Performance analysis of tail-minimization and the linear rate of convergence of a proximal algorithm for sparse signal recovery

TL;DR

This paper analyzes the recovery error bounds of tail-minimization and the convergence rate of a proximal algorithm for sparse signal recovery, showing improved bounds and relaxed RIP conditions, with an efficient algorithm demonstrating superior performance.

Contribution

It provides new error bounds for tail-minimization, relaxes RIP conditions, and introduces a proximal algorithm with proven linear convergence for sparse signal recovery.

Findings

Error bounds are improved over existing results.

RIP condition becomes more relaxed, approaching 1.

Proposed algorithm significantly outperforms state-of-the-art methods.

Abstract

Recovery error bounds of tail-minimization and the rate of convergence of an efficient proximal alternating algorithm for sparse signal recovery are considered in this article. Tail-minimization focuses on minimizing the energy in the complement $T^c$ of an estimated support $T$. Under the restricted isometry property (RIP) condition, we prove that tail-$\ell_1$ minimization can exactly recover sparse signals in the noiseless case for a given $T$. In the noisy case, two recovery results for the tail-$\ell_1$ minimization and the tail-lasso models are established. Error bounds are improved over existing results. Additionally, we show that the RIP condition becomes surprisingly relaxed, allowing the RIP constant to approach $1$ as the estimation $T$ closely approximates the true support $S$. Finally, an efficient proximal alternating minimization algorithm is introduced for solving the tail-lasso problem using Hadamard product parametrization. The linear rate of convergence is established using the Kurdyka-{\L}ojasiewicz inequality. Numerical results demonstrate that the proposed algorithm significantly improves signal recovery performance compared to state-of-the-art techniques.