# Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributions

arXiv: https://arxiv.org/abs/2605.30153 · HTML: https://mlsanity.com/paper/2605.30153/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 37 of its 43 references that are in the map.
- First posted on arXiv 28 May 2026.

## Placed in

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

## Builds on (most cited first, 30 of 37)

- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [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)
- [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024)
- [Learning Mixtures of Gaussians Using the DDPM Objective](https://mlsanity.com/paper/2307.01178/) (2023)
- [Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models](https://mlsanity.com/paper/2405.14861/) (2024)
- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions](https://mlsanity.com/paper/2402.15602/) (2024)
- [Statistical optimal transport](https://mlsanity.com/paper/2407.18163/) (2024)
- [Linear Convergence of Diffusion Models Under the Manifold Hypothesis](https://mlsanity.com/paper/2410.09046/) (2024)
- [Learning general Gaussian mixtures with efficient score matching](https://mlsanity.com/paper/2404.18893/) (2024)
- [Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality](https://mlsanity.com/paper/2410.18784/) (2024)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [Low-dimensional adaptation of diffusion models: Convergence in total variation](https://mlsanity.com/paper/2501.12982/) (2025)
- [A Good Score Does not Lead to A Good Generative Model](https://mlsanity.com/paper/2401.04856/) (2024)
- [From optimal score matching to optimal sampling](https://mlsanity.com/paper/2409.07032/) (2024)
- [Optimal Inference Schedules for Masked Diffusion Models](https://mlsanity.com/paper/2511.04647/) (2025)
- [Minimax Optimality of the Probability Flow ODE for Diffusion Models](https://mlsanity.com/paper/2503.09583/) (2025)
- [Breaking AR's Sampling Bottleneck: Provable Acceleration via Diffusion Language Models](https://mlsanity.com/paper/2505.21400/) (2025)
- [Generalization error bound for denoising score matching under relaxed manifold assumption](https://mlsanity.com/paper/2502.13662/) (2025)
- [Efficient Sampling with Discrete Diffusion Models: Sharp and Adaptive Guarantees](https://mlsanity.com/paper/2602.15008/) (2026)

## Built on by (most cited first, 3 of 3)

- [ConvergeFlow: Language Flow with Provable Convergence to Token Embeddings](https://mlsanity.com/paper/2608.23551/) (2026)
- [Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choices](https://mlsanity.com/paper/2606.23627/) (2026)
- [Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data](https://mlsanity.com/paper/2610.02663/) (2026)

## Abstract (arXiv)

Score-based diffusion models have demonstrated remarkable empirical success in learning high-dimensional distributions, particularly those exhibiting low-dimensional and multi-modal structures. However, theoretical understanding of their statistical efficiency remains limited. Existing theories typically rely on strong regularity assumptions, such as uniformly bounded densities or globally smooth score functions, which fail to capture such intrinsic structures. In this work, we study the sample complexity of diffusion models for learning distributions supported on a union of low-dimensional subspaces. Assuming that the data distribution within each subspace is subgaussian, we show that diffusion models require at most widetildeO(varepsilon-k vee 2) samples to achieve varepsilon error in 1-Wasserstein distance, where k is the intrinsic dimension. This near-optimal convergence rate depends only on the intrinsic dimension and significantly improves upon prior theoretical guarantees that suffer from the curse of dimensionality. Notably, our analysis applies to a broad collection of distributions without imposing smoothness, bounded-density, or log-concavity assumptions. Overall, our results show that diffusion models can statistically adapt to intrinsic low-dimensional structure while naturally accommodating multi-modal data, offering a rigorous theoretical justification for their success in complex high-dimensional learning tasks.