Volume 41, Issue 1
Required Number of Iterations for Sparse Signal Recovery via Orthogonal Least Squares

J. Comp. Math., 41 (2023), pp. 1-17.

Published online: 2022-11

Cited by

Export citation
• Abstract

In countless applications, we need to reconstruct  a $K$-sparse signal $\mathbf{x}\in\mathbb{R}^n$ from noisy measurements $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{v}$,  where $\mathbf{\Phi}\in\mathbb{R}^{m\times n}$ is a sensing matrix and $\mathbf{v}\in\mathbb{R}^m$ is a noise vector. Orthogonal least squares (OLS), which selects at each step the column that results in the most significant decrease in the residual power, is one of the most popular sparse recovery algorithms. In this paper,  we investigate the number of iterations required for recovering $\mathbf{x}$ with the OLS algorithm. We show that OLS provides a stable reconstruction of all $K$-sparse signals $\mathbf{x}$  in $\lceil2.8K\rceil$ iterations provided that $\mathbf{\Phi}$ satisfies the restricted isometry property (RIP). Our result provides a better recovery bound and fewer number of required iterations than those proposed by Foucart in 2013.

94A12, 65F22, 65J22

lihaifengxx@126.com (Haifeng Li)

a1293651766@126.com (Jing Zhang)

jinming.wen@mail.mcgill.ca (Jinming Wen)

dfli@mail.hust.edu.cn (Dongfang Li)

• BibTex
• RIS
• TXT
@Article{JCM-41-1, author = {Li , HaifengZhang , JingWen , Jinming and Li , Dongfang}, title = {Required Number of Iterations for Sparse Signal Recovery via Orthogonal Least Squares}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {41}, number = {1}, pages = {1--17}, abstract = {

In countless applications, we need to reconstruct  a $K$-sparse signal $\mathbf{x}\in\mathbb{R}^n$ from noisy measurements $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{v}$,  where $\mathbf{\Phi}\in\mathbb{R}^{m\times n}$ is a sensing matrix and $\mathbf{v}\in\mathbb{R}^m$ is a noise vector. Orthogonal least squares (OLS), which selects at each step the column that results in the most significant decrease in the residual power, is one of the most popular sparse recovery algorithms. In this paper,  we investigate the number of iterations required for recovering $\mathbf{x}$ with the OLS algorithm. We show that OLS provides a stable reconstruction of all $K$-sparse signals $\mathbf{x}$  in $\lceil2.8K\rceil$ iterations provided that $\mathbf{\Phi}$ satisfies the restricted isometry property (RIP). Our result provides a better recovery bound and fewer number of required iterations than those proposed by Foucart in 2013.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2104-m2020-0093}, url = {http://global-sci.org/intro/article_detail/jcm/21167.html} }
TY - JOUR T1 - Required Number of Iterations for Sparse Signal Recovery via Orthogonal Least Squares AU - Li , Haifeng AU - Zhang , Jing AU - Wen , Jinming AU - Li , Dongfang JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 17 PY - 2022 DA - 2022/11 SN - 41 DO - http://doi.org/10.4208/jcm.2104-m2020-0093 UR - https://global-sci.org/intro/article_detail/jcm/21167.html KW - Sparse signal recovery, Orthogonal least squares (OLS), Restricted isometry property (RIP). AB -

In countless applications, we need to reconstruct  a $K$-sparse signal $\mathbf{x}\in\mathbb{R}^n$ from noisy measurements $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{v}$,  where $\mathbf{\Phi}\in\mathbb{R}^{m\times n}$ is a sensing matrix and $\mathbf{v}\in\mathbb{R}^m$ is a noise vector. Orthogonal least squares (OLS), which selects at each step the column that results in the most significant decrease in the residual power, is one of the most popular sparse recovery algorithms. In this paper,  we investigate the number of iterations required for recovering $\mathbf{x}$ with the OLS algorithm. We show that OLS provides a stable reconstruction of all $K$-sparse signals $\mathbf{x}$  in $\lceil2.8K\rceil$ iterations provided that $\mathbf{\Phi}$ satisfies the restricted isometry property (RIP). Our result provides a better recovery bound and fewer number of required iterations than those proposed by Foucart in 2013.

Haifeng Li, Jing Zhang, Jinming Wen & Dongfang Li. (2022). Required Number of Iterations for Sparse Signal Recovery via Orthogonal Least Squares. Journal of Computational Mathematics. 41 (1). 1-17. doi:10.4208/jcm.2104-m2020-0093
Copy to clipboard
The citation has been copied to your clipboard