The Differential of the Exponential Map, Jacobi Fields and Exact Principal Geodesic Analysis

Details The Differential of the Exponential Map, Jacobi Fields and Exact Principal Geodesic Analysis The Differential of the Exponential Map, Jacobi Fields and Exact Principal Geodesic Analysis

2010-08-11

arxiv.org/abs/1008.1902v3

Computer Science

Computational Geometry

Important remarks on problems linked to data locality in the previous preprint has been incorporated

Scientific paper

The importance of manifolds and Riemannian geometry is spreading to applied fields in which the need to model non-linear structure has spurred wide-spread interest in geometry. The transfer of interest has created demand for methods for computing classical constructs of geometry on manifolds occurring in practical applications. This paper develops initial value problems for the computation of the differential of the exponential map and Jacobi fields on parametrically and implicitly represented manifolds. It is shown how the solution to these problems allow for determining sectional curvatures and provides upper bounds for injectivity radii. In addition, when combined with the second derivative of the exponential map, the initial value problems allow for numerical computation of Principal Geodesic Analysis, a non-linear version of the Principal Component Analysis procedure for estimating variability in datasets. The paper develops algorithms for computing Principal Geodesic Analysis without the tangent space approximation previously used and, thereby, provides an example of how the constructs of theoretical geometry apply to solving problems in statistics. By testing the algorithms on synthetic datasets, we show how curvature affects the result of PGA.

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