Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity

We study the theoretical behavior of denoising score matching--the learning task associated to diffusion models--when the data distribution is supported on a low-dimensional manifold and the score is parameterized using a random feature neural network. We derive asymptotically exact expressions for the test, train, and score errors in the high-dimensional limit. Our analysis reveals that, for linear manifolds the sample complexity required to learn the score function scales linearly with the intrinsic dimension of the manifold, rather than with the ambient dimension. Perhaps surprisingly, the benefits of low-dimensional structure starts to diminish once we have a non-linear manifold. These results indicate that diffusion models can benefit from structured data; however, the dependence on the specific type of structure is subtle and intricate.

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.

Diffusion models often generate novel samples even when the learned score is only coarse -- a phenomenon not accounted for by the standard view of diffusion training as density estimation. In this paper, we show that, under the manifold hypothesis, this behavior can instead be explained by coarse scores capturing the geometry of the data while discarding the fine-scale distributional structure of the population measure~$μ_{\scriptscriptstyle\mathrm{data}}$. Concretely, whereas estimating the full data distribution $μ_{\scriptscriptstyle\mathrm{data}}$ supported on a $k$-dimensional manifold is known to require the classical minimax rate $\tilde{\mathcal{O}}(N^{-1/k})$, we prove that diffusion models trained with coarse scores can exploit the regularity of the manifold support and attain a near-parametric rate toward a different target distribution. This target distribution has density uniformly comparable to that of~$μ_{\scriptscriptstyle\mathrm{data}}$ throughout any $\tilde{\mathcal{O}}\bigl(N^{-β/(4k)}\bigr)$-neighborhood of the manifold, where $β$ denotes the manifold regularity. Our guarantees therefore depend only on the smoothness of the underlying support, and are especially favorable when the data density itself is irregular, for instance non-differentiable. In particular, when the manifold is sufficiently smooth, we obtain that generalization -- formalized as the ability to generate novel, high-fidelity samples -- occurs at a statistical rate strictly faster than that required to estimate the full population distribution~$μ_{\scriptscriptstyle\mathrm{data}}$.