Evolution Equations with Anisotropic Distributions and Diffusion PCA
This paper presents derivations of evolution equations for the family of paths that in the Diffusion PCA framework are used for approximating data likelihood. The paths that are formally interpreted as most probable paths generalize geodesics in extremizing an energy functional on the space of differentiable curves on a manifold with connection. We discuss how the paths arise as projections of geodesics for a (non bracket-generating) sub-Riemannian metric on the frame bundle. Evolution equations in coordinates for both metric and cometric formulations of the sub-Riemannian geometry are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we use the construction to show how the evolution equations can be implemented on finite dimensional LDDMM landmark manifolds.
KeywordsChristoffel Symbol Stochastic Development Probable Path Horizontal Lift Frame Bundle
The author wishes to thank Peter W. Michor and Sarang Joshi for suggestions for the geometric interpretation of the sub-Riemannian metric on FM and discussions on diffusion processes on manifolds. This work was supported by the Danish Council for Independent Research and the Erwin Schrödinger Institute in Vienna.
- 1.Sommer, S.: Diffusion Processes and PCA on Manifolds, Mathematisches Forschungsinstitut Oberwolfach (2014)Google Scholar
- 9.Zhang, M., Fletcher, P.: Probabilistic principal geodesic analysis. In: NIPS, pp. 1178–1186 (2013)Google Scholar
- 16.Micheli, M.: The differential geometry of landmark shape manifolds: metrics, geodesics, and curvature. Ph.D. thesis, Brown University, Providence, USA (2008)Google Scholar