Wasserstein Convergence of Critically Damped Langevin Diffusions

2025/11/04 by Stanislas Strasman, Sobihan Surendran, Strasman, Stanislas +9
Computer Science · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Generative Adversarial Networks and Image Synthesis #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Statistics Theory (math.ST) #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2511.02419

openalex created_date 2025/11/04 · openalex publication_date 2025/11/04 · openalex updated_date 2026/07/28

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.

Citations

• Regularity of the score function in generative models
• Non-asymptotic Analysis of Diffusion Annealed Langevin Monte Carlo for Generative Modelling
• Bit-Level Discrete Diffusion with Markov Probabilistic Models: An Improved Framework with Sharp Convergence Bounds under Minimal Assumptions
• Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance
• Variational Diffusion Posterior Sampling with Midpoint Guidance
• Heat flow, log-concavity, and Lipschitz transport maps
• An analysis of the noise schedule for score-based generative models
• On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates
• Wasserstein Convergence Guarantees for a General Class of Score-Based Generative Models
• KL Convergence Guarantees for Score diffusion models under minimal data assumptions
• Monte Carlo guided Diffusion for Bayesian linear inverse problems
• Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization
• DiffuSeq: Sequence to Sequence Text Generation with Diffusion Models
• Convergence of score-based generative modeling for general data distributions
• Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions
• Diffusion Models: A Comprehensive Survey of Methods and Applications
• RePaint: Inpainting using Denoising Diffusion Probabilistic Models
• Score-Based Generative Modeling with Critically-Damped Langevin Diffusion
• Diffusion Schr "odinger Bridge with Applications to Score-Based\n Generative Modeling
• SRDiff: Single Image Super-Resolution with Diffusion Probabilistic Models
• Time reversal of diffusion processes under a finite entropy condition
• Improved Denoising Diffusion Probabilistic Models
• Score-Based Generative Modeling through Stochastic Differential Equations
• Almost sure contraction for diffusions on \mathbb R^d. Application to generalised Langevin diffusions
• Denoising Diffusion Probabilistic Models
• Generative Modeling by Estimating Gradients of the Data Distribution
• On sampling from a log-concave density using kinetic Langevin diffusions
• Couplings and quantitative contraction rates for Langevin dynamics
• Deep Unsupervised Learning using Nonequilibrium Thermodynamics
• Log-concavity and strong log-concavity: a review
• A Connection Between Score Matching and Denoising Autoencoders
• Hypocoercivity

Related

• Score-Based Generative Modeling with Critically-Damped Langevin Diffusion
• Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Acceleration
• Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions
• Critically-Damped Higher-Order Langevin Dynamics for Generative Modeling
• Convergence of Deterministic and Stochastic Diffusion-Model Samplers: A Simple Analysis in Wasserstein Distance