Statistical guarantees for denoising reflected diffusion models

In recent years, denoising diffusion models have become a crucial area of research due to their abundance in the rapidly expanding field of generative AI. While recent statistical advances have delivered explanations for the generation ability of idealised denoising diffusion models for high-dimensional target data, implementations introduce thresholding procedures for the generating process to overcome issues arising from the unbounded state space of such models. This mismatch between theoretical design and implementation of diffusion models has been addressed empirically by using a \emph{reflected} diffusion process as the driver of noise instead. In this paper, we study statistical guarantees of these denoising reflected diffusion models. In particular, under Sobolev smoothness assumptions, we establish rates of convergence in total variation which, up to a polylogarithmic factor, match the minimax lower bound. Our main contributions include the statistical analysis of this novel class of denoising reflected diffusion models and a refined score approximation method in both time and space, leveraging spectral decomposition and rigorous neural network analysis.

💡 Research Summary

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This paper addresses a fundamental gap between the theoretical design of denoising diffusion models (DDMs) and their practical implementations. Standard DDMs assume an unconstrained, unbounded state space and use a Gaussian Ornstein‑Uhlenbeck forward noising process. In practice, however, generated samples (e.g., pixel values) must stay within a bounded range, so implementations resort to ad‑hoc clipping or thresholding, which lacks a theoretical justification. Recent empirical work introduced reflected diffusion processes on bounded domains as the forward driver, thereby naturally respecting the bounds without explicit clipping.

The authors formalize this idea by defining Denosing Reflected Diffusion Models (DRDMs). The forward process is a reflected diffusion on a smooth bounded domain (\Omega\subset\mathbb{R}^d) generated by a self‑adjoint weighted Laplacian (\mathcal{L}=-\nabla\cdot(\kappa\nabla)) with Neumann boundary conditions. For the common choice (\kappa\equiv 1/2) the generator reduces to (\frac12\Delta), i.e., reflected Brownian motion up to a time change. The forward process admits a uniform invariant distribution on (\Omega), which can be sampled directly to initialise the backward generative dynamics.

The backward dynamics are obtained by time‑reversal of the forward process. Its drift is the score function (s(x,t)=\nabla\log\pi_t(x)), where (\pi_t) denotes the marginal density of the forward reflected diffusion at time (t). Because the transition densities of a reflected diffusion are not closed‑form Gaussians, the authors exploit the spectral decomposition of the semigroup: \