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

TY - JOUR

T1 - An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data

AU - Sommer, Stefan

PY - 2019/2/1

Y1 - 2019/2/1

N2 - 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.

AB - 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.

KW - Anisotropic normal distributions

KW - Frame bundle

KW - Manifold valued statistics

KW - Primary: 62H25

KW - Principal component analysis

KW - Probabilistic PCA

KW - Secondary: 53C99

KW - Stochastic development

UR - http://www.scopus.com/inward/record.url?scp=85051669789&partnerID=8YFLogxK

U2 - 10.1007/s13171-018-0139-5

DO - 10.1007/s13171-018-0139-5

M3 - Journal article

AN - SCOPUS:85051669789

SN - 0976-836X

VL - 81

SP - 37

EP - 62

JO - Sankhya A

JF - Sankhya A

IS - 1

ER -