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

Structure-Preserving Discretization of Incompressible Fluids

The geometric nature of Euler fluids has been clearly identified and extensively studied over the years, culminating with Lagrangian and Hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed as a consequence of Noether's theorem associated with the particle relabeling symmetry of fluid mechanics. However computational approaches to fluid mechanics have been largely derived from a numerical-analytic point of view, and are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts such as energy and circulation drift. In contrast, this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely Eulerian form. Our approach approximates the group of volume-preserving diffeomorphisms using a finite dimensional Lie group, and associated discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion yield a structure-preserving time integrator with good long-term energy behavior and for which an exact discrete Kelvin's circulation theorem holds

The vanishing of L2 harmonic one-forms on based path spaces

We prove the triviality of the first L2 cohomology class of based path spaces of Riemannian manifolds furnished with Brownian motion measure, and the consequent vanishing of L2 harmonic one-forms. We give explicit formulae for closed and co-closed one-forms expressed as differentials of functions and co-differentials of L2 two-forms, respectively; these are considered as extended Clark-Ocone formulae. A feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one-forms to the exterior differentiation operator used to construct the de Rham complex and the self-adjoint Laplacian on L2 one-forms. This Laplacian is shown to have a spectral gap

Representation and Characterization of Non-Stationary Processes by Dilation Operators and Induced Shape Space Manifolds

We have introduce a new vision of stochastic processes through the geometry induced by the dilation. The dilation matrices of a given processes are obtained by a composition of rotations matrices, contain the measure information in a condensed way. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, it coincides with the Naimark Dilation and only one rotation matrix is computed, when the process is non-stationary, a set of rotation matrices are computed. In particular, the periodicity of the correlation function that may appear in some classes of signal is transmitted to the set of dilation matrices. These rotation matrices, which can be arbitrarily close to each other depending on the sampling or the rescaling of the signal are seen as a distinctive feature of the signal. In order to study this sequence of matrices, and guided by the possibility to rescale the signal, the correct geometrical framework to use with the dilation's theoretic results is the space of curves on manifolds, that is the set of all curve that lies on a base manifold. To give a complete sight about the space of curve, a metric and the derived geodesic equation are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices. The notion of the shape of a curve can be formalized as the set of equivalence classes of curves given by the quotient space of the space of curves and the increasing diffeomorphisms. The metric in the space of curve naturally extent to the space of shapes and enable comparison between shapes.Comment: 19 pages, draft pape