Optimal score function estimation via derivatives constraints
Abstract
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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BibTeX
@misc{bonis2026optimal,
title = {{Optimal score function estimation via derivatives constraints}},
author = {Thomas Bonis and Thanh Mai Pham Ngoc and Viet Chi Tran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.19084},
eprint = {2606.19084},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.19084},
}
RIS
TY - UNPB
TI - Optimal score function estimation via derivatives constraints
AU - Thomas Bonis
AU - Thanh Mai Pham Ngoc
AU - Viet Chi Tran
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2606.19084
UR - https://arxiv.org/abs/2606.19084
AB - 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.
ER -