Weighted $\ell_1$-Minimization for Sparse Recovery under Arbitrary Prior Information

TL;DR

This paper extends weighted $ ext{l}_1$-minimization techniques for sparse signal recovery by allowing multiple, arbitrarily assigned weights based on support estimates, improving reconstruction guarantees and performance.

Contribution

It introduces a theoretical analysis for recovery guarantees with multiple weights in weighted $ ext{l}_1$-minimization, generalizing previous single-weight models.

Findings

Non-uniform weights improve reconstruction accuracy.

Theoretical recovery guarantees extend to multiple support estimates.

Numerical experiments validate benefits with synthetic and real data.

Abstract

Weighted $\ell_1$-minimization has been studied as a technique for the reconstruction of a sparse signal from compressively sampled measurements when prior information about the signal, in the form of a support estimate, is available. In this work, we study the recovery conditions and the associated recovery guarantees of weighted $\ell_1$-minimization when arbitrarily many distinct weights are permitted. For example, such a setup might be used when one has multiple estimates for the support of a signal, and these estimates have varying degrees of accuracy. Our analysis yields an extension to existing works that assume only a single support estimate set upon which a constant weight is applied. We include numerical experiments, with both synthetic signals and real video data, that demonstrate the benefits of allowing non-uniform weights in the reconstruction procedure.