On convergence of iterative thresholding algorithms to approximate sparse solution for composite nonconvex optimization

This paper aims to find an approximate true sparse solution of an underdetermined linear system. For this purpose, we propose two types of iterative thresholding algorithms with the continuation technique and the truncation technique respectively. We introduce a notion of limited shrinkage threshold...

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Veröffentlicht in:Mathematical programming Jg. 211; H. 1-2; S. 181 - 206
Hauptverfasser: Hu, Yaohua, Hu, Xinlin, Yang, Xiaoqi
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Heidelberg Springer Nature B.V 01.05.2025
Schlagworte:
Algorithms
Approximation
Convergence
Fines & penalties
Linear systems
Noise levels
Noise tolerance
Optimization algorithms
Regularization
Regularization methods
Sparsity
ISSN:0025-5610, 1436-4646
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Zusammenfassung:This paper aims to find an approximate true sparse solution of an underdetermined linear system. For this purpose, we propose two types of iterative thresholding algorithms with the continuation technique and the truncation technique respectively. We introduce a notion of limited shrinkage thresholding operator and apply it, together with the restricted isometry property, to show that the proposed algorithms converge to an approximate true sparse solution within a tolerance relevant to the noise level and the limited shrinkage magnitude. Applying the obtained results to nonconvex regularization problems with SCAD, MCP and $$\ell _p$$ ℓ p penalty ( $$0\le p \le 1$$ 0 ≤ p ≤ 1 ) and utilizing the recovery bound theory, we establish the convergence of their proximal gradient algorithms to an approximate global solution of nonconvex regularization problems. The established results include the existing convergence theory for $$\ell _1$$ ℓ 1 or $$\ell _0$$ ℓ 0 regularization problems for finding a true sparse solution as special cases. Preliminary numerical results show that our proposed algorithms can find approximate true sparse solutions that are much better than stationary solutions that are found by using the standard proximal gradient algorithm.
Bibliographie:ObjectType-Article-1
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-024-02068-1