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Linear inverse problems arise in diverse engineering fields especially insignal and image reconstruction. The development of computational methods forlinear inverse problems with sparsity is one of the recent trends in thisfield. The so-called optimal $k$-thresholding is a newly introduced method forsparse optimization and linear inverse problems. Compared to othersparsity-aware algorithms, the advantage of optimal $k$-thresholding methodlies in that it performs thresholding and error metric reduction simultaneouslyand thus works stably and robustly for solving medium-sized linear inverseproblems. However, the runtime of this method is generally high when the sizeof the problem is large. The purpose of this paper is to propose anacceleration strategy for this method. Specifically, we propose aheavy-ball-based optimal $k$-thresholding (HBOT) algorithm and its relaxedvariants for sparse linear inverse problems. The convergence of thesealgorithms is shown under the restricted isometry property. In addition, thenumerical performance of the heavy-ball-based relaxed optimal $k$-thresholdingpursuit (HBROTP) has been evaluated, and simulations indicate that HBROTPadmits robustness for signal and image reconstruction even in noisyenvironments.

Heavy-ball-based optimal thresholding algorithms for sparse linear inverse problems