Sparse matrices for weighted sparse recovery

TL;DR

This paper introduces new theoretical guarantees for weighted sparse recovery using sparse matrices derived from expander graphs, demonstrating improved sample complexity and computational efficiency.

Contribution

It provides the first sparse recovery guarantees for weighted  minimization with sparse random matrices and weighted sparse signals, extending existing theories.

Findings

Sparse matrices enable fast recovery with better computational complexity.

Sample complexity is linear in weighted sparsity and can be smaller than standard methods.

Numerical experiments support theoretical results.

Abstract

We derived the first sparse recovery guarantees for weighted $\ell_1$ minimization with sparse random matrices and the class of weighted sparse signals, using a weighted versions of the null space property to derive these guarantees. These sparse matrices from expender graphs can be applied very fast and have other better computational complexities than their dense counterparts. In addition we show that, using such sparse matrices, weighted sparse recovery with weighted $\ell_1$ minimization leads to sample complexities that are linear in the weighted sparsity of the signal and these sampling rates can be smaller than those of standard sparse recovery. Moreover, these results reduce to known results in standard sparse recovery and sparse recovery with prior information and the results are supported by numerical experiments.