Taylor expansion of geodesic triangles in Riemannian manifolds: a central tool to study the effect of curvature in geometric statistics

The impact of manifold curvature on the statistical estimation or on the accuracy of algorithms to compute on manifolds is not evident to establish. We show in this talk that Gavrilov's Taylor expansions of the double exponential can be completed by a companion neighboring log expansion which measures how the Riemannian logarithm changes when its foot-point is moving. These two tensorial and coordinate free Taylor expansions constitute a complete toolbox for the polynomial approximations of problems related to infinitesimal geodesic triangles. Moreover they are valid in general affine connection manifolds and computations can surprisingly easily be pushed to higher orders if needed.

We exemplify the use of this toolbox for two of fundamental tools for geometric statistics: the estimation of the Fréchet mean and parallel transport. We first present a new non-asymptotic (small sample) expansion in high concentration conditions which quantifies the concentration of the empirical Fréchet mean towards the population mean. This shows a statistical bias on the empirical mean in the direction of the average gradient of the curvature and a modulation of the convergence speed which could partly explain smeary means in positive curvature spaces and sticky means in stratified spaces of negative curvature.

Parallel transport is a second major tool to compare local tangent information at different points of the manifold. We previously proposed a modification of Schild's ladder called pole ladder that is surprisingly exact in only one step on symmetric spaces. Iterating geodesic parallelograms was thought to be of first order but a real convergence analysis was lacking. We show that pole and Schild's ladder naturally converges with quadratic speed even when geodesics are approximated by numerical schemes. This contrasts with Jacobi fields approximations that are bound to linear convergence. 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.