Stochastic Optimal Control for Diffusion Bridges in Function Spaces
- URL: http://arxiv.org/abs/2405.20630v2
- Date: Mon, 3 Jun 2024 03:11:45 GMT
- Title: Stochastic Optimal Control for Diffusion Bridges in Function Spaces
- Authors: Byoungwoo Park, Jungwon Choi, Sungbin Lim, Juho Lee,
- Abstract summary: We present a theory of optimal control tailored to infinite-dimensional spaces.
We show how Doob's $h$-transform can be derived from the SOC perspective and expanded to infinite dimensions.
We propose two applications: learning bridges between two infinite-dimensional distributions and generative models for sampling from an infinite-dimensional distribution.
- Score: 13.544676987441441
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: 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.
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