Optimal Statistical Guarantees for Diffusion Models on Low-Dimensional, Multi-Modal Data

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 in learning distributions supported on a union of low-dimensional subspaces. Assuming that data within each subspace follows a subgaussian distribution, 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 rate depends only on the intrinsic dimension and significantly improves upon prior works that suffer from the curse of dimensionality. Notably, our analysis applies to a broad collection of distributions without requiring smoothness or log-concavity assumptions. These results provide rigorous evidence for the effectiveness of diffusion models in learning low-dimensional, multi-modal distributions.