  Diffusion models are gaining widespread use in cutting-edge image, video, and
audio generation. Score-based diffusion models stand out among these methods,
necessitating the estimation of score function of the input data distribution.
In this study, we present a theoretical framework to analyze two-layer neural
network-based diffusion models by reframing score matching and denoising score
matching as convex optimization. We prove that training shallow neural networks
for score prediction can be done by solving a single convex program. Although
most analyses of diffusion models operate in the asymptotic setting or rely on
approximations, we characterize the exact predicted score function and
establish convergence results for neural network-based diffusion models with
finite data. Our results provide a precise characterization of what neural
network-based diffusion models learn in non-asymptotic settings.
