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Entropy 2016, 18(12), 425; doi:10.3390/e18120425

Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds

Department of Computer Science, University of Copenhagen, DK-2100 Copenhagen E, Denmark
This paper is an extended version of our paper published in the 2nd Conference on Geometric Science of Information, Paris, France, 28–30 October 2015.
Academic Editors: Frédéric Barbaresco and Frank Nielsen
Received: 1 September 2016 / Revised: 15 November 2016 / Accepted: 23 November 2016 / Published: 26 November 2016
(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)
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Abstract

We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be interpreted as most probable paths for a driving semi-martingale that through stochastic development is mapped to the manifold. We discuss how the paths are projections of geodesics for a sub-Riemannian metric on the frame bundle of the manifold, and how the curvature of the underlying connection appears in the sub-Riemannian Hamilton–Jacobi equations. Evolution equations for both metric and cometric formulations of the sub-Riemannian metric are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we relate the paths to geodesics and polynomials in Riemannian geometry. Examples from the family of paths are visualized on embedded surfaces, and we explore computational representations on finite dimensional landmark manifolds with geometry induced from right-invariant metrics on diffeomorphism groups. View Full-Text
Keywords: sub-Riemannian geometry; geodesics; most probable paths; stochastic development; non-linear data analysis; statistics sub-Riemannian geometry; geodesics; most probable paths; stochastic development; non-linear data analysis; statistics
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Sommer, S. Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds. Entropy 2016, 18, 425.

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