Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds

TL;DR

This paper develops evolution equations for anisotropically weighted paths on manifolds, connecting stochastic processes, sub-Riemannian geometry, and Riemannian polynomials, with applications to data analysis on complex geometric structures.

Contribution

It introduces a novel framework for anisotropically weighted path evolution on manifolds, integrating sub-Riemannian geometry and stochastic development, with practical computational methods.

Findings

Paths are projections of sub-Riemannian geodesics.

Curvature influences the Hamilton-Jacobi equations.

Finite-dimensional landmark models are used for visualization.

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.