Score-based diffusion models have demonstrated remarkable empirical success in learning high-dimensional distributions, particularly those exhibiting low-dimensional and multi-modal structures. However, theoretical understanding of their statistical efficiency remains limited. Existing theories typically rely on strong regularity assumptions, such as uniformly bounded densities or globally smooth score functions, which fail to capture such intrinsic structures. In this work, we study the sample complexity of diffusion models for learning distributions supported on a union of low-dimensional subspaces. Assuming that the data distribution within each subspace is subgaussian, we show that diffusion models require at most $\widetilde{O}(\varepsilon^{-k \vee 2})$ samples to achieve $\varepsilon$ error in 1-Wasserstein distance, where $k$ is the intrinsic dimension. This near-optimal convergence rate depends only on the intrinsic dimension and significantly improves upon prior theoretical guarantees that suffer from the curse of dimensionality. Notably, our analysis applies to a broad collection of distributions without imposing smoothness, bounded-density, or log-concavity assumptions. Overall, our results show that diffusion models can statistically adapt to intrinsic low-dimensional structure while naturally accommodating multi-modal data, offering a rigorous theoretical justification for their success in complex high-dimensional learning tasks.

翻译：基于得分的扩散模型在高维分布学习（尤其是具有低维和多模态结构的分布）中展现出显著的实证成功。然而，其统计效率的理论理解仍十分有限。现有理论通常依赖于强正则性假设，如一致有界密度或全局光滑得分函数，这些假设无法捕捉此类内在结构。本文研究了扩散模型在低维子空间并集支撑分布学习中的样本复杂度。假设各子空间内数据分布满足次高斯性，我们证明扩散模型在1-Wasserstein距离下达到$\varepsilon$误差所需的样本数至多为$\widetilde{O}(\varepsilon^{-k \vee 2})$，其中$k$为内在维度。该近优收敛速率仅依赖于内在维度，并显著改进了受维数灾难制约的先前理论保证。值得注意的是，我们的分析适用于广泛分布族，无需施加光滑性、有界密度或对数凹性假设。总体而言，研究结果表明扩散模型在统计上能自适应内在低维结构，同时自然处理多模态数据，为其在复杂高维学习任务中的成功提供了严格的理论依据。