# Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity

arXiv: https://arxiv.org/abs/2603.20645 · HTML: https://mlsanity.com/paper/2603.20645/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 20 of its 76 references that are in the map.
- First posted on arXiv 21 Mar 2026.

## Placed in

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

## Builds on (most cited first, 20 of 20)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [Scaling Laws for Autoregressive Generative Modeling](https://mlsanity.com/paper/2010.14701/) (2020)
- [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)
- [Generative Modeling with Denoising Auto-Encoders and Langevin Sampling](https://mlsanity.com/paper/2002.00107/) (2020)
- [Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimation](https://mlsanity.com/paper/2106.05527/) (2021)
- [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024)
- [Score-based Diffusion Models via Stochastic Differential Equations -- a Technical Tutorial](https://mlsanity.com/paper/2402.07487/) (2024)
- [Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality](https://mlsanity.com/paper/2410.18784/) (2024)
- [From optimal score matching to optimal sampling](https://mlsanity.com/paper/2409.07032/) (2024)
- [Generalization error bound for denoising score matching under relaxed manifold assumption](https://mlsanity.com/paper/2502.13662/) (2025)
- [Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive](https://mlsanity.com/paper/2510.02305/) (2025)
- [When Scores Learn Geometry: Rate Separations under the Manifold Hypothesis](https://mlsanity.com/paper/2509.24912/) (2025)
- [Towards Understanding Text Hallucination of Diffusion Models via Local Generation Bias](https://mlsanity.com/paper/2503.03595/) (2025)
- [Generalization Properties of Score-matching Diffusion Models for Intrinsically Low-dimensional Data](https://mlsanity.com/paper/2603.03700/) (2026)

## Built on by (most cited first, 7 of 7)

- [Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds](https://mlsanity.com/paper/2605.15822/) (2026)
- [Diffusion Models for Adaptive Sequential Data Generation](https://mlsanity.com/paper/2606.06007/) (2026)
- [Optimal score function estimation via derivatives constraints](https://mlsanity.com/paper/2606.19084/) (2026)
- [Recovering Weighted Tangent Geometry from a Single-Scale Score Field](https://mlsanity.com/paper/2608.22334/) (2026)
- [Matched Queries for Curvature and Density at Branching Junctions](https://mlsanity.com/paper/2609.01319/) (2026)
- [FLARE: Flow Matching with Local Axis-Angle Representations for Stochastic Micromagnetic Evolution](https://mlsanity.com/paper/2609.34070/) (2026)
- [Geometry-Aware Preference Optimization for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2610.04980/) (2026)

## Abstract (arXiv)

Diffusion models have become a leading framework in generative modeling, yet their theoretical understanding -- especially for high-dimensional data concentrated on low-dimensional structures -- remains incomplete. This paper investigates how diffusion models learn such structured data, focusing on two key aspects: statistical complexity and influence of data geometric properties. By modeling data as samples from a smooth Riemannian manifold, our analysis reveals crucial decompositions of score functions in diffusion models under different levels of injected noise. We also highlight the interplay of manifold curvature with the structures in the score function. These analyses enable an efficient neural network approximation to the score function, built upon which we further provide statistical rates for score estimation and distribution learning. Remarkably, the obtained statistical rates are governed by the intrinsic dimension of data and the manifold curvature. These results advance the statistical foundations of diffusion models, bridging theory and practice for generative modeling on manifolds.