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

TL;DR

This paper provides a theoretical foundation for MoL-Grad denoisers, showing they are proximity operators of weakly convex functions, and demonstrates their use in convergence-guaranteed algorithms without Lipschitz constant control.

Contribution

It characterizes MoL-Grad denoisers as proximity operators of weakly convex functions and extends Moreau's decomposition, enabling new explainability and convergence results.

Findings

MoL-Grad denoisers are equivalent to proximity operators of weakly convex functions.

The paper extends Moreau's decomposition to weakly convex functions.

Algorithms using MoL-Grad denoisers converge weakly to minimizers of implicit regularized cost functions.

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.