Manifold Generalization Provably Proceeds Memorization in Diffusion Models

Score-based diffusion models have achieved remarkable empirical success, motivating extensive theoretical work to establish their foundations. However, existing complexity bounds for score approximation, a vital step in diffusion modeling, rely on rigid constraints such as Lipschitz continuous scores or lower bounded densities. This severely limits their applicability to real-world perceptual data, where singularities, sharp boundaries, and disjoint clusters routinely violate such restrictive assumptions. We bridge this gap between theory and practice, presenting the first universal score approximation theorem applicable to any compactly supported distribution in $\mathbb{R}^n$. Using a novel discretization technique that directly models the underlying distribution, we prove that the neural network complexity is governed by the support's upper Minkowski dimension $d$ rather than the ambient dimension $n$. Furthermore, by leveraging the inherent smoothing of Gaussian kernels, we show that even for irregular, fractal distributions, an $ε$ approximation error can be achieved with $\mathcal{O}(ε^{-{d/(M+1)}})$ local Taylor modules each at size of $\mathcal{O}(n^M)$, an $ε$-scaling rate previously achieved only for distributions with $(M+1)$-Hölder smoothness. Thus, we reveal a possible mechanism by which score-based diffusion models represent non-smooth data distributions.

Despite the remarkable empirical success of score-based diffusion models, their statistical guarantees remain underdeveloped. Existing analyses often provide pessimistic convergence rates that do not reflect the intrinsic low-dimensional structure common in real data, such as that arising in natural images. In this work, we study the statistical convergence of score-based diffusion models for learning an unknown distribution $μ$ from finitely many samples. Under mild regularity conditions on the forward diffusion process and the data distribution, we derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-$p$ distance. Unlike prior results, our guarantees hold for all $p \ge 1$ and require only a finite-moment assumption on $μ$, without compact-support, manifold, or smooth-density conditions. Specifically, given $n$ i.i.d.\ samples from $μ$ with finite $q$-th moment and appropriately chosen network architectures, hyperparameters, and discretization schemes, we show that the expected Wasserstein-$p$ error between the learned distribution $\hatμ$ and $μ$ scales as $\mathbb{E}\, \mathbb{W}_p(\hatμ,μ) = \widetilde{O}\!\left(n^{-1 / d^\ast_{p,q}(μ)}\right),$ where $d^\ast_{p,q}(μ)$ is the $(p,q)$-Wasserstein dimension of $μ$. Our results demonstrate that diffusion models naturally adapt to the intrinsic geometry of data and mitigate the curse of dimensionality, since the convergence rate depends on $d^\ast_{p,q}(μ)$ rather than the ambient dimension. Moreover, our theory conceptually bridges the analysis of diffusion models with that of GANs and the sharp minimax rates established in optimal transport. The proposed $(p,q)$-Wasserstein dimension also extends the notion of classical Wasserstein dimension to distributions with unbounded support, which may be of independent theoretical interest.

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.