Pathway to $O(\sqrt{d})$ Complexity bound under Wasserstein metric of flow-based models

TL;DR

This paper establishes an $O(\sqrt{d})$ error bound for flow-based generative models under the Wasserstein metric, providing analytical tools to estimate sampling complexity in high dimensions.

Contribution

It introduces explicit error bounds and complexity estimates for flow-based models under Wasserstein metric, with assumptions valid for Föllmer process and 1-rectified flow.

Findings

Error can be controlled by Lipschitzness of push-forward maps

Discretization error scales as $O(\sqrt{d})$ in dimension

Sampling complexity grows with the square root of covariance trace

Abstract

We provide attainable analytical tools to estimate the error of flow-based generative models under the Wasserstein metric and to establish the optimal sampling iteration complexity bound with respect to dimension as $O(\sqrt{d})$. We show this error can be explicitly controlled by two parts: the Lipschitzness of the push-forward maps of the backward flow which scales independently of the dimension; and a local discretization error scales $O(\sqrt{d})$ in terms of dimension. The former one is related to the existence of Lipschitz changes of variables induced by the (heat) flow. The latter one consists of the regularity of the score function in both spatial and temporal directions. These assumptions are valid in the flow-based generative model associated with the F\"{o}llmer process and $1$-rectified flow under the Gaussian tail assumption. As a consequence, we show that the sampling iteration complexity grows linearly with the square root of the trace of the covariance operator, which is related to the invariant distribution of the forward process.