Wasserstein Bounds for generative diffusion models with Gaussian tail targets

TL;DR

This work establishes Wasserstein-2 guarantees for score-based diffusion models by leveraging a Gaussian-tail assumption and heat-kernel methods to obtain a global Lipschitz bound on the score. The resulting bound on the discretized backward process yields a dimension-free convergence rate, with finite-dimension complexity scaling as $\mathcal{O}(\sqrt{d})$ and infinite-dimension scaling with the trace of the forward covariance $\mathrm{Tr}(C)$. The framework accommodates bounded-support targets via early stopping and extends to Bayesian inverse problems, delivering practical error controls for high- or infinite-dimensional generative modeling. Overall, the approach provides principled, finite-sample diffusion-sampling guarantees under broad conditions, with implications for high-dimensional and functional-data applications.

Abstract

We present an estimate of the Wasserstein distance between the data distribution and the generation of score-based generative models. The sampling complexity with respect to dimension is $\mathcal{O}(\sqrt{d})$, with a logarithmic constant. In the analysis, we assume a Gaussian-type tail behavior of the data distribution and an $ε$-accurate approximation of the score. Such a Gaussian tail assumption is general, as it accommodates a practical target - the distribution from early stopping techniques with bounded support. The crux of the analysis lies in the global Lipschitz bound of the score, which is shown from the Gaussian tail assumption by a dimension-independent estimate of the heat kernel. Consequently, our complexity bound scales linearly (up to a logarithmic constant) with the square root of the trace of the covariance operator, which relates to the invariant distribution of the forward process.

Theorems & Definitions (28)

• Remark 2.1
• Theorem 3.1
• Corollary 3.2
• Theorem 3.3
• Corollary 3.4
• Remark 3.5
• Remark 3.6
• Theorem 3.7
• Corollary 3.8
• Theorem 3.9