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LaTeX source

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,graphicx,booktabs}

\newcommand{\R}{\mathbb{R}}
\DeclareMathOperator*{\argmin}{arg\,min}

\title{Spectral Methods for Sparse Recovery}
\author{A. Researcher \and B. Collaborator}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
We study recovery of a signal $x \in \R^n$ from $m \ll n$ linear measurements
and give a bound that is tight up to a logarithmic factor.
\end{abstract}

\section{Problem statement}
\label{sec:problem}
Given $A \in \R^{m \times n}$ and $y = Ax + \varepsilon$, we solve
\begin{equation}
\label{eq:lasso}
\hat{x} = \argmin_{z \in \R^n} \tfrac{1}{2}\|y - Az\|_2^2 + \lambda \|z\|_1 .
\end{equation}

\subsection{Main result}
For every $k$-sparse $x$,
\[
\mathbb{E}\left[\|\hat{x} - x\|_2^2\right]
  \leq C \sigma^2 \frac{k \log(n/k)}{m}
  + \sum_{i=1}^{n} \frac{\lambda_i}{\sqrt{m}} .
\]

\begin{itemize}
  \item Holds for sub-Gaussian designs.
  \item Constant $C$ is absolute.
  \begin{itemize}
    \item Nested items are preserved.
  \end{itemize}
\end{itemize}

\begin{table}[h]
\centering
\begin{tabular}{lrr}
\toprule
Method & Error & Time (s) \\
\midrule
LASSO & 0.041 & 1.2 \\
Ours  & \textbf{0.018} & 0.9 \\
\bottomrule
\end{tabular}
\caption{Recovery error on synthetic data.}
\end{table}

Equation~\eqref{eq:lasso} is discussed in Section~\ref{sec:problem}.\footnote{See appendix for proofs.}

\end{document}

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