Regularity of the score function in generative models

TL;DR

This paper investigates the regularity properties of the score function in score-based generative models, providing theoretical bounds that enhance understanding of convergence, stability, and neural network approximation.

Contribution

It establishes Lipschitz and higher-order regularity bounds for the score function, improving theoretical understanding and practical approximation in generative modeling.

Findings

Lipschitz estimates support convergence analysis

Higher-order bounds simplify neural network approximation

Regularity adapts to data distribution smoothness

Abstract

We study the regularity of the score function in score-based generative models and show that it naturally adapts to the smoothness of the data distribution. Under minimal assumptions, we establish Lipschitz estimates that directly support convergence and stability analyses in both diffusion and ODE-based generative models. In addition, we derive higher-order regularity bounds, which simplify existing arguments for optimally approximating the score function using neural networks.