@inproceedings{
park2024stochastic,
title={Stochastic Optimal Control for Diffusion Bridges in Function Spaces},
author={Byoungwoo Park and Jungwon Choi and Sungbin Lim and Juho Lee},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=WyQW4G57Zd}
}
Recent advancements in diffusion models and diffusion bridges primarily focus on finite-dimensional spaces, yet many real-world problems necessitate operations in infinite-dimensional function spaces for more natural and interpretable formulations. 
In this paper, we present a theory of stochastic optimal control (SOC) tailored to infinite-dimensional spaces, aiming to extend diffusion-based algorithms to function spaces. 
Specifically, we demonstrate how Doob’s $h$-transform, the fundamental tool for constructing diffusion bridges, can be derived from the SOC perspective and expanded to infinite dimensions. 
This expansion presents a challenge, as infinite-dimensional spaces typically lack closed-form densities. 
Leveraging our theory, we establish that solving the optimal control problem with a specific objective function choice is equivalent to learning diffusion-based generative models. 
We propose two applications: 1) learning bridges between two infinite-dimensional distributions and 2) generative models for sampling from an infinite-dimensional distribution. 
Our approach proves effective for diverse problems involving continuous function space representations, such as resolution-free images, time-series data, and probability density functions.
Stochastic Optimal Control for Diffusion Bridges in Function Spaces
Byoungwoo Park Jungwon Choi Sungbin Lim Juho Lee