# Reflected diffusion models adapt to low-dimensional data

arXiv: https://arxiv.org/abs/2603.24495 · HTML: https://mlsanity.com/paper/2603.24495/

## 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 5 of its 37 references that are in the map.
- First posted on arXiv 25 Mar 2026.

## Placed in

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

## Builds on (most cited first, 5 of 5)

- [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024)
- [From optimal score matching to optimal sampling](https://mlsanity.com/paper/2409.07032/) (2024)
- [Nonparametric estimation of a factorizable density using diffusion models](https://mlsanity.com/paper/2501.01783/) (2025)
- [Statistical guarantees for denoising reflected diffusion models](https://mlsanity.com/paper/2411.01563/) (2024)
- [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, 1 of 1)

- [Statistical Convergence of Spherical First Hitting Diffusion Models](https://mlsanity.com/paper/2605.07625/) (2026)

## Abstract (arXiv)

While the mathematical foundations of score-based generative models are increasingly well understood for unconstrained Euclidean spaces, many practical applications involve data restricted to bounded domains. This paper provides a statistical analysis of reflected diffusion models on the hypercube [0,1]D for target distributions supported on d-dimensional linear subspaces. A primary challenge in this setting is the absence of Gaussian transition kernels, which play a central role in standard theory in ℝD. By employing an easily implementable infinite series expansion of the transition densities, we develop analytic tools to bound the score function and its approximation by sparse ReLU networks. For target densities with Sobolev smoothness α, we establish a convergence rate in the 1-Wasserstein distance of order n^-fracα+1-δ2α+d for arbitrarily small δ > 0, demonstrating that the generative algorithm fully adapts to the intrinsic dimension d. These results confirm that the presence of reflecting boundaries does not degrade the fundamental statistical efficiency of the diffusion paradigm, matching the almost optimal rates known for unconstrained settings.