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

TL;DR

This work addresses the theoretical convergence gap for Plug-and-Play methods when using advanced denoisers by recasting the discrete PnP iterations as a continuous Ito-type stochastic differential equation. It provides two derivations—a direct SDE mapping of a simplified PnP form and a backward Markov process/Fokker–Planck proof—to connect PnP dynamics to drift and diffusion terms determined by the measurement map and denoiser. The authors establish a unified convergence framework based on SDE solvability, showing strong convergence under Lipschitz conditions on both the data-fidelity map and denoiser, and weak convergence under a milder regime where the denoiser is bounded while the measurement map remains Lipschitz. This leads to a practical takeaway: modern bounded denoisers can be theoretically justified within the weak convergence regime, helping to reconcile theoretical guarantees with the empirical success of PnP in imaging tasks.

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.

Figures (4)

• Figure 1: PSNR curves during PnP iteration comparing two methods. Orange lines show the results of PnP with Lipschitz continuous denoisers (PnP-RSN-CNN), while Indigo lines are with Lipschitz discontinuous denoisers (PnP-CNN). Lipschitz continuous denoiser ensures a smoother trajectory. In general, both Lipschitz continuous and discontinuous denoiser converges.
• Figure 2: PSNR curves during PnP iteration. Purple lines are under strong convergence, and they clearly demonstrate pathwise consistency. These lines with multiple colors are under weak convergence so that they go through different trajectories. Weak convergence also converges at the end of each trajectory, and achieve a bonus on performance.
• Figure 3: SSIM curves during PnP iteration. The instability at the beginning is due to the independence of SSIM value and noise level. Purple lines are under strong convergence and are pathwise unique. These lines with multiple colors are under weak convergence, with different trajectories. Weak convergence also gains a bonus on performance.
• Figure 4: Performance curves during PnP iteration under weak convergence. Orange lines show the results of PnP with Lipschitz continuous denoisers, while indigo lines are with Lipschitz discontinuous denoisers. Lipschitz continuous denoiser ensures a slightly smoother trajectory and also shows a tiny performance gain. In general, both Lipschitz continuous and discontinuous denoiser converges under weak probability uniqueness.

Theorems & Definitions (25)

• Definition 2.1: PnP iteration
• Theorem 2.3: SDE description of PnP
• proof
• proof : Proof of Theorem \ref{['main_theorem']}
• Proposition 3.1: Bridge between PnP convergence and SDE solvability.
• Definition 3.2: Strong convergence
• Theorem 3.3: Strong Convergence Conditions
• proof
• Definition 3.4: Weak Convergence
• Theorem 3.5: Weak convergence conditions