Wasserstein Bounds for generative diffusion models with Gaussian tail targets
This provides theoretical guarantees for diffusion models, which is incremental but important for understanding their convergence and efficiency in high-dimensional settings.
The paper tackles the problem of bounding the Wasserstein distance between data and generated distributions in score-based generative models, achieving a sampling complexity of O(√d) with a logarithmic constant under Gaussian tail assumptions and accurate score approximations.
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.