Manifold Generalization Provably Proceeds Memorization in Diffusion Models

Abstract

Diffusion models often generate novel samples even when the learned score is only \emph{coarse} -- a phenomenon not accounted for by the standard view of diffusion training as density estimation. In this paper, we show that, under the \emph{manifold hypothesis}, this behavior can instead be explained by coarse scores capturing the \emph{geometry} of the data while discarding the fine-scale distributional structure of the population measure~$\mu_{\scriptscriptstyle\mathrm{data}}$. Concretely, whereas estimating the full data distribution $\mu_{\scriptscriptstyle\mathrm{data}}$ 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 \emph{regularity of the manifold support} and attain a near-parametric rate toward a \emph{different} target distribution. This target distribution has density uniformly comparable to that of~$\mu_{\scriptscriptstyle\mathrm{data}}$ throughout any $\tilde{\mathcal{O}}\bigl(N^{-\beta/(4k)}\bigr)$-neighborhood of the manifold, where $\beta$ 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 \emph{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~$\mu_{\scriptscriptstyle\mathrm{data}}$.