Improving the Thresholds of Sparse Recovery: An Analysis of a Two-Step Reweighted Basis Pursuit Algorithm

TL;DR

This paper proposes a two-step reweighted $ ext{l}_1$ recovery algorithm that improves the weak recovery thresholds for sparse signals, especially with certain distributions, verified through theoretical analysis and numerical simulations.

Contribution

Introduction of a novel two-step reweighted $ ext{l}_1$ algorithm that strictly enhances weak recovery thresholds for specific signal distributions.

Findings

Strict improvement in weak recovery thresholds for certain distributions.

Threshold improvement depends on distribution behavior at the origin.

Numerical results show over 20% threshold enhancement for Gaussian nonzero entries.

Abstract

It is well known that $\ell_1$ minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity, as a function of the ratio between the system dimensions, so that with high probability almost all sparse signals can be recovered from i.i.d. Gaussian measurements, have been computed and are referred to as "weak thresholds" \cite{D}. In this paper, we introduce a reweighted $\ell_1$ recovery algorithm composed of two steps: a standard $\ell_1$ minimization step to identify a set of entries where the signal is likely to reside, and a weighted $\ell_1$ minimization step where entries outside this set are penalized. For signals where the non-sparse component entries are independent and identically drawn from certain classes of distributions, (including most well known continuous distributions), we prove a \emph{strict} improvement in the weak recovery threshold. Our analysis suggests that the level of improvement in the weak threshold depends on the behavior of the distribution at the origin. Numerical simulations verify the distribution dependence of the threshold improvement very well, and suggest that in the case of i.i.d. Gaussian nonzero entries, the improvement can be quite impressive---over 20% in the example we consider.