Monotone Lipschitz-Gradient Denoiser: Explainability of Operator Regularization Approaches Free From Lipschitz Constant Control

This paper addresses explainability of the operator-regularization approach under the use of monotone Lipschitz-gradient (MoL-Grad) denoiser - an operator that can be expressed as the Lipschitz continuous gradient of a differentiable convex function. We prove that an operator is a MoL-Grad denoiser...

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Published in:IEEE transactions on signal processing Vol. 73; pp. 3378 - 3393
Main Authors: Yukawa, Masahiro, Yamada, Isao
Format: Journal Article
Language:English
Published: New York IEEE 2025
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Convergence
Convex functions
convex optimization
Cost function
Estimation
Linear programming
Lipschitz condition
nonexpansive operator
Operators (mathematics)
proximity operator
Regularization
Signal processing
Signal processing algorithms
Splitting
Sufficient conditions
Training
Vectors
Weakly convex function
ISSN:1053-587X, 1941-0476
Online Access:Get full text
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Summary:This paper addresses explainability of the operator-regularization approach under the use of monotone Lipschitz-gradient (MoL-Grad) denoiser - an operator that can be expressed as the Lipschitz continuous gradient of a differentiable convex function. We prove that an operator is a MoL-Grad denoiser if and only if it is the "single-valued" proximity operator of a weakly convex function. An extension of Moreau's decomposition is also shown with respect to a weakly convex function and the conjugate of its convexified function. Under these arguments, two specific algorithms, the forward-backward splitting algorithm and the primal-dual splitting algorithm, are considered, both employing MoL-Grad denoisers. These algorithms generate a sequence of vectors converging weakly, under conditions, to a minimizer of a certain cost function which involves an "implicit regularizer" induced by the denoiser. Unlike the previous studies of operator regularization, our framework requires no control of the Lipschitz constant in learning the denoiser. The theoretical findings are supported by simulations.
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ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2025.3580667