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This paper provides a theoretical analysis of diffusion models, demonstrating their statistical optimality for learning distributions supported on a union of low-dimensional subspaces. The authors prove a sample complexity bound of $\widetilde{O}(\varepsilon^{-k \vee 2})$ for achieving $\varepsilon$ error in 1-Wasserstein distance, where $k$ is the intrinsic dimension, assuming subgaussian data within each subspace. This result establishes that diffusion models can adapt to intrinsic low-dimensional structure and handle multi-modal data without strong regularity assumptions, offering a theoretical foundation for their empirical success.
Diffusion models can provably learn complex, multi-modal data in high dimensions with near-optimal sample complexity, finally justifying their real-world performance.
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.
