Document Type

Article

Publication Date

2024

DOI

10.1007/s10915-024-02600-7

Publication Title

Journal of Scientific Computing

Volume

100

Issue

2

Pages

58 (1-43)

Abstract

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ₀ norm. Specifically, the ℓ₀ model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ₀ norm regularization term. Such an ℓ₀ model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the ℓ₀ models found by the proposed inexact algorithm outperform global minimizers of the corresponding ℓ₁ models, in terms of approximation accuracy and sparsity of the solutions.

Rights

© 2024 The Authors

This article is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Data Availability

"Data Availability Statement": Article states: "The computer codes that generate the numerical results presented in this paper can be found in the following website: https://github.com/msyan2023/InexactFPPA-L0"

Original Publication Citation

Fang, R., Xu, Y., & Yan, M. (2024). Inexact fixed-point proximity algorithm for the ℓ₀ sparse regularization problem. Journal of Scientific Computing, 100(2), 1-43, Article 58. https://doi.org/10.1007/s10915-024-02600-7

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