Conceitos essenciais
The core message of this article is that the discrete Plug-and-Play (PnP) algorithm can be described by a continuous stochastic differential equation (SDE), and the convergence properties of PnP can be analyzed through the solvability of the corresponding SDE. The authors propose a unified framework for PnP convergence analysis, showing that a much weaker condition of only bounded denoisers and Lipschitz continuous measurement models is sufficient for PnP convergence, in contrast to the previously required Lipschitz continuous denoiser condition.
Resumo
The article presents a novel approach to analyzing the convergence of Plug-and-Play (PnP) algorithms, which are popular for solving inverse image problems. The authors demonstrate that the discrete PnP iteration can be described by a continuous stochastic differential equation (SDE). They provide two approaches to transform the PnP iteration into an SDE formulation.
The authors then construct a unified framework for PnP convergence analysis by relating the solvability of the corresponding SDE to the convergence properties of the PnP algorithm. They show that the strong convergence of PnP is related to the strong solvability of the SDE, which requires Lipschitz continuous conditions on both the measurement model and the denoiser.
In contrast, the authors propose a weaker condition for the convergence of PnP, which only requires the measurement model to be Lipschitz continuous and the denoiser to be bounded. This relaxed condition is more applicable to practical denoisers, which are often not Lipschitz continuous but still perform well in experiments.
The authors provide examples to illustrate their theoretical findings, comparing the performance of PnP algorithms with Lipschitz continuous and bounded denoisers. They also discuss the relationship between the convergence conditions for PnP and the solvability of the corresponding SDE.