Global Well-posedness and Convergence Analysis of Score-based Generative Models via Sharp Lipschitz Estimates

TL;DR

This paper proves the global well-posedness and convergence of score-based generative models using sharp Lipschitz estimates, addressing both smooth and non-smooth cases with minimal assumptions.

Contribution

It introduces a novel analysis based on local Lipschitz conditions, enabling convergence results without time scale separation for non-log-concave distributions.

Findings

Global well-posedness of score-based models established.

Convergence proven under minimal assumptions.

Optimal Lipschitz bounds characterized for different cases.

Abstract

We establish global well-posedness and convergence of the score-based generative models (SGM) under minimal general assumptions of initial data for score estimation. For the smooth case, we start from a Lipschitz bound of the score function with optimal time length. The optimality is validated by an example whose Lipschitz constant of scores is bounded at initial but blows up in finite time. This necessitates the separation of time scales in conventional bounds for non-log-concave distributions. In contrast, our follow up analysis only relies on a local Lipschitz condition and is valid globally in time. This leads to the convergence of numerical scheme without time separation. For the non-smooth case, we show that the optimal Lipschitz bound is O(1/t) in the point-wise sense for distributions supported on a compact, smooth and low-dimensional manifold with boundary.