Mixture Probabilistic Principal Geodesic Analysis

TL;DR

This paper introduces a mixture probabilistic principal geodesic analysis model that extends existing methods to handle multiple data modalities on Riemannian manifolds, enabling automatic subspace estimation and dimensionality reduction.

Contribution

It proposes a novel Gaussian mixture model framework for probabilistic principal geodesic analysis, with an EM algorithm for parameter estimation and automatic dimensionality reduction.

Findings

Effective clustering and shape analysis demonstrated on synthetic and real data.

Automatic estimation of principal subspaces improves analysis accuracy.

Model handles complex nonlinear data structures on manifolds.

Abstract

Dimensionality reduction on Riemannian manifolds is challenging due to the complex nonlinear data structures. While probabilistic principal geodesic analysis~(PPGA) has been proposed to generalize conventional principal component analysis (PCA) onto manifolds, its effectiveness is limited to data with a single modality. In this paper, we present a novel Gaussian latent variable model that provides a unique way to integrate multiple PGA models into a maximum-likelihood framework. This leads to a well-defined mixture model of probabilistic principal geodesic analysis (MPPGA) on sub-populations, where parameters of the principal subspaces are automatically estimated by employing an Expectation Maximization algorithm. We further develop a mixture Bayesian PGA (MBPGA) model that automatically reduces data dimensionality by suppressing irrelevant principal geodesics. We demonstrate the advantages of our model in the contexts of clustering and statistical shape analysis, using synthetic sphere data, real corpus callosum, and mandible data from human brain magnetic resonance~(MR) and CT images.