Mixture Probabilistic Principal Geodesic Analysis
This work addresses the problem of handling complex, multi-modal data in fields like medical imaging, offering an incremental improvement over existing single-modality methods.
The paper tackles the challenge of dimensionality reduction for multi-modal data on Riemannian manifolds by introducing a mixture model of probabilistic principal geodesic analysis (MPPGA) and a Bayesian variant (MBPGA), which automatically estimate parameters and suppress irrelevant dimensions, demonstrating advantages in clustering and shape analysis with synthetic and real medical imaging data.
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.