Volume 41, Issue 1
Reconstruction of Sparse Polynomials via Quasi-Orthogonal Matching Pursuit Method

J. Comp. Math., 41 (2023), pp. 18-38.

Published online: 2022-11

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• Abstract

In this paper, we propose a Quasi-Orthogonal Matching Pursuit (QOMP) algorithm for constructing a sparse approximation of functions in terms of expansion by orthonormal polynomials. For the two kinds of sampled data, data with noises and without noises, we apply the mutual coherence of measurement matrix to establish the convergence of  the QOMP algorithm which can reconstruct $s$-sparse Legendre polynomials, Chebyshev polynomials and  trigonometric polynomials in $s$ step iterations. The results are also extended to general bounded orthogonal system including tensor product of these three univariate orthogonal polynomials. Finally, numerical experiments will be presented to verify the effectiveness of the QOMP method.

41A10, 41A05, 65D15

fengrz@buaa.edu.cn (Renzhong Feng)

huangaitong@buaa.edu.cn (Aitong Huang)

mjlai@uga.edu (Ming-Jun Lai)

Zhaiming.Shen@uga.edu (Zhaiming Shen)

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@Article{JCM-41-18, author = {Feng , RenzhongHuang , AitongLai , Ming-Jun and Shen , Zhaiming}, title = {Reconstruction of Sparse Polynomials via Quasi-Orthogonal Matching Pursuit Method}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {41}, number = {1}, pages = {18--38}, abstract = {

In this paper, we propose a Quasi-Orthogonal Matching Pursuit (QOMP) algorithm for constructing a sparse approximation of functions in terms of expansion by orthonormal polynomials. For the two kinds of sampled data, data with noises and without noises, we apply the mutual coherence of measurement matrix to establish the convergence of  the QOMP algorithm which can reconstruct $s$-sparse Legendre polynomials, Chebyshev polynomials and  trigonometric polynomials in $s$ step iterations. The results are also extended to general bounded orthogonal system including tensor product of these three univariate orthogonal polynomials. Finally, numerical experiments will be presented to verify the effectiveness of the QOMP method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2104-m2020-0250}, url = {http://global-sci.org/intro/article_detail/jcm/21168.html} }
TY - JOUR T1 - Reconstruction of Sparse Polynomials via Quasi-Orthogonal Matching Pursuit Method AU - Feng , Renzhong AU - Huang , Aitong AU - Lai , Ming-Jun AU - Shen , Zhaiming JO - Journal of Computational Mathematics VL - 1 SP - 18 EP - 38 PY - 2022 DA - 2022/11 SN - 41 DO - http://doi.org/10.4208/jcm.2104-m2020-0250 UR - https://global-sci.org/intro/article_detail/jcm/21168.html KW - Reconstruction of sparse polynomial, Compressive sensing, Mutual coherence, Quasi-orthogonal matching pursuit algorithm. AB -

In this paper, we propose a Quasi-Orthogonal Matching Pursuit (QOMP) algorithm for constructing a sparse approximation of functions in terms of expansion by orthonormal polynomials. For the two kinds of sampled data, data with noises and without noises, we apply the mutual coherence of measurement matrix to establish the convergence of  the QOMP algorithm which can reconstruct $s$-sparse Legendre polynomials, Chebyshev polynomials and  trigonometric polynomials in $s$ step iterations. The results are also extended to general bounded orthogonal system including tensor product of these three univariate orthogonal polynomials. Finally, numerical experiments will be presented to verify the effectiveness of the QOMP method.

Renzhong Feng, Aitong Huang, Ming-Jun Lai & Zhaiming Shen. (2022). Reconstruction of Sparse Polynomials via Quasi-Orthogonal Matching Pursuit Method. Journal of Computational Mathematics. 41 (1). 18-38. doi:10.4208/jcm.2104-m2020-0250
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