通过部分最优传输刻画有限混合估计中的异质速率
Characterizing Heterogeneous Rates in Finite Mixture Estimation via Partial Optimal Transport
浏览论文内容
中文总结 AI 辅助
本文提出基于Voronoi的部分最优传输（VPOT）框架，用于刻画有限混合模型最大似然估计的异质收敛速率，在任意固定维度下建立配置自适应的局部与全局上界及极小极大最优下界，揭示不同局部奇异性对应的不同收敛速度。
中文摘要 AI 辅助
有限混合模型中的参数估计可以表现出高度异质的收敛行为：局部孤立的成分可能比相互竞争的成分组被估计得更快。现有的基于Wasserstein距离的分析通常只刻画最坏情况下的速率，因此不能完全捕捉这种局部异质性。在本文中，我们引入了一个基于Voronoi的部分最优传输（VPOT）框架，以获得混合度量最大似然估计器的精细局部和全局收敛保证。关键的几何思想是将两个混合度量的比较局部化到扩展的Voronoi邻域，并使用部分最优传输来适应其局部限制的不等质量。在每个邻域内，一阶POT差异被提升到由局部竞争原子数量决定的幂，使得由此产生的损失能够适应局部的奇异性程度。在适当的正则性和强可识别性条件下，我们建立了在VPOT损失下最大似然估计器的统一局部和全局上界。这些上界揭示了一种依赖于配置的参数估计形式：较不奇异的局部配置允许更快的收敛，而最奇异的配置则恢复了由基于Wasserstein的分析所刻画的经典最坏情况行为。我们进一步建立了极小极大下界，表明在VPOT损失下估计混合度量的收敛速率是最优的。我们的结果在任意固定维度下成立，不需要混合比例一致地远离零，也不需要预先知道混合成分的真实数量。总体而言，VPOT提供了一个配置自适应的框架，用于捕捉有限混合模型中的异质参数估计行为。
英文摘要
Parameter estimation in finite mixture models can exhibit highly heterogeneous convergence behavior: locally isolated components may be estimated substantially faster than groups of competing components. Existing analyses based on Wasserstein distances typically characterize only the worst-case rate and therefore do not fully capture this local heterogeneity. In this paper, we introduce a Voronoi-based partial optimal transport (VPOT) framework for obtaining refined local and global convergence guarantees for the maximum likelihood estimator of the mixing measure. The key geometric idea is to localize the comparison of two mixing measures to extended Voronoi neighborhoods and use partial optimal transport to accommodate the unequal masses of their local restrictions. Within each neighborhood, the first-order POT discrepancy is raised to a power determined by the number of locally competing atoms, allowing the resulting loss to adapt to the local degree of singularity. Under suitable regularity and strong identifiability conditions, we establish uniform local and global upper bounds for a maximum likelihood estimator under the VPOT loss. These bounds reveal a configuration-dependent form of parameter estimation: less singular local configurations admit faster convergence, whereas the most singular configuration recovers the classical worst-case behavior characterized by Wasserstein-based analyses. We further establish a minimax lower bound showing that the convergence rate for estimating the mixing measure under the VPOT loss is optimal. Our results hold in arbitrary fixed dimension without requiring mixing proportions to be uniformly bounded away from zero or prior knowledge of the true number of mixture components. Overall, VPOT provides a configuration-adaptive framework for capturing heterogeneous parameter-estimation behavior in finite mixture models.
发表机构
- The University of Texas at Austin（德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。
