Reflected diffusion models adapt to low-dimensional data

TL;DR

This paper analyzes reflected diffusion models on bounded domains, showing they adapt to low-dimensional structures and achieve near-optimal convergence rates similar to unconstrained models.

Contribution

It introduces a statistical framework for reflected diffusion models on bounded domains, providing convergence rates that adapt to the intrinsic data dimension.

Findings

Convergence rate of order $n^{-(rac{eta+1- heta}{2eta+d})}$ for Sobolev smoothness $eta$.

Reflected boundaries do not impair the statistical efficiency of diffusion models.

The approach extends theoretical understanding of score-based models to constrained domains.

Abstract

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 $\mathbb{R}^D$. 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 $\alpha$, we establish a convergence rate in the $1$-Wasserstein distance of order $n^{-\frac{\alpha+1-\delta}{2\alpha+d}}$ for arbitrarily small $\delta > 0$, 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.