4,459 research outputs found
An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data
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
Manifold interpolation and model reduction
One approach to parametric and adaptive model reduction is via the
interpolation of orthogonal bases, subspaces or positive definite system
matrices. In all these cases, the sampled inputs stem from matrix sets that
feature a geometric structure and thus form so-called matrix manifolds. This
work will be featured as a chapter in the upcoming Handbook on Model Order
Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A.
Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the
numerical treatment of the most important matrix manifolds that arise in the
context of model reduction. Moreover, the principal approaches to data
interpolation and Taylor-like extrapolation on matrix manifolds are outlined
and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model
Order Reduction
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