Mixtures of Geodesic Factor Analyzers on Riemannian Homogeneous Spaces

Hengchao Chen, Yuanyao Tan, Chao Huang, Hongtu Zhu, Qiang Sun
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:16823-16874, 2026.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-chen26eb, title = {Mixtures of Geodesic Factor Analyzers on {R}iemannian Homogeneous Spaces}, author = {Chen, Hengchao and Tan, Yuanyao and Huang, Chao and Zhu, Hongtu and Sun, Qiang}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {16823--16874}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/chen26eb/chen26eb.pdf}, url = {https://proceedings.mlr.press/v306/chen26eb.html}, 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.} }
Endnote
%0 Conference Paper %T Mixtures of Geodesic Factor Analyzers on Riemannian Homogeneous Spaces %A Hengchao Chen %A Yuanyao Tan %A Chao Huang %A Hongtu Zhu %A Qiang Sun %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-chen26eb %I PMLR %P 16823--16874 %U https://proceedings.mlr.press/v306/chen26eb.html %V 306 %X 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.
APA
Chen, H., Tan, Y., Huang, C., Zhu, H. & Sun, Q.. (2026). Mixtures of Geodesic Factor Analyzers on Riemannian Homogeneous Spaces. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:16823-16874 Available from https://proceedings.mlr.press/v306/chen26eb.html.

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