Heavy-Ball-Based Hard Thresholding Algorithms for Sparse Signal Recovery

TL;DR

This paper introduces heavy-ball-based hard thresholding algorithms for sparse signal recovery, demonstrating improved convergence and efficiency under certain conditions, with empirical validation showing competitive performance and reduced recovery time.

Contribution

The paper proposes novel heavy-ball-based hard thresholding algorithms with theoretical recovery guarantees and demonstrates their efficiency and effectiveness through empirical comparisons.

Findings

Successfully recover sparse signals under specified RIP conditions.

Finite convergence and stability of the algorithms are established.

Empirical results show comparable or better performance with less recovery time.

Abstract

The hard thresholding technique plays a vital role in the development of algorithms for sparse signal recovery. By merging this technique and heavy-ball acceleration method which is a multi-step extension of the traditional gradient descent method, we propose the so-called heavy-ball-based hard thresholding (HBHT) and heavy-ball-based hard thresholding pursuit (HBHTP) algorithms for signal recovery. It turns out that the HBHT and HBHTP can successfully recover a $k$-sparse signal if the restricted isometry constant of the measurement matrix satisfies $\delta_{3k}<0.618 $ and $\delta_{3k}<0.577,$ respectively. The guaranteed success of HBHT and HBHTP is also shown under the conditions $\delta_{2k}<0.356$ and $\delta_{2k}<0.377,$ respectively. Moreover, the finite convergence and stability of the two algorithms are also established in this paper. Simulations on random problem instances are performed to compare the performance of the proposed algorithms and several existing ones. Empirical results indicate that the HBHTP performs very comparably to a few existing algorithms and it takes less average time to achieve the signal recovery than these existing methods.