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Diffusion models achieve state-of-the-art performance in various generationtasks. However, their theoretical foundations fall far behind. This paperstudies score approximation, estimation, and distribution recovery of diffusionmodels, when data are supported on an unknown low-dimensional linear subspace.Our result provides sample complexity bounds for distribution estimation usingdiffusion models. We show that with a properly chosen neural networkarchitecture, the score function can be both accurately approximated andefficiently estimated. Furthermore, the generated distribution based on theestimated score function captures the data geometric structures and convergesto a close vicinity of the data distribution. The convergence rate depends onthe subspace dimension, indicating that diffusion models can circumvent thecurse of data ambient dimensionality.

Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data