Various thresholds for $\ell_1$-optimization in compressed sensing

TL;DR

This paper analyzes the success thresholds of $ ext{l}_1$-optimization in compressed sensing, providing alternative bounds that match or improve existing theoretical results in high-dimensional sparse recovery.

Contribution

It offers an alternative analysis method for $ ext{l}_1$-optimization success thresholds, deriving constants that match or surpass previous bounds.

Findings

Derived new proportionality constants for $ ext{l}_1$-optimization success.

Provided bounds that match or improve upon existing results.

Enhanced understanding of sparse recovery thresholds in high dimensions.

Abstract

Recently, \cite{CRT,DonohoPol} theoretically analyzed the success of a polynomial $\ell_1$-optimization algorithm in solving an under-determined system of linear equations. In a large dimensional and statistical context \cite{CRT,DonohoPol} proved that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that $\ell_1$-optimization succeeds in solving the system. In this paper, we provide an alternative performance analysis of $\ell_1$-optimization and obtain the proportionality constants that in certain cases match or improve on the best currently known ones from \cite{DonohoPol,DT}.