A convergence analysis for iterative sparsification projection with soft-thresholding

The recently proposed iterative sparsification projection (ISP), a fast and robust sparse signal recovery algorithm framework, can be classified as smooth-ISP and nonsmooth-ISP. However, no convergence analysis has been established for the nonsmooth-ISP in the previous works. Motivated by this absen...

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Published in:	Signal, image and video processing Vol. 15; no. 8; pp. 1705 - 1712
Main Author:	Zhu, Tao
Format:	Journal Article
Language:	English
Published:	London Springer London 01.11.2021 Springer Nature B.V
Subjects:	Accumulation Algorithms Computer Imaging Computer Science Convergence Critical point Image Processing and Computer Vision Iterative methods Multimedia Information Systems Original Paper Pattern Recognition and Graphics Projection Robustness (mathematics) Signal reconstruction Signal,Image and Speech Processing Vision Iterative sparsification projection Fixed point mapping Convergence
ISSN:	1863-1703, 1863-1711
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Abstract	The recently proposed iterative sparsification projection (ISP), a fast and robust sparse signal recovery algorithm framework, can be classified as smooth-ISP and nonsmooth-ISP. However, no convergence analysis has been established for the nonsmooth-ISP in the previous works. Motivated by this absence, the present paper provides a convergence analysis for ISP with soft-thresholding (ISP-soft) which is an instance of the nonsmooth-ISP. In our analysis, the composite operator of soft-thresholding and proximal projection is viewed as a fixed point mapping, whose nonexpansiveness plays a key role. Specifically, our convergence analysis for the sequence generated by ISP-soft can be summarized as follows: 1) For each inner loop, we prove that the sequence has a unique accumulation point which is a fixed point, and show that it is a Cauchy sequence; 2) for the last inner loop, we prove that the accumulation point of the sequence is a critical point of the objective function if the final value of the threshold satisfies a condition, and show that the corresponding objective values are monotonically nonincreasing. A numerical experiment is given to validate some of our results and intuitively present the convergence.
AbstractList	The recently proposed iterative sparsification projection (ISP), a fast and robust sparse signal recovery algorithm framework, can be classified as smooth-ISP and nonsmooth-ISP. However, no convergence analysis has been established for the nonsmooth-ISP in the previous works. Motivated by this absence, the present paper provides a convergence analysis for ISP with soft-thresholding (ISP-soft) which is an instance of the nonsmooth-ISP. In our analysis, the composite operator of soft-thresholding and proximal projection is viewed as a fixed point mapping, whose nonexpansiveness plays a key role. Specifically, our convergence analysis for the sequence generated by ISP-soft can be summarized as follows: 1) For each inner loop, we prove that the sequence has a unique accumulation point which is a fixed point, and show that it is a Cauchy sequence; 2) for the last inner loop, we prove that the accumulation point of the sequence is a critical point of the objective function if the final value of the threshold satisfies a condition, and show that the corresponding objective values are monotonically nonincreasing. A numerical experiment is given to validate some of our results and intuitively present the convergence.
Author	Zhu, Tao
Author_xml	– sequence: 1 givenname: Tao orcidid: 0000-0002-3244-6459 surname: Zhu fullname: Zhu, Tao email: ee_zt_21@mail.scut.edu.cn organization: School of Electronic and Information Engineering, South China University of Technology
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SubjectTerms	Accumulation Algorithms Computer Imaging Computer Science Convergence Critical point Image Processing and Computer Vision Iterative methods Multimedia Information Systems Original Paper Pattern Recognition and Graphics Projection Robustness (mathematics) Signal reconstruction Signal,Image and Speech Processing Vision
Title	A convergence analysis for iterative sparsification projection with soft-thresholding
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