Sparse recovery based on q-ratio constrained minimal singular values

TL;DR

This paper introduces new verifiable conditions and performance bounds for sparse recovery algorithms, utilizing a novel family of quality measures for measurement matrices, with theoretical analysis and numerical validation.

Contribution

It proposes a new family of quality measures based on q-ratio constrained minimal singular values for analyzing sparse recovery performance.

Findings

Bounds are tighter than previous restricted isotropic constant analysis.

High probability bounds for subgaussian matrices are established.

Numerical experiments confirm theoretical results.

Abstract

We study verifiable sufficient conditions and computable performance bounds for sparse recovery algorithms such as the Basis Pursuit, the Dantzig selector and the Lasso estimator, in terms of a newly defined family of quality measures for the measurement matrices. With high probability, the developed measures for subgaussian random matrices are bounded away from zero as long as the number of measurements is reasonably large. Comparing to the restricted isotropic constant based performance analysis, the arguments in this paper are much more concise and the obtained bounds are tighter. Numerical experiments are presented to illustrate our theoretical results.