Partial gradient optimal thresholding algorithms for a class of sparse optimization problems

Abstract

The optimization problems with a sparsity constraint is a class of important global optimization problems. A typical type of thresholding algorithms for solving such a problem adopts the traditional full steepest descent direction or Newton-like direction as a search direction to generate an iterate on which a certain thresholding is performed. Traditional hard thresholding discards a large part of a vector, and thus some important information contained in a dense vector has been lost in such a thresholding process. Recent study (Zhao in SIAM J Optim 30(1): 31–55, 2020) shows that the hard thresholding should be applied to a compressible vector instead of a dense vector to avoid a big loss of information. On the other hand, the optimal k-thresholding as a novel thresholding technique may overcome the intrinsic drawback of hard thresholding, and performs thresholding and objective function minimization simultaneously. This motivates us to propose the so-called partial gradient optimal thresholding (PGOT) method and its relaxed versions in this paper. The PGOT is an integration of the partial gradient and the optimal k-thresholding technique. The solution error bound and convergence for the proposed algorithms have been established in this paper under suitable conditions. Application of our results to the sparse optimization problems arising from signal recovery is also discussed. Experiment results from synthetic data indicate that the proposed algorithm is efficient and comparable to several existing algorithms.

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

“…i.e., b + ηβ < 1. Therefore, applying Lemma 3.1 to the recursive relation (26), we immediately conclude that τ < 1 and the desired estimation (19) holds.…”

“…The thresholding technique was introduced by Donoho and Johnstone [13]. At present, there are three main classes of thresholding algorithms: hard thresholding [2,3,4,5,6,8,16,22,23], soft thresholding [10,12,14], and optimal thresholding [26,38,40]. A huge amount of work has been carried out for the class of hard thresholding algorithms.…”

Natural Thresholding Algorithms for Signal Recovery With Sparsity

“…If (x S − u (p) ) S * ∪S 2 2 = 0, then all terms on the right-hand of ( 35) are zero, and hence (36) and (37) are still valid in this case. Substituting (36) and (37) By (36) and ( 38), the inequality above can be written as Denote by ω = ( √ 5 + 1)/2. Merging the above relation with (34), we obtain…”

“…If (x S − u (p) ) S * ∪S 2 2 = 0, then all terms on the right-hand of ( 35) are zero, and hence (36) and (37) are still valid in this case. Substituting (36) and (37) Denote by ω = ( √ 5 + 1)/2. Merging the above relation with (34), we obtain…”

“…Based on the OT concept, the relaxed optimal k-thresholding (ROT) algorithm and their variants were also developed in [34]. A further analysis of these algorithms was performed in [35], and the algorithm using both OT technique and truncated gradient descent direction was studied in [37]. Compared to existing hard-thresholding algorithms, the strength of OT-type methods lies in their observed robustness for sparse signal recovery and steadiness in objective reduction in the course of iterations.…”