We provide a statistical theory explaining why empirical score matching can achieve minimax optimal performance for diffusion generative models when the data distribution is supported on a low-dimensional manifold. More precisely, we establish convergence rates for empirical risk minimization over suitably constrained Sobolev classes and show that, under appropriate regularity conditions, the resulting score estimators lead to minimax optimal Wasserstein-1 rates for the generated distribution, up to logarithmic factors. Our results provide sufficient conditions for score-based generative modeling to generalize from the data instead of overfitting, while the resulting statistical rates depend on the intrinsic dimension of the data manifold rather than its ambient dimension.

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