Plug-and-Play Algorithm Convergence Analysis From The Standpoint of Stochastic Differential Equation

TL;DR

This paper provides a theoretical convergence analysis of Plug-and-Play algorithms by modeling them as stochastic differential equations, revealing weaker conditions for convergence than previously known.

Contribution

It introduces a unified framework for analyzing PnP convergence via SDEs, relaxing the denoiser conditions needed for guarantees.

Findings

Discrete PnP can be described by continuous SDEs.

Weaker bounded denoiser condition suffices for convergence.

Unified SDE-based framework for PnP analysis.

Abstract

The Plug-and-Play (PnP) algorithm is popular for inverse image problem-solving. However, this algorithm lacks theoretical analysis of its convergence with more advanced plug-in denoisers. We demonstrate that discrete PnP iteration can be described by a continuous stochastic differential equation (SDE). We can also achieve this transformation through Markov process formulation of PnP. Then, we can take a higher standpoint of PnP algorithms from stochastic differential equations, and give a unified framework for the convergence property of PnP according to the solvability condition of its corresponding SDE. We reveal that a much weaker condition, bounded denoiser with Lipschitz continuous measurement function would be enough for its convergence guarantee, instead of previous Lipschitz continuous denoiser condition.