Convergence of proximal algorithms with stepsize controls for non-linear inverse problems and application to sparse non-negative matrix factorization

We consider a general ill-posed inverse problem in a Hilbert space setting by minimizing a misfit functional coupling with a multi-penalty regularization for stabilization. For solving this minimization problem, we investigate two proximal algorithms with stepsize controls: a proximal fixed point al...

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Bibliographic Details
Published in:Numerical algorithms Vol. 85; no. 4; pp. 1255 - 1279
Main Authors: Pham, Quy Muoi, Lachmund, Delf, Hào, Dinh Nho
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2020
Springer Nature B.V
Subjects:
Algebra
Algorithms
Computer Science
Convergence
Exact solutions
Factorization
Hilbert space
Inverse problems
Mathematical analysis
Nonlinear control
Numeric Computing
Numerical Analysis
Operators (mathematics)
Original Paper
Regularization
Regularization methods
Sparse matrices
Sparsity
Theory of Computation
Non-smooth Optimization
Proximal fixed point iteration
Alternating proximal iteration
Sparse non-negative matrix factorization
Non-linear inverse problems
Sparsity regularization
Sparse matrix factorization
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Non-negative sparsity regularization
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ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:We consider a general ill-posed inverse problem in a Hilbert space setting by minimizing a misfit functional coupling with a multi-penalty regularization for stabilization. For solving this minimization problem, we investigate two proximal algorithms with stepsize controls: a proximal fixed point algorithm and an alternating proximal algorithm. We prove the decrease of the objective functional and the convergence of both update schemes to a stationary point under mild conditions on the stepsizes. These algorithms are then applied to the sparse and non-negative matrix factorization problems. Based on a priori information of non-negativity and sparsity of the exact solution, the problem is regularized by corresponding terms. In both cases, the implementation of our proposed algorithms is straight-forward since the evaluation of the proximal operators in these problems can be done explicitly. Finally, we test the proposed algorithms for the non-negative sparse matrix factorization problem with both simulated and real-world data and discuss reconstruction performance, convergence, as well as achieved sparsity.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-019-00864-x