Wasserstein Convergence of Critically Damped Langevin Diffusions

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. 39th Conference on Neural Information Processing Systems (NeurIPS 2025). addition, SGMs provide a particularly interesting class of prior distributions to solve Bayesian inverse problems. Although they lack an explicit and tractable probability density function, a very active research area focuses on combining Monte Carlo guidance and SGMs to solve posterior sampling problems, Wu et al. (2023); Moufad et al. (2025); Victorino Cardoso et al. (2024).

In Dockhorn et al. (2022), the authors proposed Critically-damped Langevin Diffusion as a second-order extension of conventional diffusion models. By introducing velocity variables alongside the usual state variables -much like in Hamiltonian Monte Carlo-CLD accelerates exploration of high-dimensional spaces and often yields better sample quality in practice. Although empirical work demonstrates the benefit of CLD over standard score-based models (Dockhorn et al., 2022), its theoretical underpinnings remain incomplete. Existing convergence guarantees are only expressed in terms of Kullback-Leibler divergence (Conforti et al., 2025;Chen et al., 2023) and fail to capture any computational advantage for kinetic dynamics, leaving a gap between observed performance and formal analysis.

Contributions. We first discuss the challenges of establishing Wasserstein convergence under the standard assumptions used for Variance-Preserving (VP) or Variance-Exploding (VE) SGMs, where the forward process is elliptic (Gao et al., 2025; Strasman et al., 2025;Gentiloni-Silveri and Ocello, 2025;Bruno et al., 2025). We then provide, to the best of our knowledge, the first upper bound for CLD in the Wasserstein metric through coupling techniques under weaker assumptions, achieving convergence rates comparable to those of other SGMs. Crucially, this result is not implied by previous Kullback-Leibler divergence bounds (Conforti et al., 2025;Chen et al., 2023), and our proof technique differs significantly from existing Wasserstein analyses of diffusion models.

However, it is possible to introduce a modified dynamics that includes an additional hyperparameter controlling the noise on the data coordinate of CLD, thereby restoring ellipticity and enabling an analysis closely aligned with that of VP and VE models, but formulated on an extended phase space with matrix-valued drifts and diffusions. This hyperparameter governs the smoothness of sample paths, allowing a detailed analysis of the generative error as a function of this smoothness parameter. Such analysis offers practical guidance for tuning this hyperparameter and potentially improves sampling performance compared to standard SGMs and CLD methods. The benefits of this additional parameterization are demonstrated numerically on challenging synthetic datasets.