Beyond $\ell_1$-norm minimization for sparse signal recovery

TL;DR

This paper introduces WSPGL1, an algorithm that improves sparse signal recovery over traditional $ ext{l}_1$ minimization methods like BPDN, achieving better accuracy with similar computational costs by using weighted LASSO subproblems and support updates.

Contribution

The paper presents WSPGL1, a novel modification of SPGL1 that incorporates weighted LASSO and support updates, outperforming BPDN in sparse recovery without extra computational cost.

Findings

WSPGL1 outperforms BPDN in sparse recovery accuracy.

WSPGL1 approaches the performance of IRWL1 in simulations.

WSPGL1 maintains the computational cost of a single BPDN solve.

Abstract

Sparse signal recovery has been dominated by the basis pursuit denoise (BPDN) problem formulation for over a decade. In this paper, we propose an algorithm that outperforms BPDN in finding sparse solutions to underdetermined linear systems of equations at no additional computational cost. Our algorithm, called WSPGL1, is a modification of the spectral projected gradient for $\ell_1$ minimization (SPGL1) algorithm in which the sequence of LASSO subproblems are replaced by a sequence of weighted LASSO subproblems with constant weights applied to a support estimate. The support estimate is derived from the data and is updated at every iteration. The algorithm also modifies the Pareto curve at every iteration to reflect the new weighted $\ell_1$ minimization problem that is being solved. We demonstrate through extensive simulations that the sparse recovery performance of our algorithm is superior to that of $\ell_1$ minimization and approaches the recovery performance of iterative re-weighted $\ell_1$ (IRWL1) minimization of Cand{\`e}s, Wakin, and Boyd, although it does not match it in general. Moreover, our algorithm has the computational cost of a single BPDN problem.