Error bounds of block sparse signal recovery based on q-ratio block constrained minimal singular values

Jianfeng Wang,Zhiyong Zhou,Jun Yu

doi:10.1186/s13634-019-0653-1

Abstract

In this paper, we introduce the q-ratio block constrained minimal singular values (BCMSV) as a new measure of measurement matrix in compressive sensing of block sparse/compressive signals and present an algorithm for computing this new measure. Both the mixed ℓ2/ℓq and the mixed ℓ2/ℓ1 norms of the reconstruction errors for stable and robust recovery using block basis pursuit (BBP), the block Dantzig selector (BDS), and the group lasso in terms of the q-ratio BCMSV are investigated. We establish a sufficient condition based on the q-ratio block sparsity for the exact recovery from the noise-free BBP and developed a convex-concave procedure to solve the corresponding non-convex problem in the condition. Furthermore, we prove that for sub-Gaussian random matrices, the q-ratio BCMSV is bounded away from zero with high probability when the number of measurements is reasonably large. Numerical experiments are implemented to illustrate the theoretical results. In addition, we demonstrate that the q-ratio BCMSV-based error bounds are tighter than the block-restricted isotropic constant-based bounds.

Highlights

Compressive sensing (CS) [1, 2] aims to recover an unknown sparse signal x ∈ RN from m noisy measurements y ∈ Rm: y = Ax +, (1)where A ∈ Rm×N is a measurement matrix with m N, and ∈ Rm is additive noise such that 2 ≤ ζ for some ζ ≥ 0
We focus on three renowned convex relaxation algorithms for block sparse signal recovery from (1): the block basis pursuit (BBP), the block Dantzig selector (BDS), and the group lasso
3 Numerical experiments and results we introduce a convex-concave method to solve the sufficient condition (8) so as to achieve the maximal block sparsity k and present an algorithm to compute the q-ratio block constrained minimal singular values (BCMSV)

Summary

Introduction

Compressive sensing (CS) [1, 2] aims to recover an unknown sparse signal x ∈ RN from m noisy measurements y ∈ Rm:. This work includes four main contributions to block sparse signal recovery in compressive sensing: (i) we establish a sufficient condition based on the q-ratio block sparsity for the exact recovery from the noise-free block BP (BBP) and develop a convex-concave procedure to solve the corresponding non-convex problem in the condition; (ii) we introduce the q-ratio BCMSV and derive both the mixed 2/ q and the mixed 2/ 1 norms of the reconstruction errors for stable and robust recovery using the BBP, the block DS (BDS), and the group lasso in terms of the q-ratio BCMSV; (iii) we prove that for sub-Gaussian random matrices, the q-ratio BCMSV is bounded away from zero with high probability when the number of measurements is reasonably large; and (iv) we present an algorithm to compute the q-ratio BCMSV for an arbitrary measurement matrix and investigate its properties.

Theoretical methodology

Random matrices

Comparing error bounds

Conclusion and discussion