An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data
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An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data. / Sommer, Stefan.
I: Sankhya A, Bind 81, Nr. 1, 2019, s. 37-62.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data
AU - Sommer, Stefan
PY - 2019
Y1 - 2019
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
U2 - 10.1007/s13171-018-0139-5
DO - 10.1007/s13171-018-0139-5
M3 - Journal article
AN - SCOPUS:85051669789
VL - 81
SP - 37
EP - 62
JO - Sankhya A
JF - Sankhya A
SN - 0976-836X
IS - 1
ER -
ID: 203834137