Parametrized quasi-soft thresholding operator for compressed sensing and matrix completion
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Parametrized quasi-soft thresholding operator for compressed sensing and matrix completion Dongxiu Xie1,2 · Hugo J. Woerdeman3 · An-Bao Xu4 Received: 25 July 2019 / Revised: 16 February 2020 / Accepted: 25 April 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract Compressed sensing and matrix completion are two new approaches to signal acquisition and processing. Even though the two approaches are different, there is a close connection between them. We introduce a parametrized quasi-soft thresholding operator and use it to obtain new algorithms for compressed sensing and matrix completion. Furthermore, by updating the parametrized quasi-soft thresholding operator in every iteration, the varied parametric quasisoft thresholding algorithm is obtained. The convergence of the algorithms is analyzed, and numerical results show that the new algorithms can effectively improve the accuracy. Keywords Compressed sensing · Matrix completion · Sparse recovery · Rank minimization · Iterative soft thresholding · Iterative hard thresholding
1 Introduction The major problem of compressed sensing is to find a solution vector with the fewest nonzero entries in the infinite solution set of an underdetermined system of linear equations. The assumption is that the actual signal, which is not uniquely determined by the relatively few measurements, is sparse (in the basis chosen). Analogously, the purpose of matrix completion
Communicated by Jinyun Yuan. H. J. Woerdeman: Research supported by Simons Foundation grant (355645). A. Xu: Research supported by the National Natural Science Foundation of China (11801418).
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An-Bao Xu [email protected] Hugo J. Woerdeman [email protected]
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College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
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School of Science, Beijing Information Science and Technology University, Beijing 100192, China
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Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA
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College of Mathematics and Physics, Wenzhou University, Zhejiang 325035, China 0123456789().: V,-vol
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is to construct a low-rank matrix based on some given elements of the true underlying matrix. The design of the algorithms for solving these two reconstruction problems is the focus of compressed sensing and matrix completion. In earlier papers (see, e.g., Recht et al. 2010; Cai et al. 2010; Xu and Xie 2017), compressed sensing techniques were used to solve matrix completion problems. In this paper, we propose, among other things, a new matrix completion algorithm based on the firm thresholding algorithm introduced in Gao and Bruce (1997), Fornasier and Rauhut (2008), Woodworth and Chartrand (2016). In compressed sensing, we are interested in method of solving the l0 ’norm’ minimization problem of compressed sensing: min x0 s.t. Ax = b,
x∈Rn
(1.1)
where x ∈ Rn is the decision variable, and the linear map A : Rn → Rm and vector b ∈ Rm are given. x0 is defined to be the number of nonzero e
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