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Perturbation Analysis of Orthogonal Least Squares

Part of: Algorithms - Computer Science Numerical analysis in abstract spaces Numerical approximation and computational geometry

Published online by Cambridge University Press:  22 March 2019

Pengbo Geng ,
Wengu Chen  and
Huanmin Ge
Pengbo Geng
Affiliation:
Graduate School, China Academy of Engineering Physics, Beijing 100088, China Email: gpb1990@126.com
Wengu Chen
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China Email: chenwg@iapcm.ac.cn
Huanmin Ge
Affiliation:
Sports Engineering College, Beijing Sport University, Beijing, 100088, China Email: gehuanmin@163.com
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Abstract

The Orthogonal Least Squares (OLS) algorithm is an efficient sparse recovery algorithm that has received much attention in recent years. On one hand, this paper considers that the OLS algorithm recovers the supports of sparse signals in the noisy case. We show that the OLS algorithm exactly recovers the support of $K$-sparse signal $\boldsymbol{x}$ from $\boldsymbol{y}=\boldsymbol{\unicode[STIX]{x1D6F7}}\boldsymbol{x}+\boldsymbol{e}$ in $K$ iterations, provided that the sensing matrix $\boldsymbol{\unicode[STIX]{x1D6F7}}$ satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) $\unicode[STIX]{x1D6FF}_{K+1}<1/\sqrt{K+1}$, and the minimum magnitude of the nonzero elements of $\boldsymbol{x}$ satisfies some constraint. On the other hand, this paper demonstrates that the OLS algorithm exactly recovers the support of the best $K$-term approximation of an almost sparse signal $\boldsymbol{x}$ in the general perturbations case, which means both $\boldsymbol{y}$ and $\boldsymbol{\unicode[STIX]{x1D6F7}}$ are perturbed. We show that the support of the best $K$-term approximation of $\boldsymbol{x}$ can be recovered under reasonable conditions based on the restricted isometry property (RIP).

Keywords

orthogonal least square (OLS)sparse signalrestricted isometry property (RIP)general perturbation

MSC classification

Primary: 65D15: Algorithms for functional approximation
Secondary: 65J22: Inverse problems 68W40: Analysis of algorithms
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 780 - 797
Copyright
© Canadian Mathematical Society 2019 

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Footnotes

W. Chen is the corresponding author. This work was supported by the NSF of China (No. 11871109) and NSAF (Grant No. U1830107).

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