  Diffusion models have become the most popular approach to deep generative
modeling of images, largely due to their empirical performance and reliability.
From a theoretical standpoint, a number of recent
works~\cite{chen2022,chen2022improved,benton2023linear} have studied the
iteration complexity of sampling, assuming access to an accurate diffusion
model. In this work, we focus on understanding the \emph{sample complexity} of
training such a model; how many samples are needed to learn an accurate
diffusion model using a sufficiently expressive neural network? Prior
work~\cite{BMR20} showed bounds polynomial in the dimension, desired Total
Variation error, and Wasserstein error. We show an \emph{exponential
improvement} in the dependence on Wasserstein error and depth, along with
improved dependencies on other relevant parameters.
