A generalized forward-backward splitting operator: Degenerate analysis and applications

TL;DR

This paper investigates the properties and convergence of a generalized forward-backward splitting operator under degenerate metrics, providing a unifying framework for analyzing various operator splitting algorithms.

Contribution

It introduces a generalized operator with degenerate metric analysis, extending descent properties and unifying multiple existing algorithms under a common framework.

Findings

Convergence results established for the G-FBS operator in degenerate metrics

Extension of descent lemma and decrease property to degenerate cases

Unification and simplification of various operator splitting algorithms

Abstract

In this paper, we study the nonexpansive properties of a generalized forward-backward splitting (G-FBS) operator, particularly under the setting of degenerate metric, from which follow the convergence results in terms of degenerate metric of the associated fixed-point iterations. The descent lemma and sufficient decrease property are also extended to degenerate case. It is further shown that the G-FBS operator provides a simplifying and unifying framework to model and analyze a great variety of operator splitting algorithms, many existing results are exactly recovered or relaxed from our general results.