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[2602.12683] Flow Matching from Viewpoint of Proximal Operators
Summary
This article presents a reformulation of Optimal Transport Conditional Flow Matching (OT-CFM) through proximal operators, demonstrating its application without density assumptions on target distributions.
Why It Matters
The findings provide a significant advancement in generative modeling by offering a new perspective on OT-CFM, which could enhance the efficiency and applicability of machine learning models in various domains, particularly in scenarios with complex target distributions.
Key Takeaways
- OT-CFM can be reformulated using proximal operators, enhancing its theoretical foundation.
- The approach does not require the target distribution to have a density, broadening its applicability.
- Convergence of minibatch OT-CFM to population formulations is established as batch size increases.
- The dynamics of OT-CFM exhibit normal hyperbolicity for manifold-supported targets, indicating stability in certain directions.
- The use of second epi-derivatives of convex potentials provides deeper insights into the model's behavior.
Computer Science > Machine Learning arXiv:2602.12683 (cs) [Submitted on 13 Feb 2026] Title:Flow Matching from Viewpoint of Proximal Operators Authors:Kenji Fukumizu, Wei Huang, Han Bao, Shuntuo Xu, Nisha Chandramoothy View a PDF of the paper titled Flow Matching from Viewpoint of Proximal Operators, by Kenji Fukumizu and 4 other authors View PDF Abstract:We reformulate Optimal Transport Conditional Flow Matching (OT-CFM), a class of dynamical generative models, showing that it admits an exact proximal formulation via an extended Brenier potential, without assuming that the target distribution has a density. In particular, the mapping to recover the target point is exactly given by a proximal operator, which yields an explicit proximal expression of the vector field. We also discuss the convergence of minibatch OT-CFM to the population formulation as the batch size increases. Finally, using second epi-derivatives of convex potentials, we prove that, for manifold-supported targets, OT-CFM is terminally normally hyperbolic: after time rescaling, the dynamics contracts exponentially in directions normal to the data manifold while remaining neutral along tangential directions. Comments: Subjects: Machine Learning (cs.LG); Machine Learning (stat.ML) Cite as: arXiv:2602.12683 [cs.LG] (or arXiv:2602.12683v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2602.12683 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Kenji ...
