A Two-step Krasnosel'skii-Mann Algorithm with Adaptive Momentum and Its Applications to Image Denoising and Matrix Completion

TL;DR

This paper introduces a novel two-step Krasnosel'skii-Mann algorithm with adaptive momentum that accelerates convergence for convex optimization problems in image processing, demonstrating superior performance in denoising and matrix completion tasks.

Contribution

The paper develops a new two-step KM algorithm with adaptive momentum, providing convergence guarantees and improved efficiency over existing methods in image processing applications.

Findings

TKMA outperforms existing algorithms like FPPA, PGA, and Halpern in image denoising.

The algorithm achieves an o(1/k^{1/2}) convergence rate under certain conditions.

Numerical experiments confirm the effectiveness of TKMA in practical tasks.

Abstract

In this paper, we propose a Two-step Krasnosel'skii-Mann (KM) Algorithm (TKMA) with adaptive momentum for solving convex optimization problems arising in image processing. Such optimization problems can often be reformulated as fixed-point problems for certain operators, which are then solved using iterative methods based on the same operator, including the KM iteration, to ultimately obtain the solution to the original optimization problem. Prior to developing TKMA, we first introduce a KM iteration enhanced with adaptive momentum, derived from geometric properties of an averaged nonexpansive operator T, KM acceleration technique, and information from the composite operator T^2. The proposed TKMA is constructed as a convex combination of this adaptive-momentum KM iteration and the Picard iteration of T^2. We establish the convergence of the sequence generated by TKMA to a fixed point of T. Moreover, under specific assumptions on the adaptive momentum parameters, we prove that the algorithm achieves an o(1/k^{1/2}) convergence rate in terms of the distance between successive iterates. Numerical experiments demonstrate that TKMA outperforms the FPPA, PGA, Fast KM algorithm, and Halpern algorithm on tasks such as image denoising and low-rank matrix completion.