发表机构
University of Toronto; Florida State University; University of Georgia; University of North Carolina at Chapel Hill（多伦多大学; 佛罗里达州立大学; 佐治亚大学; 北卡罗来纳大学教堂山分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出黎曼齐性空间上的MGFA模型，证明其MLE的根-n一致性，提出迭代算法，经实验及案例研究表明该模型在流形值数据聚类及形状分析中表现优异。
AI 中文摘要
本文在黎曼齐性空间上引入了测地因子分析器混合模型（Mixtures of Geodesic Factor Analyzers, MGFA）。MGFA在每个混合分量内采用测地因子模型，相比黎曼径向分布混合模型具有更强的表达能力，可对具有各向异质子群的流形值数据进行聚类。我们证明了MGFA最大似然估计器（Maximum Likelihood Estimator, MLE）的根-n一致性，从而填补了黎曼径向分布混合模型这一特殊情形的理论空白。我们还提出了一种迭代估计算法，并将其应用于球面、形状空间和双曲空间。数值实验表明，MGFA在模型设定正确的场景中显著优于对比方法，同时在模型设定错误时仍保持鲁棒性。最后，在胼胝体和左海马体形状数据集上的案例研究证明了MGFA在二维轮廓和三维形状分析中的有效性。
英文摘要
This paper introduces Mixtures of Geodesic Factor Analyzers (MGFA) on Riemannian homogeneous spaces. MGFA uses a geodesic factor model within each mixture component, providing greater expressiveness than mixtures of Riemannian radial distributions and enabling clustering of manifold-valued data with anisotropic subpopulations. We establish root-$n$ consistency for the MGFA maximum likelihood estimator (MLE), thereby filling a theoretical gap for mixtures of Riemannian radial distributions as a special case. We also propose an iterative estimation algorithm and implement it on spheres, shape spaces, and hyperbolic spaces. Numerical experiments show that MGFA substantially outperforms competing methods in well-specified regimes while remaining robust under model misspecification. Finally, case studies on corpus callosum and left hippocampus shape datasets demonstrate MGFA's effectiveness for both 2D contour and 3D shape analysis.
Comments22 pages main, 49 pages appendix
