Mixtures of geodesic factor analyzers on Riemannian homogeneous spaces

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.

Many modern datasets are not ordinary tables of numbers. For example, brain shapes, directions on a sphere, and certain network representations live on curved spaces called manifolds. This makes standard clustering methods less reliable, because they ignore the geometry of the space. We introduce Mixtures of Geodesic Factor Analyzers (MGFA), a method for clustering data on these curved spaces. MGFA models each cluster using a low-dimensional hidden structure, allowing it to capture directional variation within a group rather than treating every group as a simple round cloud. We also develop theory showing how complex this model is and how accurately its probability distribution can be estimated from data. Experiments on simulated spherical, shape, and hyperbolic data show that MGFA recovers clusters more accurately than existing approaches when hidden factor structure is present. Applications to brain-shape data from Alzheimer's disease studies suggest that MGFA can identify meaningful patient subgroups.