Wasserstein convergence of critically damped Langevin diffusions
Stanislas Strasman, Sobihan Surendran, Claire Boyer, Sylvain Le Corff, Vincent Lemaire, Antonio Ocello
Score-based Generative Models (SGMs) have achieved impressive performance in data generation across a wide range of applications and benefit from strong theoretical guarantees. Recently, methods inspired by statistical mechanics, in particular Hamiltonian dynamics, have introduced Critically-damped Langevin Diffusions (CLD), which define diffusion processes in extended spaces by coupling the data with auxiliary variables. These approaches, along with their associated score-matching and sampling procedures, have been shown to outperform standard diffusion-based samplers numerically. In this paper, we propose an upper bound on the sampling error for CLD-based generative models in the Wasserstein metric. To better exploit the extended space, we also propose a modified dynamic that introduces an additional hyperparameter controlling the noise applied to the data coordinates. This hyperparameter influences the smoothness of sample paths, and our discretization error analysis offers practical guidance for tuning, leading to improved sampling performance.