  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.
