Numerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds

Abstract Parallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae do not exist in general, practitioners often have resort numerical schemes. Ladder methods are popular class of algorithms that rely iterative constructions geodesic parallelograms. And yet, the literature lacks clear analysis their convergence performance. In this work, we give Taylor approximations elementary Schild’s ladder and pole with respect Riemann curvature underlying space. We then prove these can be iterated converge quadratic speed, even when geodesics approximated by also contribute new link between Fanning scheme which explains why latter naturally converges only linearly. The extra computational cost thus easily compensated drastic reduction number steps needed achieve requested accuracy. Illustrations 2-sphere, space symmetric positive definite matrices special Euclidean group show theoretical errors established measured high accuracy practice. an anisotropic left-invariant metric particular interest as it tractable example non-symmetric reduces case. As secondary contribution, compute covariant derivative