On the Relation between Rectified Flows and Optimal Transport

2025-05-26

AI Summary

This work investigates the theoretical relationship between Rectified Flows (RF) and Optimal Transport (OT), clarifying conditions under which they are equivalent—and exposing key limitations of prevailing claims. Method: Through rigorous theoretical analysis and constructive counterexamples, we examine the asymptotic convergence of RF to OT maps under gradient constraints, characterize the intrinsic velocity-field invariance of RF, and derive closed-form solutions for Gaussian and Gaussian mixture distributions. Contribution/Results: We prove that the widely cited result—“RF converges asymptotically to the OT map under gradient constraints”—holds only under stronger assumptions than previously stated (e.g., existence of jointly convex potential functions); enforcing such constraints generally fails to recover the true OT map. We establish the first systematic characterization of RF’s velocity-field invariance and provide explicit analytical solutions for canonical distributions. Finally, we derive necessary and sufficient conditions for RF–OT equivalence, delivering critical theoretical guidance—and precise boundary conditions—for integrating RF-based generative modeling with OT theory.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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 those previously acknowledged. In particular, we present several counter-examples that invalidate earlier equivalence results in the literature, and we argue that enforcing a gradient constraint on rectified flows is, in general, not a reliable method for computing optimal transport maps.

Problem

• Explores connections between rectified flows and optimal transport
• Analyzes invariance properties and explicit velocity fields in rectified flows
• Challenges claims linking gradient-constrained rectified flows to optimal transport

Innovation

• Rectified flows straighten transport paths
• Analyze Gaussian and mixture settings
• Gradient constraints not reliable for optimal transport