A problem dependent analysis of SOCP algorithms in noisy compressed sensing

Abstract: Under-determined systems of linear equations with sparse solutions have been the subject of an extensive research in last several years above all due to results of \cite{CRT,CanRomTao06,DonohoPol}. In this paper we will consider \emph{noisy} under-determined linear systems. In a breakthrough \cite{CanRomTao06} it was established that in \emph{noisy} systems for any linear level of under-determinedness there is a linear sparsity that can be \emph{approximately} recovered through an SOCP (second order cone programming) optimization algorithm so that the approximate solution vector is (in an $\ell_2$-norm sense) guaranteed to be no further from the sparse unknown vector than a constant times the noise. In our recent work \cite{StojnicGenSocp10} we established an alternative framework that can be used for statistical performance analysis of the SOCP algorithms. To demonstrate how the framework works we then showed in \cite{StojnicGenSocp10} how one can use it to precisely characterize the \emph{generic} (worst-case) performance of the SOCP. In this paper we present a different set of results that can be obtained through the framework of \cite{StojnicGenSocp10}. The results will relate to \emph{problem dependent} performance analysis of SOCP's. We will consider specific types of unknown sparse vectors and characterize the SOCP performance when used for recovery of such vectors. We will also show that our theoretical predictions are in a solid agreement with the results one can get through numerical simulations.