\[
% Made for MathJax
%mathbf letters for matrices/vectors
\renewcommand{\vec}[1]{\mathbf{#1}}
\newcommand{\avec}{\vec{a}}
\newcommand{\bvec}{\vec{b}}
\newcommand{\cvec}{\vec{c}}
\newcommand{\dvec}{\vec{d}}
\newcommand{\evec}{\vec{e}}
\newcommand{\fvec}{\vec{f}}
\newcommand{\gvec}{\vec{g}}
\newcommand{\hvec}{\vec{h}}
\newcommand{\ivec}{\vec{i}}
\newcommand{\jvec}{\vec{j}}
\newcommand{\kvec}{\vec{k}}
\newcommand{\lvec}{\vec{l}}
\newcommand{\mvec}{\vec{m}}
\newcommand{\nvec}{\vec{n}}
\newcommand{\ovec}{\vec{o}}
\newcommand{\pvec}{\vec{p}}
\newcommand{\qvec}{\vec{q}}
\newcommand{\rvec}{\vec{r}}
\newcommand{\svec}{\vec{s}}
\newcommand{\tvec}{\vec{t}}
\newcommand{\uvec}{\vec{u}}
\newcommand{\vvec}{\vec{v}}
\newcommand{\wvec}{\vec{w}}
\newcommand{\xvec}{\vec{x}}
\newcommand{\yvec}{\vec{y}}
\newcommand{\zvec}{\vec{z}}
\newcommand{\abf}{\mathbf{a}}
\newcommand{\bbbf}{\mathbf{b}}
\newcommand{\cbf}{\mathbf{c}}
\newcommand{\dbf}{\mathbf{d}}
\newcommand{\ebf}{\mathbf{e}}
\newcommand{\fbf}{\mathbf{f}}
\newcommand{\gbf}{\mathbf{g}}
\newcommand{\hbf}{\mathbf{h}}
\newcommand{\ibf}{\mathbf{i}}
\newcommand{\jbf}{\mathbf{j}}
\newcommand{\kbf}{\mathbf{k}}
\newcommand{\lbf}{\mathbf{l}}
\newcommand{\mbf}{\mathbf{m}}
\newcommand{\nbf}{\mathbf{n}}
\newcommand{\obf}{\mathbf{o}}
\newcommand{\pbf}{\mathbf{p}}
\newcommand{\qbf}{\mathbf{q}}
\newcommand{\rbf}{\mathbf{r}}
\newcommand{\sbf}{\mathbf{s}}
\newcommand{\tbf}{\mathbf{t}}
\newcommand{\ubf}{\mathbf{u}}
\newcommand{\vbf}{\mathbf{v}}
\newcommand{\wbf}{\mathbf{w}}
\newcommand{\xbf}{\mathbf{x}}
\newcommand{\ybf}{\mathbf{y}}
\newcommand{\zbf}{\mathbf{z}}
\newcommand{\alphabf}{\boldsymbol{\alpha}}
\newcommand{\Betabf}{\boldsymbol{\Beta}}
\newcommand{\betabf}{\boldsymbol{\beta}}
\newcommand{\Gammabf}{\boldsymbol{\Gamma}}
\newcommand{\gammabf}{\boldsymbol{\gamma}}
\newcommand{\Deltabf}{\boldsymbol{\Delta}}
\newcommand{\deltabf}{\boldsymbol{\delta}}
\newcommand{\epsilontbf}{\boldsymbol{\epsilon}}
\newcommand{\varepsilontbf}{\boldsymbol{\varepsilon}}
\newcommand{\zetabf}{\boldsymbol{\zeta}}
\newcommand{\etabf}{\boldsymbol{\eta}}
\newcommand{\Thetabf}{\boldsymbol{\Theta}}
\newcommand{\thetabf}{\boldsymbol{\theta}}
\newcommand{\iotabf}{\boldsymbol{\iota}}
\newcommand{\kappabf}{\boldsymbol{\kappa}}
\newcommand{\Lambdabf}{\boldsymbol{\Lambda}}
\newcommand{\lambdabf}{\boldsymbol{\lambda}}
\newcommand{\mubf}{\boldsymbol{\mu}}
\newcommand{\nubf}{\boldsymbol{\nu}}
\newcommand{\Xibf}{\boldsymbol{\Xi}}
\newcommand{\xibf}{\boldsymbol{\xi}}
\newcommand{\Pibf}{\boldsymbol{\Pi}}
\newcommand{\pibf}{\boldsymbol{\pi}}
\newcommand{\rhobf}{\boldsymbol{\rho}}
\newcommand{\Sigmabf}{\boldsymbol{\Sigma}}
\newcommand{\sigmabf}{\boldsymbol{\sigma}}
\newcommand{\taubf}{\boldsymbol{\tau}}
\newcommand{\Upsilonbf}{\boldsymbol{\Upsilon}}
\newcommand{\upsilonbf}{\boldsymbol{\upsilon}}
\newcommand{\Phibf}{\boldsymbol{\Phi}}
\newcommand{\phibf}{\boldsymbol{\phi}}
\newcommand{\varphibf}{\boldsymbol{\varphi}}
\newcommand{\chibf}{\boldsymbol{\chi}}
\newcommand{\Psibf}{\boldsymbol{\Psi}}
\newcommand{\psibf}{\boldsymbol{\psi}}
\newcommand{\Omegabf}{\boldsymbol{\Omega}}
\newcommand{\omegabf}{\boldsymbol{\omega}}
\newcommand{\Abf}{\mathbf{A}}
\newcommand{\Bbf}{\mathbf{B}}
\newcommand{\Cbf}{\mathbf{C}}
\newcommand{\Dbf}{\mathbf{D}}
\newcommand{\Ebf}{\mathbf{E}}
\newcommand{\Fbf}{\mathbf{F}}
\newcommand{\Gbf}{\mathbf{G}}
\newcommand{\Hbf}{\mathbf{H}}
\newcommand{\Ibf}{\mathbf{I}}
\newcommand{\Jbf}{\mathbf{J}}
\newcommand{\Kbf}{\mathbf{K}}
\newcommand{\Lbf}{\mathbf{L}}
\newcommand{\Mbf}{\mathbf{M}}
\newcommand{\Nbf}{\mathbf{N}}
\newcommand{\Obf}{\mathbf{O}}
\newcommand{\Pbf}{\mathbf{P}}
\newcommand{\Qbf}{\mathbf{Q}}
\newcommand{\Rbf}{\mathbf{R}}
\newcommand{\Sbf}{\mathbf{S}}
\newcommand{\Tbf}{\mathbf{T}}
\newcommand{\Ubf}{\mathbf{U}}
\newcommand{\Vbf}{\mathbf{V}}
\newcommand{\Wbf}{\mathbf{W}}
\newcommand{\Xbf}{\mathbf{X}}
\newcommand{\Ybf}{\mathbf{Y}}
\newcommand{\Zbf}{\mathbf{Z}}
% mathsf letters
\newcommand{\asf}{\mathsf{a}}
\newcommand{\bsf}{\mathsf{b}}
\newcommand{\csf}{\mathsf{c}}
\newcommand{\dsf}{\mathsf{d}}
\newcommand{\esf}{\mathsf{e}}
\newcommand{\fsf}{\mathsf{f}}
\newcommand{\gsf}{\mathsf{g}}
\newcommand{\hsf}{\mathsf{h}}
\newcommand{\isf}{\mathsf{i}}
\newcommand{\jsf}{\mathsf{j}}
\newcommand{\ksf}{\mathsf{k}}
\newcommand{\lsf}{\mathsf{l}}
\newcommand{\msf}{\mathsf{m}}
\newcommand{\nsf}{\mathsf{n}}
\newcommand{\osf}{\mathsf{o}}
\newcommand{\psf}{\mathsf{p}}
\newcommand{\qsf}{\mathsf{q}}
\newcommand{\rsf}{\mathsf{r}}
\newcommand{\ssf}{\mathsf{s}}
\newcommand{\tsf}{\mathsf{t}}
\newcommand{\usf}{\mathsf{u}}
\newcommand{\vsf}{\mathsf{v}}
\newcommand{\wsf}{\mathsf{w}}
\newcommand{\xsf}{\mathsf{x}}
\newcommand{\ysf}{\mathsf{y}}
\newcommand{\zsf}{\mathsf{z}}
\newcommand{\Asf}{\mathsf{A}}
\newcommand{\Bsf}{\mathsf{B}}
\newcommand{\Csf}{\mathsf{C}}
\newcommand{\Dsf}{\mathsf{D}}
\newcommand{\Esf}{\mathsf{E}}
\newcommand{\Fsf}{\mathsf{F}}
\newcommand{\Gsf}{\mathsf{G}}
\newcommand{\Hsf}{\mathsf{H}}
\newcommand{\Isf}{\mathsf{I}}
\newcommand{\Jsf}{\mathsf{J}}
\newcommand{\Ksf}{\mathsf{K}}
\newcommand{\Lsf}{\mathsf{L}}
\newcommand{\Msf}{\mathsf{M}}
\newcommand{\Nsf}{\mathsf{N}}
\newcommand{\Osf}{\mathsf{O}}
\newcommand{\Psf}{\mathsf{P}}
\newcommand{\Qsf}{\mathsf{Q}}
\newcommand{\Rsf}{\mathsf{R}}
\newcommand{\Ssf}{\mathsf{S}}
\newcommand{\Tsf}{\mathsf{T}}
\newcommand{\Usf}{\mathsf{U}}
\newcommand{\Vsf}{\mathsf{V}}
\newcommand{\Wsf}{\mathsf{W}}
\newcommand{\Xsf}{\mathsf{X}}
\newcommand{\Ysf}{\mathsf{Y}}
\newcommand{\Zsf}{\mathsf{Z}}
% mathbb letters
\newcommand{\Abb}{\mathbb{A}}
\newcommand{\Cbb}{\mathbb{C}}
\newcommand{\Dbb}{\mathbb{D}}
\newcommand{\Ebb}{\mathbb{E}}
\newcommand{\Fbb}{\mathbb{F}}
\newcommand{\Gbb}{\mathbb{G}}
\newcommand{\Hbb}{\mathbb{H}}
\newcommand{\Ibb}{\mathbb{I}}
\newcommand{\Jbb}{\mathbb{J}}
\newcommand{\Kbb}{\mathbb{K}}
\newcommand{\Lbb}{\mathbb{L}}
\newcommand{\Mbb}{\mathbb{M}}
\newcommand{\Nbb}{\mathbb{N}}
\newcommand{\Obb}{\mathbb{O}}
\newcommand{\Pbb}{\mathbb{P}}
\newcommand{\Qbb}{\mathbb{Q}}
\newcommand{\Rbb}{\mathbb{R}}
\newcommand{\Sbb}{\mathbb{S}}
\newcommand{\Tbb}{\mathbb{T}}
\newcommand{\Ubb}{\mathbb{U}}
\newcommand{\Vbb}{\mathbb{V}}
\newcommand{\Wbb}{\mathbb{W}}
\newcommand{\Xbb}{\mathbb{X}}
\newcommand{\Ybb}{\mathbb{Y}}
\newcommand{\Zbb}{\mathbb{Z}}
% mathcal letters
\newcommand{\Acal}{\mathcal{A}}
\newcommand{\Bcal}{\mathcal{B}}
\newcommand{\Ccal}{\mathcal{C}}
\newcommand{\Dcal}{\mathcal{D}}
\newcommand{\Ecal}{\mathcal{E}}
\newcommand{\Fcal}{\mathcal{F}}
\newcommand{\Gcal}{\mathcal{G}}
\newcommand{\Hcal}{\mathcal{H}}
\newcommand{\Ical}{\mathcal{I}}
\newcommand{\Jcal}{\mathcal{J}}
\newcommand{\Kcal}{\mathcal{K}}
\newcommand{\Lcal}{\mathcal{L}}
\newcommand{\Mcal}{\mathcal{M}}
\newcommand{\Ncal}{\mathcal{N}}
\newcommand{\Ocal}{\mathcal{O}}
\newcommand{\Pcal}{\mathcal{P}}
\newcommand{\Qcal}{\mathcal{Q}}
\newcommand{\Rcal}{\mathcal{R}}
\newcommand{\Scal}{\mathcal{S}}
\newcommand{\Tcal}{\mathcal{T}}
\newcommand{\Ucal}{\mathcal{U}}
\newcommand{\Vcal}{\mathcal{V}}
\newcommand{\Wcal}{\mathcal{W}}
\newcommand{\Xcal}{\mathcal{X}}
\newcommand{\Ycal}{\mathcal{Y}}
\newcommand{\Zcal}{\mathcal{Z}}
% shorthands
\newcommand{\constant}{\textnormal{constant}}
% topology
\newcommand{\scomp}[1]{#1^{\mathsf c}} % set complement
\newcommand{\closure}[1]{\overline{#1}}
\newcommand{\conv}{*}
\newcommand{\circconv}{\circledast}
\DeclareMathOperator{\affinehull}{aff}
\DeclareMathOperator{\convexhull}{conv}
\DeclareMathOperator{\conichull}{conic}
\newcommand{\pdiv}[3][]{\operatorname{D}_{#1}\left(#2\:\vert\vert\:#3 \right)}
\newcommand{\kldiv}[2]{\pdiv[\text{KL}]{#1}{#2}}
\newcommand{\bigo}[1]{\mathcal{O}\left( #1 \right)}
\newcommand{\littleo}[1]{o\left(#1 \right)}
\newcommand{\LHS}{\operatorname{LHS}}
\newcommand{\RHS}{\operatorname{RHS}}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\posreals}{\reals^+}
\newcommand{\nonnegreals}{\reals^{\geq 0}}
\newcommand{\naturals}{\mathbb{N}}
\newcommand{\integers}{\mathbb{Z}}
\newcommand{\complex}{\mathbb{C}}
% operators
\newcommand{\textif}{\text{if}\ }
\newcommand{\textelse}{\text{else}\ }
%\DeclareMathOperator* is used for defining operators that have limits typeset beneath them instead of to the right (at least when in a display)
\DeclareMathOperator*{\maximizeop}{maximize}
\DeclareMathOperator*{\minimizeop}{minimize}
\newcommand{\minimize}[1]{\minimizeop\limits_{#1}\ }
\newcommand{\maximize}[1]{\maximizeop\limits_{#1}\ }
\newcommand{\subjectto}{\text{subject to}\ }
\DeclareMathOperator*{\argmax}{\arg\!\max}
\DeclareMathOperator*{\argmin}{\arg\!\min}
\DeclareMathOperator*{\argsup}{\arg\!\sup}
\DeclareMathOperator*{\arginf}{\arg\!\inf}
% use \command* to get autoscaling with paired delimiter commands
% the unstarred version takes an optional argument that can be \big, \Big, etc
% WARNING: Mathjax has a bug where there is an extra s in DeclarePairedDelimiter
% See: https://github.com/mathjax/MathJax/issues/2758
\DeclarePairedDelimiters{\ceil}{\lceil}{\rceil} % requires mathtools package
\DeclarePairedDelimiters{\floor}{\lfloor}{\rfloor} % requires mathtools package
\DeclarePairedDelimitersX{\inner}[2]{\langle}{\rangle}{#1, #2} % requires mathtools package
\DeclarePairedDelimiters{\abs}{\lvert}{\rvert}%
\DeclarePairedDelimiters{\norm}{\lVert}{\rVert}%
\DeclarePairedDelimiters{\parens}{(}{)}
\DeclarePairedDelimiters{\braks}{[}{]}
\newcommand{\image}[1]{\operatorname{img}\parens*{#1}}
\newcommand{\kernel}[1]{\operatorname{ker}\parens*{#1}}
\newcommand{\diameter}[1]{\operatorname{diam}\parens*{#1}}
\newcommand{\radius}[1]{\operatorname{radius}\parens*{#1}}
\newcommand{\vecmat}[1]{\operatorname{vec}\parens*{#1}}
\newcommand{\diag}[1]{\operatorname{diag}\parens*{#1}}
\newcommand{\trace}[1]{\operatorname{tr}\parens*{#1}}
% note: span is a latex primitive
\newcommand{\vspan}[1]{\operatorname{span}\parens*{#1}}
\newcommand{\rank}[1]{\operatorname{rank}\parens*{#1}}
\newcommand{\sign}[1]{\operatorname{sign}\parens*{#1}}
\newcommand{\tdef}{\triangleq}
% middle bar, for use in delimited expressions like sets
\newcommand{\mmid}{\ \middle| \ }
%\newcommand{\set}[2][]{
% \ifx\\#1\\
% \{ #2 \}
% \else
% \{ #1 \mid #2 \}
% \fi
%}
\newcommand{\onevec}{\mathbf{1}}
\newcommand*{\zerovec}{\mathbf{0}}
\newcommand{\neginfty}{{-\infty}}
% Positive semidefinite, positive definite cones
\newcommand{\sym}[1]{\Sbb^{#1}}
\newcommand{\psd}[1]{\Sbb^{#1}_+}
\newcommand{\pd}[1]{\Sbb^{#1}_{++}}
% up/down arrows for limits
\newcommand{\dto}{\downarrow}
\newcommand{\uto}{\uparrow}
\newcommand{\T}{\top}
\newcommand{\inverse}{^{-1}}
\newcommand{\invsqrt}{^{-1/2}}
\newcommand{\pinv}{^{\dagger}}
\newcommand{\transpose}{{}^\top}
% unit ball
\newcommand{\ball}{\mathbb{B}}
% n-sphere
\newcommand{\sphere}[1]{S^{#1 - 1}}
% Manifolds
\newcommand{\mnf}[1]{\manifold{#1}}
%\newcommand{\manifold}[1]{\mathcal{\MakeUppercase{#1}}}
\newcommand{\manifold}[1]{\mathcal{#1}}
\newcommand{\Mmanifold}{\manifold{M}}
\newcommand{\tangent}{\mathrm{T}} % Tangent space
\newcommand{\cotangent}{\mathrm{T}^*} % Cotangent space
\newcommand{\scalarfields}[1]{\mathfrak{F}(#1)}
\newcommand{\vectorfields}[1]{\mathfrak{X}(#1)}
% Probability
% Random variables
%\newcommand{\rv}[1]{\mathbf{\MakeUppercase{#1}}}
\newcommand{\rv}[1]{\mathbf{#1}}
\newcommand{\arv}{\rv{A}}
\newcommand{\brv}{\rv{B}}
\newcommand{\crv}{\rv{C}}
\newcommand{\drv}{\rv{D}}
\newcommand{\erv}{\rv{E}}
\newcommand{\frv}{\rv{F}}
\newcommand{\grv}{\rv{G}}
\newcommand{\hrv}{\rv{H}}
\newcommand{\irv}{\rv{I}}
\newcommand{\jrv}{\rv{J}}
\newcommand{\krv}{\rv{K}}
\newcommand{\lrv}{\rv{L}}
\newcommand{\mrv}{\rv{M}}
\newcommand{\nrv}{\rv{N}}
\newcommand{\orv}{\rv{O}}
\newcommand{\prv}{\rv{P}}
\newcommand{\qrv}{\rv{Q}}
\newcommand{\rrv}{\rv{R}}
\newcommand{\srv}{\rv{S}}
\newcommand{\trv}{\rv{T}}
\newcommand{\urv}{\rv{U}}
\newcommand{\vrv}{\rv{V}}
\newcommand{\wrv}{\rv{W}}
\newcommand{\xrv}{\rv{X}}
\newcommand{\yrv}{\rv{Y}}
\newcommand{\zrv}{\rv{Z}}
\newcommand{\xspace}{\Xcal}
\newcommand{\yspace}{\Ycal}
\newcommand{\zspace}{\Zcal}
\newcommand\given[1][]{\:#1\vert\:}
\newcommand{\simplex}{\Delta}
\newcommand{\ev}[2][]{\mathbb{E}_{#1}\left[ #2\right]}
\newcommand{\var}[2][]{\operatorname{Var}_{#1} \braks{#2}}
\newcommand{\cov}[2]{\operatorname{Cov}\parens*{#1, #2}}
\newcommand{\prob}[2][]{\mathbb{P}_{#1}\left(#2\right)}
\newcommand{\ind}[1]{\unicode{x1D7D9} _{\left[ #1\right]}} % https://stackoverflow.com/questions/43886576/how-to-do-a-bold-1-mathbbm1-in-mathjax
\DeclareMathOperator{\Binom}{Binom}
\newcommand{\Normal}[1]{\operatorname{Normal}\parens*{#1}}
\newcommand{\Bernoulli}[1]{\operatorname{Bernoulli}\parens*{#1}}
\newcommand{\Binomial}[1]{\operatorname{Binomial}\parens*{#1}}
\newcommand{\stdnormal}{\Normal{0, 1}}
\newcommand{\stdmvnormal}{\Normal(0, I)}
% convergence symbols
\newcommand{\distto}{\overset{d}{\longrightarrow}}
\newcommand{\asto}{\overset{a.s.}{\longrightarrow}}
\newcommand{\probto}{\overset{\Pbb}{\longrightarrow}}
\newcommand{\lpto}[1]{\overset{L^{#1}}{\longrightarrow}}
\newcommand{\weakto}{\overset{\mathrm{weak}}{\longrightarrow}}
\newcommand{\normto}{\overset{\mathrm{norm}}{\longrightarrow}}
% convex analysis
\newcommand{\gph}[1]{\operatorname{gph}\parens*{#1}}
\newcommand{\epigraph}[1]{\operatorname{epi}\parens*{#1}}
\newcommand{\domain}[1]{\operatorname{dom}\parens*{#1}}
\newcommand{\intr}[1]{\mathrm{int}\left( #1 \right)}
\newcommand{\clos}[1]{\mathrm{cl}\left( #1 \right)}
\newcommand{\relint}[1]{{#1}^{\circ}}
% groups
\newcommand{\orth}[1]{\mathsf{O}(#1)}
\newcommand{\sorth}[1]{\mathsf{SO}(#1)}
\newcommand{\sunit}[1]{\mathsf{SU}(#1)}
\newcommand{\unit}[1]{\mathsf{U}(#1)}
% derivatives
\newcommand{\dd}[1][]{\mathrm{d}{#1}} % differential
\newcommand{\diff}{\textnormal{d}}
\newcommand{\deriv}{\textnormal{D}}
\newcommand{\grad}{\nabla}
\renewcommand{\dfrac}[2]{\frac{\dd #1}{\dd #2}}
\newcommand{\pfrac}[2]{\frac{\partial #1}{\partial #2}}
% matrices
\newcommand{\pmat}[1]{\begin{pmatrix} #1 \end{pmatrix}}
\newcommand{\bmat}[1]{\begin{bmatrix} #1 \end{bmatrix}}
\newcommand{\Bmat}[1]{\begin{Bmatrix} #1 \end{Bmatrix}}
\]
