Wasserstein Bounds for generative diffusion models with Gaussian tail targets

TL;DR

This paper derives a Wasserstein distance estimate for score-based generative models with Gaussian tail data, showing sampling complexity scales as the square root of data dimension with a logarithmic factor.

Contribution

It introduces a dimension-independent analysis of Wasserstein bounds for diffusion models under Gaussian tail assumptions, with a new Lipschitz bound of the score function.

Findings

Sampling complexity scales as O(√d) with dimension d.

The analysis applies to distributions with Gaussian tail behavior.

The bounds depend on the trace of the covariance operator.

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 $\epsilon$-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.