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This paper introduces Quantile AlignTree Flow Matching (QAT-FM), a novel coupling strategy that efficiently constructs a hierarchical coupling between a Gaussian prior and target distributions, addressing the limitations of independent and optimal transport-based couplings. By leveraging a quantile-aligned tree structure, QAT-FM achieves a coupling construction time of $\mathcal{O}(Nd\log N)$ and supports scalable training for high-dimensional generative tasks. Experimental results show that QAT-FM not only enhances generative performance but also significantly reduces coupling construction costs while ensuring non-crossing interpolation paths and improved path separation.
QAT-FM reduces coupling construction costs while achieving competitive generative performance, transforming how we approach high-dimensional generative tasks.
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.
