{\diam}{\mathrm{diam}} \newcommand{\diag}{\mathrm{diag}} \newcommand{\dist}{\mathrm{dist}} \newcommand{\ERM}{\mathrm{ERM}} \newcommand{\Lip}{\mathrm{Lip}} \newcommand{\ReLU}{\mathrm{ReLU}} \newcommand{\Rep}{\mathrm{Rep}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\Rad}{\mathrm{Rad}} \newcommand{\DKL}{\mathrm{D}_{KL}} \newcommand{\conv}{\mathrm{conv}} \newcommand{\csch}{\mathrm{csch}} \newcommand{\sech}{\mathrm{sech}} \newcommand{\Sigmoid}{\mathrm{Sigmoid}} %define theorems \newtheorem{thm}{Theorem} \newtheorem{cor}{Corollary} \newtheorem{defi}{Definition} \newtheorem{lem}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{rmk}{Remark} \newtheorem{assump}{Assumption} \newtheorem{exam}{Example} \newtheorem{conj}{Conjecture} \newtheorem{algo}{Algorithm} % \renewcommand{\abstractname}{Summary}