A sharp sufficient condition of block signal recovery via $l_2/l_1$-minimization

TL;DR

This paper establishes a precise condition based on the block restricted isometry property that guarantees accurate recovery of block sparse signals using $l_2/l_1$-minimization, even in noisy environments.

Contribution

It provides a sharp sufficient condition for block signal recovery via $l_2/l_1$-minimization, improving existing bounds and demonstrating the condition's optimality.

Findings

The condition guarantees stable recovery in noisy cases.

Exact reconstruction is possible in noise-free scenarios.

The condition is shown to be sharp through an example.

Abstract

This work gains a sharp sufficient condition on the block restricted isometry property for the recovery of sparse signal. Under the certain assumption, the signal with block structure can be stably recovered in the present of noisy case and the block sparse signal can be exactly reconstructed in the noise-free case. Besides, an example is proposed to exhibit the condition is sharp. As byproduct, when $t=1$, the result improves the bound of block restricted isometry constant $\delta_{s|\mathcal{I}}$ in Lin and Li (Acta Math. Sin. Engl. Ser. 29(7): 1401-1412, 2013).