\usepackage{amsfonts}       % blackboard math symbols
\usepackage{nicefrac}       % compact symbols for 1/2, etc.
\usepackage{microtype}      % microtypography
\usepackage{xcolor}         % colors
\usepackage{mathtools} 
\usepackage{tikz}
\usepackage{graphicx}
\usepackage{subfigure}
% \usepackage{caption}
\usepackage{diagbox}
% \usepackage{scalerel}

\newcommand{\theHalgorithm}{\arabic{algorithm}}

\usepackage{amsmath}
\usepackage{dsfont}
\usepackage{amssymb,amsthm,commath}
\usepackage{bm,xfrac,xcolor}
\usepackage{nccmath}
\usepackage[toc,page,header]{appendix}

\hypersetup{colorlinks,urlcolor=black, citecolor=blue}

\usepackage{algorithm}
\usepackage[noend]{algorithmic}
\usepackage{bbm}
\usepackage{enumitem}
\usepackage{gensymb}
\usepackage{refcount}
\usepackage{mathtools}
\usepackage{stackengine}
\usepackage{multirow}
\usepackage{makecell}

\usepackage{comment}
\usepackage{mathrsfs}
% \usepackage{floatrow}
% Table float box with bottom caption, box width adjusted to content

\makeatletter
    \setlength\@fptop{0\p@}
\makeatother

\newcommand{\INDSTATE}[1][1]{\STATE\hspace{-0.75em}{\algorithmicindent}}

\newcommand\barbelow[1]{\stackunder[1.2pt]{$#1$}{\rule{1.5ex}{.1ex}}}
\newcommand*\samethanks[1][\value{footnote}]{\footnotemark[#1]}

\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}
% \usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}
% \makeatletter
\def\myfnt{\ifx\protect\@typeset@protect\expandafter\footnote\else\expandafter\@gobble\fi}
\makeatother

\makeatletter
\def\BState{\State\hskip-\ALG@thistlm}
\makeatother

\makeatletter
\newcommand*{\skipnumber}[2][1]{%
   {\renewcommand*{\alglinenumber}[1]{}\State #2}%
   \addtocounter{ALG@line}{-#1}}
\makeatother

\renewcommand\thealgorithm{}
\makeatletter
\newcommand{\setalglineno}[1]{%
  \setcounter{ALC@line}{\numexpr#1-1}}
\makeatother

\usepackage{placeins,afterpage}

\makeatletter
\newcommand*\bigcdot{\mathpalette\bigcdot@{.5}}
\newcommand*\bigcdot@[2]{\mathbin{\vcenter{\hbox{\scalebox{#2}{$\m@th#1\bullet$}}}}}
\makeatother

%---------------------------------------
% \usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}
% \makeatletter
% \def\myfnt{\ifx\protect\@typeset@protect\expandafter\footnote\else\expandafter\@gobble\fi}
% \makeatother

% \makeatletter
% \def\BState{\State\hskip-\ALG@thistlm}
% \makeatother

% \makeatletter
% \newcommand*{\skipnumber}[2][1]{%
%   {\renewcommand*{\alglinenumber}[1]{}\State #2}%
%   \addtocounter{ALG@line}{-#1}}
% \makeatother

% \renewcommand\thealgorithm{}
% \makeatletter
% \newcommand{\setalglineno}[1]{%
%   \setcounter{ALC@line}{\numexpr#1-1}}
% \makeatother


% \renewcommand{\appendixpagename}{\centering Appendices}
% \newcommand\barbelow[1]{\stackunder[1.2pt]{$#1$}{\rule{1.5ex}{.1ex}}}
% \newcommand*\samethanks[1][\value{footnote}]{\footnotemark[#1]}

\usepackage{tikz}
\tikzset{
  c/.style={every coordinate/.try}
}

\newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}}

% \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}

\mathchardef\mhyphen="2D

\newcommand{\nondiag}{\mathit{non\mhyphen diag}}
\newcommand{\offdiag}{\mathit{off\mhyphen diag}}
\newcommand{\vecm}{\mathit{vec}}
\newcommand{\diag}{\mathit{diag}}
\newcommand{\tup}{\mathit{tup}}
\newcommand{\tupr}{\rmbf^{1:m}}
\newcommand{\tuprhat}{\hat{\rmbf}^{1:m}}

\newcommand{\TP}{\emph{TP}}
\newcommand{\TN}{\emph{TN}}
\newcommand{\FP}{\emph{FP}}
\newcommand{\FN}{\emph{FN}}

\newcommand{\TPs}{\emph{TP }}
\newcommand{\TNs}{\emph{TN }}
\newcommand{\FPs}{\emph{FP }}
\newcommand{\FNs}{\emph{FN }}

% \newcommand{\lin}{\textrm{\textup{lin}}}
\newcommand{\fair}{\textrm{\textup{fair}}}
\newcommand{\quadr}{\textrm{\textup{quad}}}
\newcommand{\qmean}{\textrm{\textup{qmean}}}
\newcommand{\cov}{\textrm{\textup{cov}}}
\newcommand{\eo}{\textrm{\textup{EO}}}
\newcommand{\bcov}{\textrm{\textup{\mathbf{cov}}}}

\newcommand{\Acal}{{\cal A}}
\newcommand{\Bcal}{{\cal B}}
\newcommand{\Ccal}{{\cal C}}
\newcommand{\Dcal}{{\cal D}}
\newcommand{\Ecal}{{\cal E}}
\newcommand{\Fcal}{{\cal F}}
\newcommand{\Gcal}{{\cal G}}
\newcommand{\Hcal}{{\cal H}}
\newcommand{\Ical}{{\cal I}}
\newcommand{\Jcal}{{\cal J}}
\newcommand{\Kcal}{{\cal K}}
\newcommand{\Lcal}{{\cal L}}
\newcommand{\Mcal}{{\cal M}}
\newcommand{\Ncal}{{\cal N}}
\newcommand{\Ocal}{{\cal O}}
\newcommand{\Pcal}{{\cal P}}
\newcommand{\Qcal}{{\cal Q}}
\newcommand{\Rcal}{{\cal R}}
\newcommand{\Scal}{{\cal S}}
\newcommand{\Tcal}{{\cal T}}
\newcommand{\Ucal}{{\cal U}}
\newcommand{\Vcal}{{\cal V}}
\newcommand{\Wcal}{{\cal W}}
\newcommand{\Xcal}{{\cal X}}
\newcommand{\Ycal}{{\cal Y}}
\newcommand{\Zcal}{{\cal Z}}


\newcommand{\Ambb}{\mathbb{A}}
\newcommand{\Bmbb}{\mathbb{B}}
\newcommand{\Cmbb}{\mathbb{C}}
\newcommand{\Dmbb}{\mathbb{D}}
\newcommand{\Embb}{\mathbb{E}}
\newcommand{\Fmbb}{\mathbb{F}}
\newcommand{\Gmbb}{\mathbb{G}}
\newcommand{\Hmbb}{\mathbb{H}}
\newcommand{\Imbb}{\mathbb{I}}
\newcommand{\Jmbb}{\mathbb{J}}
\newcommand{\Kmbb}{\mathbb{K}}
\newcommand{\Lmbb}{\mathbb{L}}
\newcommand{\Mmbb}{\mathbb{M}}
\newcommand{\Nmbb}{\mathbb{N}}
\newcommand{\Ombb}{\mathbb{O}}
\newcommand{\Pmbb}{\mathbb{P}}
\newcommand{\Qmbb}{\mathbb{Q}}
\newcommand{\Rmbb}{\mathbb{R}}
\newcommand{\Smbb}{\mathbb{S}}
\newcommand{\Tmbb}{\mathbb{T}}
\newcommand{\Umbb}{\mathbb{U}}
\newcommand{\Vmbb}{\mathbb{V}}
\newcommand{\Wmbb}{\mathbb{W}}
\newcommand{\Xmbb}{\mathbb{X}}
\newcommand{\Ymbb}{\mathbb{Y}}
\newcommand{\Zmbb}{\mathbb{Z}}

\newcommand{\thetambf}{\bm{\theta}}

\newcommand{\Bcalbar}{\oline{\Bcal}}

\newcommand{\smbfbar}{\oline{\mathbf{s}}}
\newcommand{\cmbfbar}{\oline{\mathbf{c}}}
\newcommand{\rmbfbar}{\oline{\mathbf{r}}}
\newcommand{\dmbfbar}{\oline{\mathbf{d}}}

\newcommand{\Bcalhat}{\hat{\Bcal}}
\newcommand{\smbfhat}{\hat{\mathbf{s}}}
\newcommand{\cmbfhat}{\hat{\mathbf{c}}}
\newcommand{\rmbfhat}{\hat{\mathbf{r}}}
\newcommand{\ambfhat}{\hat{\mathbf{a}}}
\newcommand{\fmbfhat}{\hat{\mathbf{f}}}
\newcommand{\dmbfhat}{\hat{\mathbf{d}}}
\newcommand{\bmbfhat}{\hat{\mathbf{b}}}
\newcommand{\Bmbfhat}{\hat{\mathbf{B}}}
\newcommand{\lambdahat}{\hat{\lambda}}

\newcommand{\ambfbar}{\oline{\mathbf{a}}}
\newcommand{\zmbfbar}{\oline{\mathbf{z}}}
\newcommand{\fmbfbar}{\oline{\mathbf{f}}}
\newcommand{\bmbfbar}{\oline{\mathbf{b}}}
\newcommand{\Bmbfbar}{\oline{\mathbf{B}}}
\newcommand{\lambdabar}{\oline{\lambda}}




\newcommand{\Ambf}{\mathbf{A}}
\newcommand{\ambf}{\mathbf{a}}
\newcommand{\Bmbf}{\mathbf{B}}
\newcommand{\bmbf}{\mathbf{b}}
\newcommand{\Cmbf}{\mathbf{C}}
\newcommand{\cmbf}{\mathbf{c}}
\newcommand{\Dmbf}{\mathbf{D}}
\newcommand{\dmbf}{\mathbf{d}}
\newcommand{\Embf}{\mathbf{E}}
\newcommand{\embf}{\mathbf{e}}
\newcommand{\Fmbf}{\mathbf{F}}
\newcommand{\fmbf}{\mathbf{f}}
\newcommand{\Gmbf}{\mathbf{G}}
\newcommand{\gmbf}{\mathbf{g}}
\newcommand{\Hmbf}{\mathbf{H}}
\newcommand{\hmbf}{\mathbf{h}}
\newcommand{\Imbf}{\mathbf{I}}
\newcommand{\imbf}{\mathbf{i}}
\newcommand{\Jmbf}{\mathbf{J}}
\newcommand{\jmbf}{\mathbf{j}}
\newcommand{\Kmbf}{\mathbf{K}}
\newcommand{\kmbf}{\mathbf{k}}
\newcommand{\Lmbf}{\mathbf{L}}
\newcommand{\lmbf}{\mathbf{l}}
\newcommand{\Mmbf}{\mathbf{M}}
\newcommand{\mmbf}{\mathbf{m}}
\newcommand{\Nmbf}{\mathbf{N}}
\newcommand{\nmbf}{\mathbf{n}}
\newcommand{\Ombf}{\mathbf{O}}
\newcommand{\ombf}{\mathbf{o}}
\newcommand{\Pmbf}{\mathbf{P}}
\newcommand{\pmbf}{\mathbf{p}}
\newcommand{\Qmbf}{\mathbf{Q}}
\newcommand{\qmbf}{\mathbf{q}}
\newcommand{\Rmbf}{\mathbf{R}}
\newcommand{\rmbf}{\mathbf{r}}
\newcommand{\Smbf}{\mathbf{S}}
\newcommand{\smbf}{\mathbf{s}}
\newcommand{\Tmbf}{\mathbf{T}}
\newcommand{\tmbf}{\mathbf{t}}
\newcommand{\Umbf}{\mathbf{U}}
\newcommand{\umbf}{\mathbf{u}}
\newcommand{\Vmbf}{\mathbf{V}}
\newcommand{\vmbf}{\mathbf{v}}
\newcommand{\Wmbf}{\mathbf{W}}
\newcommand{\wmbf}{\mathbf{w}}
\newcommand{\Xmbf}{\mathbf{X}}
\newcommand{\xmbf}{\mathbf{x}}
\newcommand{\Ymbf}{\mathbf{Y}}
\newcommand{\ymbf}{\mathbf{y}}
\newcommand{\Zmbf}{\mathbf{Z}}
\newcommand{\zmbf}{\mathbf{z}}


\newcommand{\oline}[1]{\mkern 1.5mu\overline{\mkern-1.5mu#1}}
\newcommand{\Abar}{\oline{A}}
\newcommand{\abar}{\oline{a}}
\newcommand{\Bbar}{\oline{B}}
\newcommand{\bbar}{\oline{b}}
\newcommand{\Cbar}{\oline{C}}
\newcommand{\cbar}{\oline{c}}
\newcommand{\Dbar}{\oline{D}}
\newcommand{\dbar}{\oline{d}}
\newcommand{\Ebar}{\oline{E}}
\newcommand{\ebar}{\oline{e}}
\newcommand{\Fbar}{\oline{F}}
\newcommand{\fbar}{\oline{f}}
\newcommand{\Gbar}{\oline{G}}
\newcommand{\gbar}{\oline{g}}
\newcommand{\Hbar}{\oline{H}}
\renewcommand{\hbar}{\oline{h}}
\newcommand{\Ibar}{\oline{I}}
\newcommand{\ibar}{\oline{i}}
\newcommand{\Jbar}{\oline{J}}
\newcommand{\jbar}{\oline{j}}
\newcommand{\Kbar}{\oline{K}}
\newcommand{\kbar}{\oline{k}}
\newcommand{\Lbar}{\oline{L}}
\newcommand{\lbar}{\oline{l}}
\newcommand{\Mbar}{\skew{4}\overline{M}}
\newcommand{\mbar}{\oline{m}}
\newcommand{\Nbar}{\oline{N}}
\newcommand{\nbar}{\oline{n}}
\newcommand{\Obar}{\oline{O}}
\newcommand{\obar}{\oline{o}}
\newcommand{\Pbar}{\oline{P}}
\newcommand{\pbar}{\oline{p}}
\newcommand{\Qbar}{\oline{Q}}
\newcommand{\qbar}{\oline{q}}
\newcommand{\Rbar}{\oline{R}}
\newcommand{\rbar}{\oline{r}}
\newcommand{\Sbar}{\oline{S}}
\newcommand{\sbar}{\oline{s}}
\newcommand{\Tbar}{\oline{T}}
\newcommand{\tbar}{\oline{t}}
\newcommand{\Ubar}{\oline{U}}
\newcommand{\ubar}{\oline{u}}
\newcommand{\Vbar}{\oline{V}}
\newcommand{\vbar}{\oline{v}}
\newcommand{\Wbar}{\oline{W}}
\newcommand{\wbar}{\oline{w}}
\newcommand{\Xbar}{\oline{X}}
\newcommand{\xbar}{\oline{x}}
\newcommand{\Ybar}{\oline{Y}}
\newcommand{\ybar}{\oline{y}}
\newcommand{\Zbar}{\oline{Z}}
\newcommand{\zbar}{\oline{z}}
\newcommand{\Atilde}{\tilde{A}}
\newcommand{\atilde}{\tilde{a}}
\newcommand{\Btilde}{\tilde{B}}
\newcommand{\btilde}{\tilde{b}}
\newcommand{\Ctilde}{\tilde{C}}
\newcommand{\ctilde}{\tilde{c}}
\newcommand{\Dtilde}{\tilde{D}}
\newcommand{\dtilde}{\tilde{d}}
\newcommand{\Etilde}{\tilde{E}}
\newcommand{\etilde}{\tilde{e}}
\newcommand{\Ftilde}{\tilde{F}}
\newcommand{\ftilde}{\tilde{f}}
\newcommand{\Gtilde}{\tilde{G}}
\newcommand{\gtilde}{\tilde{g}}
\newcommand{\Htilde}{\tilde{H}}
\newcommand{\htilde}{\tilde{h}}
\newcommand{\Itilde}{\tilde{I}}
\newcommand{\itilde}{\tilde{i}}
\newcommand{\Jtilde}{\tilde{J}}
\newcommand{\jtilde}{\tilde{j}}
\newcommand{\Ktilde}{\tilde{K}}
\newcommand{\ktilde}{\tilde{k}}
\newcommand{\Ltilde}{\tilde{L}}
\newcommand{\ltilde}{\tilde{l}}
\newcommand{\Mtilde}{\tilde{M}}
\newcommand{\mtilde}{\tilde{m}}
\newcommand{\Ntilde}{\tilde{N}}
\newcommand{\ntilde}{\tilde{n}}
\newcommand{\Otilde}{\tilde{O}}
\newcommand{\otilde}{\tilde{o}}
\newcommand{\Ptilde}{\tilde{P}}
\newcommand{\ptilde}{\tilde{p}}
\newcommand{\Qtilde}{\tilde{Q}}
\newcommand{\qtilde}{\tilde{q}}
\newcommand{\Rtilde}{\tilde{R}}
\newcommand{\rtilde}{\tilde{r}}
\newcommand{\Stilde}{\tilde{S}}
\newcommand{\stilde}{\tilde{s}}
\newcommand{\Ttilde}{\tilde{T}}
\newcommand{\ttilde}{\tilde{t}}
\newcommand{\Utilde}{\tilde{U}}
\newcommand{\utilde}{\tilde{u}}
\newcommand{\Vtilde}{\tilde{V}}
\newcommand{\vtilde}{\tilde{v}}
\newcommand{\Wtilde}{\tilde{W}}
\newcommand{\wtilde}{\tilde{w}}
\newcommand{\Xtilde}{\tilde{X}}
\newcommand{\xtilde}{\tilde{x}}
\newcommand{\Ytilde}{\tilde{Y}}
\newcommand{\ytilde}{\tilde{y}}
\newcommand{\Ztilde}{\tilde{Z}}
\newcommand{\ztilde}{\tilde{z}}

\newcommand{\Ahat}{\hat{A}}
\newcommand{\ahat}{\hat{a}}
\newcommand{\Bhat}{\hat{B}}
\newcommand{\bhat}{\hat{b}}
\newcommand{\Chat}{\hat{C}}
\newcommand{\chat}{\hat{c}}
\newcommand{\Dhat}{\hat{D}}
\newcommand{\dhat}{\hat{d}}
\newcommand{\Ehat}{\hat{E}}
\newcommand{\ehat}{\hat{e}}
\newcommand{\Fhat}{\hat{F}}
\newcommand{\fhat}{\hat{f}}
\newcommand{\Ghat}{\hat{G}}
\newcommand{\ghat}{\hat{g}}
\newcommand{\Hhat}{\hat{H}}
\newcommand{\hhat}{\hat{h}}
\newcommand{\Ihat}{\hat{I}}
\newcommand{\ihat}{\hat{i}}
\newcommand{\Jhat}{\hat{J}}
\newcommand{\jhat}{\hat{j}}
\newcommand{\Khat}{\hat{K}}
\newcommand{\khat}{\hat{k}}
\newcommand{\Lhat}{\widehat{L}}
\newcommand{\lhat}{\hat{l}}
\newcommand{\Mhat}{\hat{M}}
\newcommand{\mhat}{\hat{m}}
\newcommand{\Nhat}{\hat{N}}
\newcommand{\nhat}{\hat{n}}
\newcommand{\Ohat}{\hat{O}}
\newcommand{\ohat}{\hat{o}}
\newcommand{\Phat}{\hat{P}}
\newcommand{\phat}{\hat{p}}
\newcommand{\Qhat}{\hat{Q}}
\newcommand{\qhat}{\hat{q}}
\newcommand{\Rhat}{\hat{R}}
\newcommand{\rhat}{\hat{r}}
\newcommand{\Shat}{\hat{S}}
\newcommand{\shat}{\hat{s}}
\newcommand{\That}{\hat{T}}
\newcommand{\that}{\hat{t}}
\newcommand{\Uhat}{\hat{U}}
\newcommand{\uhat}{\hat{u}}
\newcommand{\Vhat}{\hat{V}}
\newcommand{\vhat}{\hat{v}}
\newcommand{\What}{\hat{W}}
\newcommand{\what}{\hat{w}}
\newcommand{\Xhat}{\hat{X}}
\newcommand{\xhat}{\hat{x}}
\newcommand{\Yhat}{\hat{Y}}
\newcommand{\yhat}{\hat{y}}
\newcommand{\Zhat}{\hat{Z}}
\newcommand{\zhat}{\hat{z}}
\newcommand{\ophi}{\overline{\phi}}

\newcommand\inner[2]{\langle #1, #2 \rangle}
\newcommand{\Cdcal}{\mathcal{C}^{d}}
\newcommand{\Cd}{{C}^{d}}
\newcommand{\Cdistr}{C^d_{i\cancel{i}}}
\newcommand{\Cdonlyistr}{C^d_{\cancel{i}}}
\newcommand{\Cii}{C_{ii}}
\newcommand{\Cij}{C_{ij}}
\newcommand{\1}{\mathbbm{1}}

\newcommand{\hphi}{\hat{\phi}}
\newcommand{\hvarphi}{\hat{\varphi}}
\newcommand{\hPhi}{\hat{\Phi}}
\newcommand{\hPsi}{\hat{\Psi}}

\newcommand{\bphi}{\oline{\phi}}
\newcommand{\bvarphi}{\oline{\varphi}}
\newcommand{\bPhi}{\oline{\Phi}}
\newcommand{\bPsi}{\oline{\Psi}}

\newcommand{\scl}[1]{\scalebox{0.9}{#1}}

\newcommand{\alphambf}{\bm{\alpha}}
\newcommand{\taumbf}{\bm{\tau}}
\newcommand{\ellmbf}{\bm{\ell}}
\newcommand{\etambf}{\bm{\eta}}
\newcommand{\gammambf}{\bm{\gamma}}

\newcommand{\ellmbfbar}{\oline{\bm{\ell}}}
\newcommand{\ellmbfhat}{\hat{\bm{\ell}}}
\newcommand{\etambfbar}{\oline{\bm{\eta}}}
\newcommand{\etambfhat}{\hat{\bm{\eta}}}


\newcommand{\ellbar}{\oline{\ell}}
\newcommand{\ellhat}{\hat{\ell}}
\newcommand{\etabar}{\oline{\eta}}
\newcommand{\etahat}{\hat{\eta}}


\newcommand{\bmmbf}{\oline{\mathbf{m}}}
\newcommand{\bmone}{\oline{m}_{11}}
\newcommand{\bmzero}{\oline{m}_{00}}
\newcommand{\bpone}{\oline{p}_{11}}
\newcommand{\bpzero}{\oline{p}_{00}}
\newcommand{\bqone}{\oline{q}_{11}}
\newcommand{\bqzero}{\oline{q}_{00}}
\newcommand{\bqnot}{\oline{q}_{0}}
\newcommand{\bell}{\oline{\ell}}
\newcommand{\btau}{\oline{\tau}}

%\newcommand{\hphi}{\hat{\phi}}
\newcommand{\hmmbf}{\hat{\mathbf{m}}}
\newcommand{\hmone}{\hat{m}_{11}}
\newcommand{\hmzero}{\hat{m}_{00}}
\newcommand{\hpone}{\hat{p}_{11}}
\newcommand{\hpzero}{\hat{p}_{00}}
\newcommand{\hqone}{\hat{q}_{11}}
\newcommand{\hqzero}{\hat{q}_{00}}
\newcommand{\hqnot}{\hat{q}_{0}}
\newcommand{\hell}{\hat{\ell}}

\newcommand{\sphi}{\phi^*}
\newcommand{\smmbf}{\mathbf{m}^*}
\newcommand{\smone}{{m_{11}^*}}
\newcommand{\smzero}{{m_{00}^*}}
\newcommand{\spone}{{p_{11}^*}}
\newcommand{\spzero}{{p_{00}^*}}
\newcommand{\sqone}{{q_{11}^*}}
\newcommand{\sqzero}{{q_{00}^*}}
\newcommand{\sqnot}{{q_{0}^*}}
\newcommand{\sell}{{\ell^*}}
\newcommand{\stau}{{\tau^*}}


\newcommand{\tphi}{\barbelow{\phi}}
\newcommand{\tmmbf}{\barbelow{\mathbf{m}}}
\newcommand{\tsmbf}{\barbelow{\mathbf{s}}}
\newcommand{\tmone}{\barbelow{m}_{11}}
\newcommand{\tmzero}{\barbelow{m}_{00}}
\newcommand{\tpone}{\barbelow{p}_{11}}
\newcommand{\tpzero}{\barbelow{p}_{00}}
\newcommand{\tqone}{\barbelow{q}_{11}}
\newcommand{\tqzero}{\barbelow{q}_{00}}
\newcommand{\tqnot}{\barbelow{q}_{0}}
\newcommand{\tell}{\barbelow{\ell}}
\newcommand{\ttau}{\barbelow{\tau}}
\newcommand{\tC}{\barbelow{C}}

\newcommand{\pphi}{{\phi'}}
\newcommand{\pmmbf}{{\mathbf{m}'}}
\newcommand{\pmone}{{m}_{11}'}
\newcommand{\pmzero}{{m}_{00}'}
\newcommand{\ppone}{{p}_{11}'}
\newcommand{\ppzero}{{p}_{00}'}
\newcommand{\pqone}{{q}_{11}'}
\newcommand{\pqzero}{{q}_{00}'}
\newcommand{\pqnot}{{q}_{0}'}
\newcommand{\pell}{{\ell}'}
\newcommand{\ptau}{{\tau}'}

\newcommand{\ppphi}{{\phi''}}
\newcommand{\ppmmbf}{{\mathbf{m}''}}
\newcommand{\ppmone}{{m}_{11}''}
\newcommand{\ppmzero}{{m}_{00}''}
\newcommand{\pppone}{{p}_{11}''}
\newcommand{\pppzero}{{p}_{00}''}
\newcommand{\ppqone}{{q}_{11}''}
\newcommand{\ppqzero}{{q}_{00}''}
\newcommand{\ppqnot}{{q}_{0}''}
\newcommand{\ppell}{{\ell}''}
\newcommand{\pptau}{{\tau}''}

\DeclareMathOperator{\aff}      {aff}
\DeclareMathOperator{\cl}       {cl}
\DeclareMathOperator{\cone}     {cone}
\DeclareMathOperator{\coni}     {coni}
\DeclareMathOperator{\conv}     {conv}
\DeclareMathOperator{\dom}      {dom}
\DeclareMathOperator{\epi}      {epi}
\DeclareMathOperator{\inertia}  {inertia}
\DeclareMathOperator{\interior} {int}
\DeclareMathOperator{\linspan}  {span}
\DeclareMathOperator{\lin}      {lin}
\DeclareMathOperator{\minimize} {minimize}
\DeclareMathOperator{\Null}     {Null}
\DeclareMathOperator{\paraff}   {par}
\DeclareMathOperator{\proj}     {Proj}
\DeclareMathOperator{\recc}     {recc}
\DeclareMathOperator{\recf}     {recf}
\DeclareMathOperator{\ri}       {ri}
\DeclareMathOperator{\st}       {s.\!t.}
\DeclareMathOperator{\subjectto}{subject~to}
\DeclareMathOperator{\trace}    {trace}

% \DeclareMathOperator{\argmin}   {arg\,min}
% \DeclareMathOperator{\argmax}   {arg\,max}

\DeclareMathOperator*{\argmin}{argmin}
\DeclareMathOperator*{\argmax}{argmax}

\newcommand{\half} {\frac12}
%\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\thalf}{\tfrac12}
\newcommand{\var}{{\mathbb{V}}}
\newcommand{\sgn}{\mbox{{sgn}}}
% Redefined math commands:
\renewcommand{\(}{\left)}
\renewcommand{\)}{\right)}
% Math commands with arguments
%\newcommand{\C}[1]{\mathbb{C}^{#1}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\eR}[1]{\skew{-2}\overline{\mathbb{R}}^{#1}}
\newcommand{\N}[1]{\mathbb{N}^{#1}}
\renewcommand{\P}{\mathbb{P}}
\newcommand{\Q}[1]{\mathbb{Q}^{#1}}
\newcommand{\R}[1]{\mathbb{R}^{#1}}
\newcommand{\Z}[1]{\mathbb{Z}^{#1}}

\newcommand{\baligned}     {\begin{aligned}}
	\newcommand{\ealigned}     {\end{aligned}}
\newcommand{\barray}       {\begin{array}}
	\newcommand{\earray}       {\end{array}}
\newcommand{\bbmatrix}     {\begin{bmatrix}}
	\newcommand{\ebmatrix}     {\end{bmatrix}}
\newcommand{\bcases}       {\begin{cases}}
	\newcommand{\ecases}       {\end{cases}}
\newcommand{\bcenter}      {\begin{center}}
	\newcommand{\ecenter}      {\end{center}}
\newcommand{\bcolumn}      {\begin{column}}
	\newcommand{\ecolumn}      {\end{column}}
\newcommand{\bcolumns}     {\begin{columns}}
	\newcommand{\ecolumns}     {\end{columns}}
\newcommand{\benumerate}   {\begin{enumerate}}
	\newcommand{\eenumerate}   {\end{enumerate}}
\newcommand{\bequation}    {\begin{equation}}
	\newcommand{\eequation}    {\end{equation}}
\newcommand{\bequationn}   {\begin{equation*}}
	\newcommand{\eequationn}   {\end{equation*}}
\newcommand{\bfigure}      {\begin{figure}}
	\newcommand{\efigure}      {\end{figure}}
\newcommand{\bflushright}  {\begin{flushright}}
	\newcommand{\eflushright}  {\end{flushright}}
\newcommand{\bitemize}     {\begin{itemize}}
	\newcommand{\eitemize}     {\end{itemize}}
\newcommand{\bpmatrix}     {\begin{pmatrix}}
	\newcommand{\epmatrix}     {\end{pmatrix}}
\newcommand{\bsubequations}{\begin{subequations}}
	\newcommand{\esubequations}{\end{subequations}}
\newcommand{\btable}       {\begin{table}}
	\newcommand{\etable}       {\end{table}}
\newcommand{\btabular}     {\begin{tabular}}
	\newcommand{\etabular}     {\end{tabular}}
\newcommand{\bvmatrix}     {\begin{vmatrix}}
	\newcommand{\evmatrix}     {\end{vmatrix}}

\newcommand{\bequalin}     {\bequationn\baligned}
\newcommand{\eequalin}     {\ealigned\eequationn}
\newcommand{\bequali}      {\bsubequations\begin{align}}
	\newcommand{\eequali}      {\end{align}\esubequations}


\newtheorem{assumption}{Assumption}
\newtheorem{prop}{Proposition}
\newtheorem{example}{Example}
\newtheorem{remark}{Remark}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem{definition}{Definition}
\newtheorem{lemma}{Lemma}

\newcommand{\balgorithm}  {\begin{algorithm}}
	\newcommand{\ealgorithm}  {\end{algorithm}}
\newcommand{\balgorithmic}{\begin{algorithmic}}
	\newcommand{\ealgorithmic}{\end{algorithmic}}
\newcommand{\bassumption} {\begin{assumption}}
	\newcommand{\eassumption} {\end{assumption}}
\newcommand{\bcorollary}  {\begin{corollary}}
	\newcommand{\ecorollary}  {\end{corollary}}
\newcommand{\bdefinition} {\begin{definition}}
	\newcommand{\edefinition} {\end{definition}}
\newcommand{\bexample}    {\begin{example}}
	\newcommand{\eexample}    {\end{example}}
\newcommand{\bproposition}    {\begin{prop}}
	\newcommand{\eproposition}    {\end{prop}}
\newcommand{\blemma}      {\begin{lemma}}
	\newcommand{\elemma}      {\end{lemma}}
\newcommand{\bproblem}    {\begin{problem}}
	\newcommand{\eproblem}    {\end{problem}}
\newcommand{\bproof}      {\begin{proof}}
	\newcommand{\eproof}      {\end{proof}}
\newcommand{\bremark}     {\begin{remark}}
	\newcommand{\eremark}     {\end{remark}}
\newcommand{\btheorem}    {\begin{theorem}}
	\newcommand{\etheorem}    {\end{theorem}}
	
\newcommand{\prodRcal}{\Rcal^{1:m}}

\renewcommand\theadalign{bl}
\renewcommand\theadfont{\bfseries}
\renewcommand\theadgape{\Gape[4pt]}
\renewcommand\cellgape{\Gape[4pt]}

\usepackage{chngcntr}
\counterwithout{equation}{section}
\counterwithout{table}{section}
\counterwithout{figure}{section}
\counterwithout{algorithm}{section}
\usepackage[customcolors]{hf-tikz}
\usepackage{adjustbox}
\usepackage[normalem]{ulem}
\usepackage{cancel}
\usetikzlibrary{calc,shapes.geometric}
\renewcommand\cite{\citep}

% \captionsetup[subfigure]{font=scriptsize,labelfont=scriptsize}
% \algrenewcommand\algorithmicindent{0.1cm}%

\usepackage{colortbl}
\newcommand{\highlight}[1]{{\cellcolor{gray!25}#1}}