Denoising Score Matching in Diffusion Models: The Effect of Rank Constraints on Generalization

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.

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.