Q1. Can neural networks well approximate and learn score functions, especially when data have intrinsic geometric structures? If so, how should one choose the neural network architectures, and what is the sample complexity of learning? Q2. Can diffusion models estimate the data distribution using the learned score functions? If so, how are the data intrinsic geometric structures being captured and how do they affect the sample complexity?

We increasingly see data is very low dimensional

We find complexity bounds for diffusion models supported on low dimensional linear subspaces