The high-order block RIP for non-convex block-sparse compressed sensing

TL;DR

This paper introduces high-order block RIP conditions for exact and stable recovery of non-convex block-sparse signals using mixed $l_2/l_p$ minimization, supported by theoretical bounds and numerical validation.

Contribution

It establishes new high-order block RIP conditions for non-convex block-sparse compressed sensing and provides measurement bounds ensuring these conditions hold with high probability.

Findings

Exact recovery guaranteed under new block RIP conditions.

Stable recovery in noisy scenarios demonstrated.

Numerical experiments validate the proposed method's effectiveness.

Abstract

This paper concentrates on the recovery of block-sparse signals, which is not only sparse but also nonzero elements are arrayed into some blocks (clusters) rather than being arbitrary distributed all over the vector, from linear measurements. We establish high-order sufficient conditions based on block RIP to ensure the exact recovery of every block $s$-sparse signal in the noiseless case via mixed $l_2/l_p$ minimization method, and the stable and robust recovery in the case that signals are not accurately block-sparse in the presence of noise. Additionally, a lower bound on necessary number of random Gaussian measurements is gained for the condition to be true with overwhelming probability. Furthermore, the numerical experiments conducted demonstrate the performance of the proposed algorithm.