Inexact Fixed-Point Proximity Algorithm for the ℓ0 Sparse Regularization Problem

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ 0 norm. Specifically, the ℓ 0 model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ 0 norm regularization term. Such an ℓ 0 model...

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Vydáno v:Journal of scientific computing Ročník 100; číslo 2; s. 58
Hlavní autoři: Fang, Ronglong, Xu, Yuesheng, Yan, Mingsong
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.08.2024
Springer Nature B.V
Témata:
Algorithms
Availability
Computational Mathematics and Numerical Analysis
Convergence
Convex analysis
Fixed points (mathematics)
Image classification
Iterative algorithms
Machine learning
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical analysis
Operators (mathematics)
Optimization
Proximity
Regularization
Sparsity
Theoretical
Wavelet transforms
The
68U10
Inexact fixed-point proximity algorithm
Non-convex optimization
65K10
Sparse regularization
90C26
norm
ISSN:0885-7474, 1573-7691
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Popis
Shrnutí:We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ 0 norm. Specifically, the ℓ 0 model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ 0 norm regularization term. Such an ℓ 0 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 ℓ 0 models found by the proposed inexact algorithm outperform global minimizers of the corresponding ℓ 1 models, in terms of approximation accuracy and sparsity of the solutions.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-024-02600-7