Random linear under-determined systems with block-sparse solutions -- asymptotics, large deviations, and finite dimensions

TL;DR

This paper analyzes random under-determined linear systems with block-sparse solutions, providing asymptotic, large deviation, and finite-dimensional insights into the performance of $ ext{l}_1$ and $ ext{l}_2/ ext{l}_1$ optimization techniques.

Contribution

It extends previous asymptotic analyses to include novel finite-dimensional considerations for block-sparse systems.

Findings

Characterized phase transition thresholds for block-sparse recovery.

Derived large deviation bounds for solution success probabilities.

Provided finite-dimensional performance estimates.

Abstract

In this paper we consider random linear under-determined systems with block-sparse solutions. A standard subvariant of such systems, namely, precisely the same type of systems without additional block structuring requirement, gained a lot of popularity over the last decade. This is of course in first place due to the success in mathematical characterization of an $\ell_1$ optimization technique typically used for solving such systems, initially achieved in \cite{CRT,DOnoho06CS} and later on perfected in \cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}. The success that we achieved in \cite{StojnicCSetam09,StojnicUpper10} characterizing the standard sparse solutions systems, we were then able to replicate in a sequence of papers \cite{StojnicCSetamBlock09,StojnicUpperBlock10,StojnicICASSP09block,StojnicJSTSP09} where instead of the standard $\ell_1$ optimization we utilized its an $\ell_2/\ell_1$ variant as a better fit for systems with block-sparse solutions. All of these results finally settled the so-called threshold/phase transitions phenomena (which naturally assume the asymptotic/large dimensional scenario). Here, in addition to a few novel asymptotic considerations, we also try to raise the level a bit, step a bit away from the asymptotics, and consider the finite dimensions scenarios as well.