Beyond Straightness: Non-Crossing Flow Matching via Quantile AlignTree Coupling

2026-09-03

AI Summary

为解决流匹配中路径交叉和速度模糊问题，提出基于分位数对齐树结构的QAT-FM方法，有效构建源与目标分布间的层次化耦合。

Abstract

The performance of Flow Matching largely depends on the quality of the coupling between the source and target distributions. However, independent coupling often leads to path crossings and local velocity ambiguity, while OT-based couplings typically incur high construction costs. To address this challenge, we propose Quantile AlignTree Flow Matching (QAT-FM), an efficient structured coupling strategy that constructs a hierarchical coupling between a Gaussian prior and the target data distribution via a quantile-aligned tree structure. QAT-FM constructs the coupling in $\mathcal{O}(Nd\log N)$ time and supports per-pair source sampling with $\mathcal{O}(d)$ complexity, enabling scalable training for large-scale high-dimensional generative tasks. Theoretically, we prove that the QAT coupling satisfies marginal consistency, induces non-crossing linear interpolation paths, and consistently improves path separation at intermediate times compared with independent coupling, thereby alleviating local velocity ambiguity. QAT-FM further extends naturally to conditional generation, enabling structured conditional coupling while preserving global Gaussian alignment. Experiments across diverse benchmark datasets demonstrate that QAT-FM achieves competitive generative performance while substantially reducing coupling construction cost.

Problem

• Flow Matching
• coupling quality
• path crossings
• local velocity ambiguity
• construction cost

Innovation

• Quantile AlignTree
• Flow Matching
• non-crossing paths
• marginal consistency
• conditional generation