| Autor: |
Séguin, Axel, Kressner, Daniel |
| Rok vydání: |
2022 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| Popis: |
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. |
| Databáze: |
arXiv |
| Externí odkaz: |
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