Hermite interpolation with retractions on manifolds
Axel Séguin, Daniel Kressner
TL;DR
This work addresses Hermite interpolation of manifold-valued data by introducing a retraction-based RH interpolation that avoids the need for explicit exponential/Log maps. It develops a general framework using endpoint retraction curves and retraction-convex sets to ensure well-posedness, proves a manifold analogue of the classical $O(h^4)$ interpolation-error rate, and validates the approach through extensive experiments on the Stiefel and fixed-rank matrix manifolds. The method offers high-order accuracy while remaining applicable to manifolds where the exponential/log maps are difficult to compute, demonstrated through practical examples in Q-factor and SVD interpolation and extended to Riemannian continuation and dynamical low-rank approximation. Overall, RH interpolation provides a flexible, efficient building block for manifold-based numerical methods, with potential impact on manifold ODEs, optimization, and model-order reduction.
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.