A Two-step Krasnosel'skii-Mann Algorithm with Adaptive Momentum and Its Applications to Image Denoising and Matrix Completion
For researchers in convex optimization and image processing, this algorithm offers improved convergence speed and practical performance on fixed-point problems.
The paper proposes a Two-step Krasnosel'skii-Mann Algorithm (TKMA) with adaptive momentum for convex optimization in image processing, achieving an o(1/√k) convergence rate and outperforming FPPA, PGA, Fast KM, and Halpern algorithms on image denoising and matrix completion tasks.
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 $α$-averaged nonexpansive operator T with $α\in(0,1)$, 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 prove that the sequence generated by TKMA converges weakly to a fixed point of T in a real Hilbert space. Moreover, under $α\in(0,1/2]$ and specific assumptions on the adaptive momentum parameters, we prove that the algorithm achieves an $o\left(1/\sqrt{k}\right)$ 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.