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[2505.19712] On the Relation between Rectified Flows and Optimal Transport
Summary
This paper explores the relationship between rectified flows and optimal transport, highlighting invariance properties and counterexamples that challenge previous equivalence claims in the literature.
Why It Matters
Understanding the connections between rectified flows and optimal transport is crucial for advancing generative modeling techniques in machine learning. This research provides deeper insights into the limitations of current methods, which can lead to more reliable applications in various fields, including data science and AI.
Key Takeaways
- Rectified flows can straighten transport paths in generative models.
- The paper presents counterexamples that challenge earlier claims about rectified flows and optimal transport equivalence.
- Gradient constraints on rectified flows are not a reliable method for computing optimal transport maps.
- The study offers explicit constructions and analysis in Gaussian settings.
- Invariance properties of rectified flows are established, contributing to the theoretical framework.
Computer Science > Machine Learning arXiv:2505.19712 (cs) [Submitted on 26 May 2025 (v1), last revised 15 Feb 2026 (this version, v3)] Title:On the Relation between Rectified Flows and Optimal Transport Authors:Johannes Hertrich, Antonin Chambolle, Julie Delon View a PDF of the paper titled On the Relation between Rectified Flows and Optimal Transport, by Johannes Hertrich and 2 other authors View PDF HTML (experimental) Abstract:This paper investigates the connections between rectified flows, flow matching, and optimal transport. Flow matching is a recent approach to learning generative models by estimating velocity fields that guide transformations from a source to a target distribution. Rectified flow matching aims to straighten the learned transport paths, yielding more direct flows between distributions. Our first contribution is a set of invariance properties of rectified flows and explicit velocity fields. In addition, we also provide explicit constructions and analysis in the Gaussian (not necessarily independent) and Gaussian mixture settings and study the relation to optimal transport. Our second contribution addresses recent claims suggesting that rectified flows, when constrained such that the learned velocity field is a gradient, can yield (asymptotically) solutions to optimal transport problems. We study the existence of solutions for this problem and demonstrate that they only relate to optimal transport under assumptions that are significantly stronger than...
