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~$μ_{\scriptscriptstyle\mathrm{data}}$. 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 \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~$μ_{\scriptscriptstyle\mathrm{data}}$ 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 \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~$μ_{\scriptscriptstyle\mathrm{data}}$.

翻译：扩散模型常能在学习到的分数函数仅为粗糙估计时生成新颖样本——这一现象无法用密度估计的标准视角解释扩散训练过程。本文证明，在流形假设下，该行为可被解释为：粗糙分数捕捉了数据的几何结构，同时舍弃了总体测度μ_data的精细尺度分布特征。具体而言，已知估计支撑于k维流形上的完整数据分布μ_data需达到经典极小极大速率Õ(N^{-1/k})，而我们证明，采用粗糙分数训练的扩散模型可利用流形支撑的正则性，以近参数速率收敛至另一个不同的目标分布。该目标分布密度在流形的任意Õ(N^{-β/(4k)})邻域内与μ_data的密度均匀可比，其中β表示流形正则性。因此，我们的保证仅依赖于底层支撑的光滑性，且当数据密度本身不规则（例如不可微）时尤为有利。特别地，当流形足够光滑时，我们得到：泛化（形式化为生成新颖高保真样本的能力）发生的统计速率严格快于估计完整总体分布μ_data所需的速率。