将概率分布运输从个体样本转向子群体混合，实现高效可解释的跨分布映射。
A Biconvex Formulation for Stable Transport of Mixture Models with a Unique Solution
- 用混合模型重构运输问题，转化为严格双凸优化，保证唯一解
- 理论证明分布扰动有界时运输方案变化也受限，具备稳定性
- 计算复杂度仅与混合组分数量相关，适合大规模数据
最优传输（OT）为概率分布间的映射提供了严谨框架。尽管已有广泛进展，其在大规模数据上的应用仍面临计算负担重、点对点运输方案难以解释的问题。本文提出最优混合传输（OMT），将运输范式从个体样本转向子群体混合，将运输问题重新表述为具有唯一全局最小值的严格双凸优化。我们进一步建立了OMT映射的稳定性理论，证明底层分布的有界扰动会导致运输计划的有界变化。通过将子群体建模为指数族分布，OMT使计算复杂度脱离样本规模，仅依赖混合组分数量。我们在多种合成基准和真实数据集（包括图像与大规模单细胞RNA测序数据）上验证了OMT的有效性与实用性。
原文摘要 · Abstract (English)
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.
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