Beyond Transport Cost: Routing Differences between Flow Matching and Optimal Transport

2026-10-05

AI Summary

This study addresses a critical limitation in existing generative models, where coupling design relies solely on transport cost while neglecting routing discrepancies, leading to evaluation biases between Flow Matching (FM) and Optimal Transport (OT). We propose a decoupled perspective of routing and cost, revealing their fundamental distinctions within FM and OT frameworks. By leveraging exact FM trajectories as an oracle to construct route-aware training couplings, our approach transcends the conventional paradigm of mere cost minimization. Both theoretical analysis and neural network-based empirical evaluations demonstrate that the proposed coupling significantly outperforms cost-equivalent baselines in directional consistency. These findings confirm that jointly optimizing transport cost and routing is essential for enhancing generation quality.

Abstract

In generative models, Optimal Transport (OT) is used to improve Flow Matching (FM) by reducing noise-data coupling cost. However, different noise-to-output assignments can yield nearly equal costs, raising a key question. Is cost alone sufficient to guide coupling design? We address this question by separating transport cost from routing, i.e., the destination reached by each noise sample. We show numerically how FM and OT can differ in routing while remaining close in cost. We examine its consequences in learned neural FM. Using the exact FM routing as an oracle, we further construct a routing-aware training coupling and find that it yields a directionally consistent improvement in generation over a cost-matched, cost-only counterpart. Our findings highlight what cost minimization can overlook and motivate using both cost and routing to evaluate the design of OT-based FM couplings. Code will be released upon acceptance.

Problem

• Flow Matching
• Optimal Transport
• Transport Cost
• Routing
• Generative Models

Innovation

• Flow Matching
• Optimal Transport
• Routing
• Generative Models
• Coupling Design