Optimal transport (OT) provides a principled framework for mapping between probability distributions. Despite extensive progress, applying OT to large-scale data remains computationally demanding, and the resulting pointwise transport plans are often difficult to interpret. We introduce Optimal Mixture Transport (OMT), a scalable framework that shifts the transport paradigm from individual samples to mixtures of subpopulations, reformulating the transport problem as a strictly biconvex optimization with a unique global minimizer. We further establish theoretical guarantees on the stability of the OMT map, showing that bounded perturbations of the underlying distributions lead to bounded changes in the transport plan. By formulating subpopulations as exponential-family distributions, OMT decouples computational complexity from the sample size, scaling solely with the number of mixture components. We demonstrate the effectiveness and practicality of OMT on a wide range of synthetic benchmarks and real-world datasets, including image data and large-scale single-cell RNA sequencing measurements.

翻译：最优传输(OT)为概率分布之间的映射提供了理论框架。尽管已有显著进展，但将OT应用于大规模数据仍面临计算高成本问题，且生成的逐点传输方案往往难以解释。本文提出最优混合传输(OMT)——一种可扩展框架，将传输范式从单个样本迁移至子群体混合，并将传输问题重构为具有唯一全局最小化器的严格双凸优化问题。进一步地，我们建立了OMT映射稳定性的理论保证，表明基础分布的有界扰动会引发传输方案的有界变化。通过将子群体建模为指数族分布，OMT实现了计算复杂度与样本量的解耦，仅随混合成分数量缩放。我们在合成基准测试和真实数据集（包括图像数据和大规模单细胞RNA测序数据）上验证了OMT的有效性与实用性。