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:这项工作考虑了Hermite 内插问题:找到一个足够平稳的多曲线,将一个里曼尼方块上的数据点汇集在一起,同时在每个点对准指定的衍生物进行匹配。我们建议采用新的程序,依靠撤回的一般概念来解决大量方块上的问题,包括计算里曼尼指数或对数地图并非直截了当的方位,例如固定式矩阵的方位。我们通过引入和显示撤回式convex元件的存在来分析这种方法的稳妥性,这是对大地曲线元件的概括化。我们扩展到对Hermite 内插体的失时跨度错误的典型结果。我们最后用对固定式矩阵的方位和有正态柱的Stiefel矩阵进行数字实验的方法来说明这些结果和这些方法的有效性。