发表机构
Institute of Statistics and Big Data, Renmin University of China; Department of Statistics and Data Science, Tsinghua University（中国人民大学统计与大数据研究院; 清华大学统计与数据科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对流匹配的路径交叉与耦合成本高问题，提出QAT-FM方法，通过分位数对齐树构建高效结构化耦合，在多基准数据集上实现了性能与成本的优化平衡。
AI 中文摘要
流匹配的性能在很大程度上取决于源分布与目标分布之间耦合的质量。然而，独立耦合常会导致路径交叉和局部速度模糊，而基于最优传输（OT）的耦合通常会产生较高的构造成本。为应对这一挑战，我们提出了分位数对齐树流匹配（Quantile AlignTree Flow Matching，简称QAT-FM），这是一种高效的结构化耦合策略，它通过分位数对齐的树结构在高斯先验与目标数据分布之间构建分层耦合。QAT-FM以O(Nd log N)的时间复杂度构建耦合，并支持每对源采样达到O(d)的复杂度，可实现大规模高维生成任务的可扩展训练。理论上，我们证明QAT耦合满足边际一致性，能诱导无交叉的线性插值路径，且与独立耦合相比，在中间时刻持续提升路径分离度，从而缓解局部速度模糊。QAT-FM还可自然扩展到条件生成，在保留全局高斯对齐的同时实现结构化条件耦合。在多个基准数据集上的实验表明，QAT-FM在达到有竞争力的生成性能的同时，大幅降低了耦合构造成本。
英文摘要
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.
