2018/01/31 by Stefan Sommer, Sommer, Stefan
Computer Science · #Anomaly Detection Techniques and Applications #Data Visualization and Analytics #Time Series Analysis and Forecasting
paper · pdf · doi:10.48550/arxiv.1801.10341
We provide a probabilistic and infinitesimal view of how the principal\ncomponent analysis procedure (PCA) can be generalized to analysis of nonlinear\nmanifold valued data. Starting with the probabilistic PCA interpretation of the\nEuclidean PCA procedure, we show how PCA can be generalized to manifolds in an\nintrinsic way that does not resort to linearization of the data space. The\nunderlying probability model is constructed by mapping a Euclidean stochastic\nprocess to the manifold using stochastic development of Euclidean\nsemimartingales. The construction uses a connection and bundles of covariant\ntensors to allow global transport of principal eigenvectors, and the model is\nthereby an example of how principal fiber bundles can be used to handle the\nlack of global coordinate system and orientations that characterizes manifold\nvalued statistics. We show how curvature implies non-integrability of the\nequivalent of Euclidean principal subspaces, and how the stochastic flows\nprovide an alternative to explicit construction of such subspaces. We describe\nestimation procedures for inference of parameters and prediction of principal\ncomponents, and we give examples of properties of the model on embedded\nsurfaces.\n