AI 中文总结
本文针对高斯混合模型，开发熵正则化部分最优传输并定义部分混合格罗莫夫-瓦瑟斯坦距离，证明其极小值存在唯一性等性质，经数值实验验证该方法对异常值匹配具有鲁棒性。
AI 中文摘要
最优传输和格罗莫夫-瓦瑟斯坦（Gromov--Wasserstein）距离是比较概率测度和度量测度空间的有用工具，但它们的平衡公式要求所有质量都必须匹配。对于存在异常值、缺失部分或仅有部分重叠的数据，这一约束通常过于严格。在本文中，我们针对高斯混合模型开发了熵正则化部分最优传输，并定义了部分混合格罗莫夫-瓦瑟斯坦距离。对于有限维熵正则化部分最优传输问题，我们证明了极小值的存在性与唯一性，并建立了定量的大惩罚估计。此外，所得的熵正则化部分分量耦合通过高斯最优映射诱导出连续的部分传输计划。我们分析了它们的大惩罚极限及随后的零熵极限，并构造了相关的位移插值和重心投影映射。另外，通过将每个高斯混合模型识别为由高斯分量构成的有限度量测度空间，我们建立了部分混合格罗莫夫-瓦瑟斯坦距离的度量性质和大惩罚极限。最后，在合成高斯混合模型和点云上的数值实验，说明了惩罚项和熵正则化的作用，以及部分匹配对异常值的鲁棒性。
英文摘要
Optimal transport and Gromov--Wasserstein distances are useful tools for comparing probability measures and metric measure spaces, but their balanced formulations force all mass to be matched. This constraint is often too strong for data with outliers, missing parts, or only partial overlap. In this paper, we develop entropic partial optimal transport for Gaussian mixture models and define a partial mixture Gromov--Wasserstein distance. For the finite entropic partial optimal transport problem, we prove the existence and uniqueness of the minimizer and establish quantitative large-penalty estimates. Moreover, the resulting entropic partial component couplings induce continuous partial transport plans through Gaussian optimal maps. We analyze their large-penalty and subsequent zero-entropy limits and construct the associated displacement interpolations and barycentric projection maps. In addition, by identifying each Gaussian mixture with a finite metric measure space of Gaussian components, we establish the metric property and large-penalty limit of the partial mixture Gromov--Wasserstein distance. Finally, numerical experiments on synthetic Gaussian mixtures and point clouds illustrate the effects of the penalty and entropic regularization and the robustness of partial matching to outliers.
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