AI 中文总结
本文针对Stiefel流形上带闭式逆的两类二阶收缩映射，分析其条件数与逆映射的局部存在性，推导插值误差界，并将其应用于Hermite插值，验证其性能可与经典方法媲美。
AI 中文摘要
收缩映射（Retractions）为流形上的实际数据处理任务提供了黎曼指数映射与对数映射的计算高效替代方案，其中带闭式逆的二阶收缩映射特别适用于流形上的插值问题。在正交标架的Stiefel流形上，仅存在两类此类收缩映射：规范度量下二阶精度的Cayley收缩映射，以及新近提出的欧氏度量下二阶精度的polar-light收缩映射。本文在Stiefel流形插值的背景下研究这些映射的性质，为获取显式插值误差界，我们分析收缩映射及其逆的条件数，证明这些收缩映射是良态的，并推导了与经典欧氏插值类似的插值误差界；逆收缩映射通常非良态，我们讨论如何通过等距群作用映射数据以确保计算稳定。与紧流形上的所有收缩映射类似，规范Cayley逆收缩映射与polar-light逆收缩映射仅局部存在，我们在任意点周围构造了法邻域，保证Cayley逆收缩映射或polar-light逆收缩映射在该邻域内可计算。作为收缩映射的应用，我们考虑Hermite插值，其目标是复现采样的函数值与导数信息，数值实例表明，基于收缩映射的插值与基于黎曼法坐标的经典方法相比具有竞争力。
英文摘要
Retractions provide a computationally efficient alternative to the Riemannian exponential and logarithm maps for practical data-processing tasks on manifolds. In particular, second-order retractions with closed-form inverse are well-suited for interpolation problems on manifolds. On the Stiefel manifold of orthogonal frames, there are only two retractions of this type: the Cayley retraction, which is second-order accurate under the canonical metric, and the recently proposed polar-light retraction, which is second-order accurate under the Euclidean metric. In this paper, we study the properties of these maps in the context of interpolation on the Stiefel manifold. To obtain explicit interpolation error bounds, we examine the conditioning of the retraction maps and their inverses. We show that the retractions are well-conditioned, and we derive interpolation error bounds similar to those of classical Euclidean interpolation. The inverse retractions are not well-conditioned in general, and we discuss how data can be mapped via an isometric group action to ensure stable computations. As with all retractions on compact manifolds, the inverse canonical Cayley retraction and the inverse polar-light retraction exist only locally, and we construct normal neighborhoods around any point in which either the inverse Cayley retraction or the invese polar-light retraction are guaranteed to be computable. As an application of the retraction maps, we consider Hermite interpolation, where the objective is to reproduce both sampled function values and derivative information. A numerical example demonstrates that retraction-based interpolation is competitive with classical methods based on Riemannian normal coordinates.
Comments21 pages 2 figures
