A tight bound of modified iterative hard thresholding algorithm for compressed sensing

• Published in: Applications of mathematics (Prague) Vol. 68; no. 5; pp. 623 - 642
• Main Authors: Ma, Jinyao, Zhang, Haibin, Yang, Shanshan, Jiang, Jiaojiao
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2023; Springer Nature B.V
• Subjects: Algorithms; Analysis; Applications of Mathematics; Classical and Continuum Physics; Classification; Compressibility; Iterative methods; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Optimization; Sparsity; Support vector machines; Theoretical; classification problem; 90C31; 34B16; 34C25; iterative hard thresholding; signal reconstruction; least squares support vector machine
• ISSN: 0862-7940; 1572-9109
• DOI: 10.21136/AM.2023.0221-22

We provide a theoretical study of the iterative hard thresholding with partially known support set (IHT-PKS) algorithm when used to solve the compressed sensing recovery problem. Recent work has shown that IHT-PKS performs better than the traditional IHT in reconstructing sparse or compressible signals. However, less work has been done on analyzing the performance guarantees of IHT-PKS. In this paper, we improve the current RIP-based bound of IHT-PKS algorithm from δ 3 s − 2 k < 1 32 ≈ 0.1768 to δ 3 s − 2 k < 5 − 1 4 , where δ 3 s −2 k is the restricted isometric constant of the measurement matrix. We also present the conditions for stable reconstruction using the IHT μ -PKS algorithm which is a general form of IHT-PKS. We further apply the algorithm on Least Squares Support Vector Machines (LS-SVM), which is one of the most popular tools for regression and classification learning but confronts the loss of sparsity problem. After the sparse representation of LS-SVM is presented by compressed sensing, we exploit the support of bias term in the LS-SVM model with the IHT μ -PKS algorithm. Experimental results on classification problems show that IHT μ -PKS outperforms other approaches to computing the sparse LS-SVM classifier.