Wasserstein Convergence of Critically Damped Langevin Diffusions

TL;DR

This paper analyzes a generalized Critically-damped Langevin Diffusion model used in score-based generative models, deriving bounds on sampling error and providing practical tuning guidance to improve performance.

Contribution

It introduces an extended hyperparameter in CLDs, derives a Wasserstein error bound, and offers discretization insights for better sampling in generative models.

Findings

Derived a new upper bound on Wasserstein sampling error.

Introduced an additional hyperparameter to control noise and smoothness.

Provided practical guidelines for hyperparameter tuning.

Abstract

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 (CLDs), which define diffusion processes on 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 analyze a generalized dynamic that extends classical CLDs by introducing an additional hyperparameter controlling the noise applied to the data coordinate, thereby better exploiting the extended space. We further derive a novel upper bound on the sampling error of CLD-based generative models in the Wasserstein metric. This additional hyperparameter influences the smoothness of sample paths, and our discretization error analysis provides practical guidance for its tuning, leading to improved sampling performance.