Hermite interpolation with retractions on manifolds

TL;DR

This paper introduces a new Hermite interpolation method on Riemannian manifolds using retractions, enabling smooth curve fitting even when exponential maps are difficult to compute, with proven convergence and numerical validation.

Contribution

It proposes a novel retraction-based Hermite interpolation technique on manifolds, extending classical error results and demonstrating effectiveness on fixed-rank and Stiefel manifolds.

Findings

Method works on manifolds where exponential maps are complex to compute

Numerical experiments confirm the accuracy and efficiency of the approach

Extension of classical interpolation error results to the manifold setting

Abstract

Interpolation of data on non-Euclidean spaces is an active research area fostered by its numerous applications. This work considers the Hermite interpolation problem: finding a sufficiently smooth manifold curve that interpolates a collection of data points on a Riemannian manifold while matching a prescribed derivative at each point. We propose a novel procedure relying on the general concept of retractions to solve this problem on a large class of manifolds, including those for which computing the Riemannian exponential or logarithmic maps is not straightforward, such as the manifold of fixed-rank matrices. We analyze the well-posedness of the method by introducing and showing the existence of retraction-convex sets, a generalization of geodesically convex sets. We extend to the manifold setting a classical result on the asymptotic interpolation error of Hermite interpolation. We finally illustrate these results and the effectiveness of the method with numerical experiments on the manifold of fixed-rank matrices and the Stiefel manifold of matrices with orthonormal columns.