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.

翻译：非欧几里得空间上的数据插值因其广泛应用而成为活跃的研究领域。本文研究Hermite插值问题：在黎曼流形上寻找足够光滑的曲线，使其插值一组数据点，并在每个点处匹配指定的导数。我们提出了一种基于回缩这一通用概念的新方法，用于解决一大类流形上的该问题，包括那些难以计算黎曼指数映射或对数映射的流形，例如固定秩矩阵流形。我们通过引入并证明回缩凸集（测地凸集的推广）的存在性，分析了该方法的适定性。我们将Hermite插值渐近插值误差的经典结果推广到流形设定中。最后，我们通过固定秩矩阵流形和正交列矩阵的Stiefel流形上的数值实验，验证了这些结果及该方法的有效性。