Optimization over Geodesics for Exact Principal Geodesic Analysis
Preprint
Abstract
In fields ranging from computer vision to signal processing and statistics, increasing computational power allows a move from classical linear models to models that incorporate non-linear phenomena. This shift has created interest in computational aspects of differential geometry, and solving optimization problems that incorporate non-linear geometry constitutes an important computational task. In this paper, we develop methods for numerically solving optimization problems over spaces of geodesics using numerical integration of Jacobi fields and second order derivatives of geodesic families. As an important application of this optimization strategy, we compute exact Principal Geodesic Analysis (PGA), a non-linear version of the PCA dimensionality reduction procedure. By applying the exact PGA algorithm to synthetic data, we exemplify the differences between the linearized and exact algorithms caused by the non-linear geometry. In addition, we use the numerically integrated Jacobi fields to determine sectional curvatures and provide upper bounds for injectivity radii.Keywords
All Related Versions
This publication has 20 references indexed in Scilit:
- Natural metrics and least-committed priors for articulated trackingImage and Vision Computing, 2012
- Manifold Valued Statistics, Exact Principal Geodesic Analysis and the Effect of Linear Approximations2010
- Curvature computations for n-manifolds in Rn+m and solution to an open problem proposed by R. GoldmanComputer Aided Geometric Design, 2007
- Globally Convergent Optimization Algorithms on Riemannian Manifolds: Uniform Framework for Unconstrained and Constrained OptimizationJournal of Optimization Theory and Applications, 2006
- Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric MeasurementsJournal of Mathematical Imaging and Vision, 2006
- Geodesics Between 3D Closed Curves Using Path-Straightening2006
- A Riemannian Framework for Tensor ComputingInternational Journal of Computer Vision, 2006
- Feature-based registration of medical images: Estimation and validation of the pose accuracy1998
- On a computational method for the second fundamental tensor and its application to bifurcation problemsNumerische Mathematik, 1990
- Riemannian center of mass and mollifier smoothingCommunications on Pure and Applied Mathematics, 1977