Compressed sensing of block-sparse positive vectors

TL;DR

This paper extends the analysis of compressed sensing to block-sparse positive vectors, proposing an adjusted $ ext{l}_2/ ext{l}_1$-optimization method and providing a precise performance characterization for this class.

Contribution

It introduces a modified $ ext{l}_2/ ext{l}_1$-optimization tailored for block-sparse positive vectors and offers a detailed performance analysis of this approach.

Findings

The adjusted method effectively recovers block-sparse positive vectors.

Precise thresholds for successful recovery are established.

The approach outperforms standard methods for this specific class.

Abstract

In this paper we revisit one of the classical problems of compressed sensing. Namely, we consider linear under-determined systems with sparse solutions. A substantial success in mathematical characterization of an $\ell_1$ optimization technique typically used for solving such systems has been achieved during the last decade. Seminal works \cite{CRT,DOnoho06CS} showed that the $\ell_1$ can recover a so-called linear sparsity (i.e. solve systems even when the solution has a sparsity linearly proportional to the length of the unknown vector). Later considerations \cite{DonohoPol,DonohoUnsigned} (as well as our own ones \cite{StojnicCSetam09,StojnicUpper10}) provided the precise characterization of this linearity. In this paper we consider the so-called structured version of the above sparsity driven problem. Namely, we view a special case of sparse solutions, the so-called block-sparse solutions. Typically one employs $\ell_2/\ell_1$-optimization as a variant of the standard $\ell_1$ to handle block-sparse case of sparse solution systems. We considered systems with block-sparse solutions in a series of work \cite{StojnicCSetamBlock09,StojnicUpperBlock10,StojnicICASSP09block,StojnicJSTSP09} where we were able to provide precise performance characterizations if the $\ell_2/\ell_1$-optimization similar to those obtained for the standard $\ell_1$ optimization in \cite{StojnicCSetam09,StojnicUpper10}. Here we look at a similar class of systems where on top of being block-sparse the unknown vectors are also known to have components of the same sign. In this paper we slightly adjust $\ell_2/\ell_1$-optimization to account for the known signs and provide a precise performance characterization of such an adjustment.