Abstract
Parallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae do not exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms that rely on iterative constructions of geodesic parallelograms. And yet, the literature lacks a clear analysis of their convergence performance. In this work, we give Taylor approximations of the elementary constructions of Schild’s ladder and the pole ladder with respect to the Riemann curvature of the underlying space. We then prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes. We also contribute a new link between Schild’s ladder and the Fanning scheme which explains why the latter naturally converges only linearly. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy. Illustrations on the 2-sphere, the space of symmetric positive definite matrices and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy in practice. The special Euclidean group with an anisotropic left-invariant metric is of particular interest as it is a tractable example of a non-symmetric space in general, which reduces to a Riemannian symmetric space in a particular case. As a secondary contribution, we compute the covariant derivative of the curvature in this space.
Highlights
In many applications, it is natural to model data as points that lie on a manifold.there has been a growing interest in defining a consistent framework to perform statistics and machine learning on manifolds [31]
As for the Fanning scheme (FS) [22], we prove that the ladder methods converge when using approximate geodesics and that all geodesics of the construction may be computed in one pass—i.e., using one integration step (e.g., Runge–Kutta) per parallelogram construction, reducing the computational cost
In accordance with our main result, a quadratic speed of convergence is reached, and the speed depends on the asymmetry of the space, itself induced by the anisotropy β of the metric
Summary
It is natural to model data as points that lie on a manifold.there has been a growing interest in defining a consistent framework to perform statistics and machine learning on manifolds [31]. It is natural to model data as points that lie on a manifold. The parallel transport of tangent vectors appears as a natural tool to compare tangent vectors across tangent spaces. In [17], it is a key ingredient to spline-fitting in a Kendall shape space. In computational anatomy, it allows to compare longitudinal, intra-subject evolution across populations [3,19,20,32]. It allows to compare longitudinal, intra-subject evolution across populations [3,19,20,32] It is used in computer vision in, e.g., [6,15]
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