  Diffusion models have become the go-to method for large-scale generative
models in real-world applications. These applications often involve data
distributions confined within bounded domains, typically requiring ad-hoc
thresholding techniques for boundary enforcement. Reflected diffusion models
(Lou23) aim to enhance generalizability by generating the data distribution
through a backward process governed by reflected Brownian motion. However,
reflected diffusion models may not easily adapt to diverse domains without the
derivation of proper diffeomorphic mappings and do not guarantee optimal
transport properties. To overcome these limitations, we introduce the Reflected
Schrodinger Bridge algorithm: an entropy-regularized optimal transport approach
tailored for generating data within diverse bounded domains. We derive elegant
reflected forward-backward stochastic differential equations with Neumann and
Robin boundary conditions, extend divergence-based likelihood training to
bounded domains, and explore natural connections to entropic optimal transport
for the study of approximate linear convergence - a valuable insight for
practical training. Our algorithm yields robust generative modeling in diverse
domains, and its scalability is demonstrated in real-world constrained
generative modeling through standard image benchmarks.
