  Diffusion-based generative models have achieved promising results recently,
but raise an array of open questions in terms of conceptual understanding,
theoretical analysis, algorithm improvement and extensions to discrete,
structured, non-Euclidean domains. This work tries to re-exam the overall
framework, in order to gain better theoretical understandings and develop
algorithmic extensions for data from arbitrary domains. By viewing diffusion
models as latent variable models with unobserved diffusion trajectories and
applying maximum likelihood estimation (MLE) with latent trajectories imputed
from an auxiliary distribution, we show that both the model construction and
the imputation of latent trajectories amount to constructing diffusion bridge
processes that achieve deterministic values and constraints at end point, for
which we provide a systematic study and a suit of tools. Leveraging our
framework, we present 1) a first theoretical error analysis for learning
diffusion generation models, and 2) a simple and unified approach to learning
on data from different discrete and constrained domains. Experiments show that
our methods perform superbly on generating images, semantic segments and 3D
point clouds.
