# Optimal score function estimation via derivatives constraints

arXiv: https://arxiv.org/abs/2606.19084 · HTML: https://mlsanity.com/paper/2606.19084/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 36 references that are in the map.
- First posted on arXiv 17 Jun 2026.

## Builds on (most cited first, 10 of 10)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024)
- [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024)
- [Gradient Dynamics of Shallow Univariate ReLU Networks](https://mlsanity.com/paper/1906.07842/) (2019)
- [Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spaces](https://mlsanity.com/paper/2505.02508/) (2025)
- [Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity](https://mlsanity.com/paper/2603.20645/) (2026)
- [Kernel-Smoothed Scores for Denoising Diffusion: A Bias-Variance Study](https://mlsanity.com/paper/2505.22841/) (2025)
- [Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds](https://mlsanity.com/paper/2605.15822/) (2026)

## Built on by (most cited first, 1 of 1)

- [Posterior contraction rates in Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families](https://mlsanity.com/paper/2608.11130/) (2026)

## Abstract (arXiv)

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.