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

Abstract

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.

17 pages, 6 figures

References in corpus (16)

• Nearly unbiased variable selection under minimax concave penalty
• Sparse Regularization via Convex Analysis
• Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation
• Plug and play methods for magnetic resonance imaging (long version)
• Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems
• Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imaging
• An Online Plug-and-Play Algorithm for Regularized Image Reconstruction
• Wavelet methods in statistics: Some recent developments and their applications
• Some sharp performance bounds for least squares regression with $L_1$ regularization
• On the Convergence of the Iterative Shrinkage/Thresholding Algorithm With a Weakly Convex Penalty
• Enhanced Sparsity by Non-Separable Regularization
• Provable Convergence of Plug-and-Play Priors with MMSE denoisers
• Linearly Involved Generalized Moreau Enhanced Models and Their Proximal Splitting Algorithm under Overall Convexity Condition
• Nonconvex Sparse Logistic Regression with Weakly Convex Regularization
• Linearly-involved Moreau-Enhanced-over-Subspace Model: Debiased Sparse Modeling and Stable Outlier-Robust Regression
• Continuous Relaxation of Discontinuous Shrinkage Operator: Proximal Inclusion and Conversion

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