Sparse Recovery of Positive Signals with Minimal Expansion

TL;DR

This paper introduces a new method for recovering non-negative sparse signals using sparse measurement matrices based on expander graphs with minimal expansion, providing theoretical guarantees, a faster algorithm, and robustness to noise.

Contribution

We construct sparse measurement matrices with small expansion coefficients for non-negative sparse recovery, and establish necessary and sufficient conditions for successful $ ext{l}_1$ recovery, along with a faster alternative algorithm.

Findings

Recovery is possible with minimal expansion matrices.

The proposed method is robust to noise and approximate sparsity.

A new, faster recovery algorithm exploits graph expansion properties.

Abstract

We investigate the sparse recovery problem of reconstructing a high-dimensional non-negative sparse vector from lower dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes are crucial in applications, such as DNA microarrays and sensor networks, where dense measurements are not practically feasible. One possible construction uses the adjacency matrices of expander graphs, which often leads to recovery algorithms much more efficient than $\ell_1$ minimization. However, to date, constructions based on expanders have required very high expansion coefficients which can potentially make the construction of such graphs difficult and the size of the recoverable sets small. In this paper, we construct sparse measurement matrices for the recovery of non-negative vectors, using perturbations of the adjacency matrix of an expander graph with much smaller expansion coefficient. We present a necessary and sufficient condition for $\ell_1$ optimization to successfully recover the unknown vector and obtain expressions for the recovery threshold. For certain classes of measurement matrices, this necessary and sufficient condition is further equivalent to the existence of a "unique" vector in the constraint set, which opens the door to alternative algorithms to $\ell_1$ minimization. We further show that the minimal expansion we use is necessary for any graph for which sparse recovery is possible and that therefore our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much faster than $\ell_1$ optimization. Finally, we demonstrate through theoretical bounds, as well as simulation, that our method is robust to noise and approximate sparsity.