Towards a better compressed sensing

TL;DR

This paper explores enhancements to $ ext{l}_1$ optimization in compressed sensing by proposing algorithms with feedback mechanisms that can recover sparser signals than traditional methods.

Contribution

It introduces new algorithms with feedback that significantly improve sparse recovery beyond standard $ ext{l}_1$ methods.

Findings

Algorithms recover higher sparsity levels

Feedback mechanisms enhance recovery performance

Potential for improved compressed sensing applications

Abstract

In this paper we look at a well known linear inverse problem that is one of the mathematical cornerstones of the compressed sensing field. In seminal works \cite{CRT,DOnoho06CS} $\ell_1$ optimization and its success when used for recovering sparse solutions of linear inverse problems was considered. Moreover, \cite{CRT,DOnoho06CS} established for the first time in a statistical context that an unknown vector of linear sparsity can be recovered as a known existing solution of an under-determined linear system through $\ell_1$ optimization. In \cite{DonohoPol,DonohoUnsigned} (and later in \cite{StojnicCSetam09,StojnicUpper10}) the precise values of the linear proportionality were established as well. While the typical $\ell_1$ optimization behavior has been essentially settled through the work of \cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}, we in this paper look at possible upgrades of $\ell_1$ optimization. Namely, we look at a couple of algorithms that turn out to be capable of recovering a substantially higher sparsity than the $\ell_1$. However, these algorithms assume a bit of "feedback" to be able to work at full strength. This in turn then translates the original problem of improving upon $\ell_1$ to designing algorithms that would be able to provide output needed to feed the $\ell_1$ upgrades considered in this papers.