Regularity of the score function in generative models

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.