Performance Analysis of Sparse Recovery Based on Constrained Minimal Singular Values

TL;DR

This paper investigates the stability of sparse signal reconstruction using the $ ext{l}_1$-constrained minimal singular value ($ ext{l}_1$-CMSV), providing efficient algorithms for verification and showing its effectiveness in predicting recovery performance.

Contribution

It introduces new algorithms to verify the $ ext{l}_1$-CMSV, demonstrating its role in sparse recovery stability and offering a less complex alternative to Restricted Isometry Constant analysis.

Findings

Algorithms efficiently verify $ ext{l}_1$-CMSV

Subgaussian matrices have bounded $ ext{l}_1$-CMSV with high probability

$ ext{l}_1$-CMSV effectively predicts sparse recovery performance

Abstract

The stability of sparse signal reconstruction is investigated in this paper. We design efficient algorithms to verify the sufficient condition for unique $\ell_1$ sparse recovery. One of our algorithm produces comparable results with the state-of-the-art technique and performs orders of magnitude faster. We show that the $\ell_1$-constrained minimal singular value ($\ell_1$-CMSV) of the measurement matrix determines, in a very concise manner, the recovery performance of $\ell_1$-based algorithms such as the Basis Pursuit, the Dantzig selector, and the LASSO estimator. Compared with performance analysis involving the Restricted Isometry Constant, the arguments in this paper are much less complicated and provide more intuition on the stability of sparse signal recovery. We show also that, with high probability, the subgaussian ensemble generates measurement matrices with $\ell_1$-CMSVs bounded away from zero, as long as the number of measurements is relatively large. To compute the $\ell_1$-CMSV and its lower bound, we design two algorithms based on the interior point algorithm and the semi-definite relaxation.