Upper-bounding $\ell_1$-optimization sectional thresholds

TL;DR

This paper develops a new method to establish upper bounds on the sectional thresholds of -minimization in sparse linear systems, advancing the theoretical understanding of its worst-case performance.

Contribution

It introduces a simple mechanism, based on previous work on Hopfield models, to derive solid upper bounds on -optimization sectional thresholds.

Findings

Provided the first solid upper bounds on sectional thresholds.

Enhanced understanding of worst-case -minimization performance.

Built on previous Hopfield model analysis for threshold estimation.

Abstract

In this paper we look at a particular problem related to under-determined linear systems of equations with sparse solutions. $\ell_1$-minimization is a fairly successful polynomial technique that can in certain statistical scenarios find sparse enough solutions of such systems. Barriers of $\ell_1$ performance are typically referred to as its thresholds. Depending if one is interested in a typical or worst case behavior one then distinguishes between the \emph{weak} thresholds that relate to a typical behavior on one side and the \emph{sectional} and \emph{strong} thresholds that relate to the worst case behavior on the other side. Starting with seminal works \cite{CRT,DonohoPol,DOnoho06CS} a substantial progress has been achieved in theoretical characterization of $\ell_1$-minimization statistical thresholds. More precisely, \cite{CRT,DOnoho06CS} presented for the first time linear lower bounds on all of these thresholds. Donoho's work \cite{DonohoPol} (and our own \cite{StojnicCSetam09,StojnicUpper10}) went a bit further and essentially settled the $\ell_1$'s \emph{weak} thresholds. At the same time they also provided fairly good lower bounds on the values on the \emph{sectional} and \emph{strong} thresholds. In this paper, we revisit the \emph{sectional} thresholds and present a simple mechanism that can be used to create solid upper bounds as well. The method we present relies on a seemingly simple but substantial progress we made in studying Hopfield models in \cite{StojnicHopBnds10}.