% Include statements
\usepackage{graphicx}
\usepackage{amsfonts,amssymb,amsmath,amsthm}
\usepackage[numbers,square]{natbib}
\usepackage[left=1in,top=1in,right=1in,bottom=1in,nohead]{geometry}
\usepackage[all]{xy}
\usepackage{multirow,rotating,array}
\usepackage[ruled,lined]{algorithm2e}
\SetKw{KwSet}{Set}
\usepackage{pdfsync}
\usepackage{setspace}
\usepackage{bbm}
\usepackage{hyperref}
\hypersetup{backref,colorlinks=true,citecolor=blue,linkcolor=blue,urlcolor=blue}
\renewcommand{\qedsymbol}{$\blacksquare$}
\setlength{\parindent}{0cm}
\setlength{\parskip}{10pt}
\bibliographystyle{mybibsty}

% Theorem environments (\autoref compatible)
\usepackage{aliascnt}

\newtheorem{theorem}{Theorem}[section]

\newaliascnt{result}{theorem}
\newtheorem{result}[theorem]{Result}
\aliascntresetthe{result}
\providecommand*{\resultautorefname}{Result}
\newaliascnt{lemma}{theorem}
\newtheorem{lemma}[lemma]{Lemma}
\aliascntresetthe{lemma}
\providecommand*{\lemmaautorefname}{Lemma}
\newaliascnt{prop}{theorem}
\newtheorem{proposition}[prop]{Proposition}
\aliascntresetthe{prop}
\providecommand*{\propautorefname}{Proposition}
\newaliascnt{cor}{theorem}
\newtheorem{corollary}[cor]{Corollary}
\aliascntresetthe{cor}
\providecommand*{\corautorefname}{Corollary}
\newaliascnt{conj}{theorem}
\newtheorem{conjecture}[conj]{Conjecture}
\aliascntresetthe{conj}
\providecommand*{\conjautorefname}{Corollary}
\newaliascnt{def}{theorem}
\newtheorem{definition}[def]{Definition}
\aliascntresetthe{def}
\providecommand*{\defautorefname}{Definition}
\newaliascnt{ex}{theorem}
\newtheorem{example}[ex]{Example}
\aliascntresetthe{ex}
\providecommand*{\exautorefname}{Example}


\newtheorem{assumption}{Assumption}
\renewcommand{\theassumption}{\Alph{assumption}}
\providecommand*{\assumptionautorefname}{Assumption}

\def\algorithmautorefname{Algorithm}
\renewcommand*{\figureautorefname}{Figure}%
\renewcommand*{\tableautorefname}{Table}%
\renewcommand*{\partautorefname}{Part}%
\renewcommand*{\chapterautorefname}{Chapter}%
\renewcommand*{\sectionautorefname}{Section}%
\renewcommand*{\subsectionautorefname}{Section}%
\renewcommand*{\subsubsectionautorefname}{Section}% 



\newcommand{\x}{\mathbf{x}}
\newcommand{\X}{\mathbb{X}}
\newcommand{\A}{\mathbb{A}}
\newcommand{\B}{\mathbb{B}}
\newcommand{\hbeta}{\hat{\beta}}
\newcommand{\MSE}{\textrm{MSE}} 
\newcommand{\AMSE}{\textrm{AMSE}} 
\newcommand{\hbetag}{\hbeta_{\textrm{good}}} 
\newcommand{\hbetar}[1]{\hat{\beta}_{\textrm{ridge},#1}}
\renewcommand{\v}{\mathbf{v}}
\renewcommand{\u}{\mathbf{u}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\V}{\mathbb{V}}
\newcommand{\R}{\mathbb{R}}

\newcommand{\data}{\mathcal{D}}
\newcommand{\F}{\mathcal{F}}

\renewcommand{\P}{\mathbb{P}}
\newcommand{\tr}{\textrm{trace}}
\newcommand{\df}{\textrm{df}}
\newcommand{\train}{\hat{R}_{\textrm{train}}}
\newcommand{\opt}{\textrm{opt}}
\newcommand{\opthat}{\widehat{\textrm{opt}}}
\newcommand{\gichat}{\hat{R}_{\textrm{gic}}}
\newcommand{\gic}{R_{\textrm{gic}}}
\newcommand{\VV}{\mathcal{V}}
\newcommand{\CVV}{\textrm{CV}_{\VV}}
\newcommand{\CV}[1]{\textrm{CV}_{#1}}

