# Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structures

arXiv: https://arxiv.org/abs/2606.19894 · HTML: https://mlsanity.com/paper/2606.19894/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 29 of its 40 references that are in the map.
- First posted on arXiv 18 Jun 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 29 of 29)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [ResNet with one-neuron hidden layers is a Universal Approximator](https://mlsanity.com/paper/1806.10909/) (2018)
- [On the Generalization Properties of Diffusion Models](https://mlsanity.com/paper/2311.01797/) (2023)
- [KL Convergence Guarantees for Score diffusion models under minimal data assumptions](https://mlsanity.com/paper/2308.12240/) (2023)
- [Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models](https://mlsanity.com/paper/2405.14861/) (2024)
- [Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions](https://mlsanity.com/paper/2402.15602/) (2024)
- [Unraveling the Smoothness Properties of Diffusion Models: A Gaussian Mixture Perspective](https://mlsanity.com/paper/2405.16418/) (2024)
- [Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients](https://mlsanity.com/paper/2505.03432/) (2025)
- [Improved Sample Complexity Bounds for Diffusion Model Training](https://mlsanity.com/paper/2311.13745/) (2023)
- [Topological Singularity Detection at Multiple Scales](https://mlsanity.com/paper/2210.00069/) (2022)
- [HADES: Fast Singularity Detection with Local Measure Comparison](https://mlsanity.com/paper/2311.04171/) (2023)
- [Transformers are Inherently Succinct](https://mlsanity.com/paper/2510.19315/) (2025)

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

- [Recovering Weighted Tangent Geometry from a Single-Scale Score Field](https://mlsanity.com/paper/2608.22334/) (2026)

## Abstract (arXiv)

Score-based diffusion models have achieved remarkable empirical success, motivating extensive theoretical work to establish their foundations. However, existing complexity bounds for score approximation, a vital step in diffusion modeling, rely on rigid constraints such as Lipschitz continuous scores or lower bounded densities. This severely limits their applicability to real-world perceptual data, where singularities, sharp boundaries, and disjoint clusters routinely violate such restrictive assumptions. We bridge this gap between theory and practice, presenting the first universal score approximation theorem applicable to any compactly supported distribution in ℝn. Using a novel discretization technique that directly models the underlying distribution, we prove that the neural network complexity is governed by the support's upper Minkowski dimension d rather than the ambient dimension n. Furthermore, by leveraging the inherent smoothing of Gaussian kernels, we show that even for irregular, fractal distributions, an ε approximation error can be achieved with O(ε^-d/(M+1)) local Taylor modules each at size of O(nM), an ε-scaling rate previously achieved only for distributions with (M+1)-H"older smoothness. Thus, we reveal a possible mechanism by which score-based diffusion models represent non-smooth data distributions.