Applications of Mathematics, Vol. 68, No. 5, pp. 623-642, 2023

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

Jinyao Ma, Haibin Zhang, Shanshan Yang, Jiaojiao Jiang

Received September 13, 2022.   Published online March 21, 2023.

Abstract:  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 $\delta_{3s-2k}<\frac1{\sqrt{32}}\approx0.1768$ to $\delta_{3s-2k}<\frac{\sqrt5-1}4\approx0.309$, where $\delta_{3s-2k}$ is the restricted isometric constant of the measurement matrix. We also present the conditions for stable reconstruction using the ${\rm IHT}^{\mu}$-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 ${\rm IHT}^{\mu}$-PKS algorithm. Experimental results on classification problems show that ${\rm IHT}^{\mu}$-PKS outperforms other approaches to computing the sparse LS-SVM classifier.
Keywords:  iterative hard thresholding; signal reconstruction; classification problem; least squares support vector machine
Classification MSC:  34B16, 34C25, 90C31

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Affiliations:   Jinyao Ma, Haibin Zhang, Shanshan Yang, Beijing Institute for Scientific and Engineering Computing, Faculty of Science, Beijing University of Technology, No. 100 Ping Le Yuan, Chaoyang District, Beijing 100124, P. R. China, e-mail:,,; Jiaojiao Jiang (corresponding author), School of Computer Science and Engineering, University of New South Wales, High St, Sydney NSW 2052, Australia, e-mail:

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