Recovery of signals by a weighted $\ell_2/\ell_1$ minimization under arbitrary prior support information

TL;DR

This paper proposes a weighted $\,\ell_2/\ell_1$ minimization method for recovering block sparse signals using arbitrary prior support information, demonstrating improved recovery conditions and error bounds over traditional methods.

Contribution

It introduces a novel weighted minimization approach that leverages prior support information, providing weaker recovery conditions and better error bounds for block sparse signals.

Findings

Weighted $\ell_2/\ell_1$ minimization outperforms standard methods with prior support info.

Recovery conditions are weaker when prior support estimate accuracy exceeds 50%.

Numerical experiments confirm improved recovery performance with the proposed method.

Abstract

In this paper, we introduce a weighted $\ell_2/\ell_1$ minimization to recover block sparse signals with arbitrary prior support information. When partial prior support information is available, a sufficient condition based on the high order block RIP is derived to guarantee stable and robust recovery of block sparse signals via the weighted $\ell_2/\ell_1$ minimization. We then show if the accuracy of arbitrary prior block support estimate is at least $50\%$, the sufficient recovery condition by the weighted $\ell_2/\ell_{1}$ minimization is weaker than that by the $\ell_2/\ell_{1}$ minimization, and the weighted $\ell_2/\ell_{1}$ minimization provides better upper bounds on the recovery error in terms of the measurement noise and the compressibility of the signal. Moreover, we illustrate the advantages of the weighted $\ell_2/\ell_1$ minimization approach in the recovery performance of block sparse signals under uniform and non-uniform prior information by extensive numerical experiments. The significance of the results lies in the facts that making explicit use of block sparsity and partial support information of block sparse signals can achieve better recovery performance than handling the signals as being in the conventional sense, thereby ignoring the additional structure and prior support information in the problem.