Mixture of Geodesic Factor Analyzers on Riemannian Homogeneous Spaces

2026-08-07

AI Summary

This work addresses the challenge of clustering manifold-valued data in Riemannian homogeneous spaces when anisotropic subpopulations are present. We propose a Mixture of Geodesic Factor Analyzers (MGFA), which, for the first time, incorporates geodesic factor models into this setting to flexibly capture local geometric structures. The method bridges a theoretical gap by establishing √n-consistency of maximum likelihood estimators for mixtures of Riemannian radial distributions—a property previously unattained—and is efficiently optimized via an EM-type algorithm. Empirical evaluations demonstrate that MGFA substantially outperforms existing approaches under correctly specified models and remains robust under model misspecification. Its effectiveness is further validated through analyses of real brain shape data, including the corpus callosum and hippocampus, across spherical, shape, and hyperbolic manifolds.

Abstract

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.

Problem

• manifold-valued data
• anisotropic subpopulations
• Riemannian homogeneous spaces
• clustering
• mixture models

Innovation

• Mixture of Geodesic Factor Analyzers
• Riemannian homogeneous spaces
• geodesic factor model
• anisotropic clustering
• root-n consistency