{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/stochastic-optimal-control-for-diffusion","title":"Stochastic Optimal Control for Diffusion Bridges in Function Spaces","arxiv_id":"2405.20630","date":"2024-05-31","proceeding":null,"authors":["Byoungwoo Park","JungWon Choi","Sungbin Lim","Juho Lee"],"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.","url_abs":"https://arxiv.org/abs/2405.20630v3","url_pdf":"https://arxiv.org/pdf/2405.20630v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"stochastic-optimal-control-for-diffusion","repo_url":"https://github.com/bw-park/DBFS","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"image-to-image-translation","task_name":"Image-to-Image Translation"},{"task_slug":"time-series-1","task_name":"Time Series"}],"methods":[{"method_slug":"diffusion","method_name":"Diffusion"},{"method_slug":"focus","method_name":"Focus"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2405.20630","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}