Reflected Schrödinger Bridge for Constrained Generative Modeling
This work addresses constrained generative modeling for applications like image generation, offering a more principled approach than ad-hoc methods, though it builds incrementally on reflected diffusion models.
The paper tackles the problem of generating data within bounded domains, which is common in real-world applications, by introducing the Reflected Schrödinger Bridge algorithm, resulting in robust generative modeling across diverse domains and demonstrated scalability on standard image benchmarks.
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.