Plug-and-Play Algorithm Convergence Analysis From The Standpoint of Stochastic Differential Equation
This provides a theoretical foundation for convergence in inverse image problems, though it is incremental as it refines existing conditions rather than introducing a new paradigm.
The paper tackles the lack of theoretical convergence analysis for Plug-and-Play algorithms with advanced denoisers by modeling discrete iterations as a continuous stochastic differential equation, revealing that a weaker condition of bounded denoiser with Lipschitz continuous measurement function suffices for convergence instead of the previous Lipschitz continuous denoiser requirement.
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.