Various thresholds for $\ell_1$-optimization in compressed sensing
Abstract
Recently, \cite{CRT,DonohoPol} theoretically analyzed the success of a polynomial -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 -optimization succeeds in solving the system. In this paper, we provide an alternative performance analysis of -optimization and obtain the proportionality constants that in certain cases match or improve on the best currently known ones from \cite{DonohoPol,DT}.
Cite
@article{arxiv.0907.3666,
title = {Various thresholds for $\ell_1$-optimization in compressed sensing},
author = {Mihailo Stojnic},
journal= {arXiv preprint arXiv:0907.3666},
year = {2009}
}