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

Haifeng Li, Jing Zhang, Jinming Wen & Dongfang Li

DOI: 10.4208/jcm.2104-m2020-0093

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

Published online: 2022-11

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  • 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.

  • Keywords

Sparse signal recovery, Orthogonal least squares (OLS), Restricted isometry property (RIP).

  • AMS Subject Headings

94A12, 65F22, 65J22

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

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.

Li , HaifengZhang , JingWen , Jinming and Li , Dongfang. (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
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