An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data

TL;DR

This paper introduces a probabilistic, intrinsic approach to PCA on manifolds using stochastic development, allowing analysis of nonlinear manifold data without linearization, and incorporates curvature effects.

Contribution

It generalizes PCA to manifold data through a probabilistic model based on stochastic processes and connections, avoiding linearization and enabling global principal component transport.

Findings

Curvature affects the integrability of principal subspaces.

Stochastic flows offer an alternative to explicit subspace construction.

The model enables inference and prediction on embedded surfaces.

Abstract

We provide a probabilistic and infinitesimal view of how the principal component analysis procedure (PCA) can be generalized to analysis of nonlinear manifold valued data. Starting with the probabilistic PCA interpretation of the Euclidean PCA procedure, we show how PCA can be generalized to manifolds in an intrinsic way that does not resort to linearization of the data space. The underlying probability model is constructed by mapping a Euclidean stochastic process to the manifold using stochastic development of Euclidean semimartingales. The construction uses a connection and bundles of covariant tensors to allow global transport of principal eigenvectors, and the model is thereby an example of how principal fiber bundles can be used to handle the lack of global coordinate system and orientations that characterizes manifold valued statistics. We show how curvature implies non-integrability of the equivalent of Euclidean principal subspaces, and how the stochastic flows provide an alternative to explicit construction of such subspaces. We describe estimation procedures for inference of parameters and prediction of principal components, and we give examples of properties of the model on embedded surfaces.