SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A2593-A2619, January 2020.
The main contribution of this paper is twofold: On the one hand, a general framework for performing Hermite interpolation on Riemannian manifolds is presented. The method is applicable if algorithms for the associated Riemannian exponential and logarithm mappings are available. This includes many of the matrix manifolds that arise in practical Riemannian computing applications such as data analysis and signal processing, computer vision and image processing, structured matrix optimization problems, and model reduction. On the other hand, we expose a natural relation between data processing errors and the sectional curvature of the manifold in question. This provides general error bounds for manifold data processing methods that rely on Riemannian normal coordinates. Numerical experiments are conducted for the compact Stiefel manifold of rectangular column-orthogonal matrices. As use cases, we compute Hermite interpolation curves for orthogonal matrix factorizations such as the singular value decomposition and the QR-decomposition.

中文翻译：

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黎曼矩阵流形上的Hermite插值和数据处理错误

SIAM科学计算杂志，第42卷，第5期，第A2593-A2619页，2020年1月。
本文的主要贡献有两个方面：一方面，提出了在黎曼流形上执行Hermite插值的通用框架。如果相关联的黎曼指数和对数映射的算法可用，则该方法适用。这包括在实际的黎曼计算应用中出现的许多矩阵流形，例如数据分析和信号处理，计算机视觉和图像处理，结构化矩阵优化问题以及模型简化。另一方面，我们揭示了数据处理错误与所讨论歧管的截面曲率之间的自然关系。这为依赖黎曼正态坐标的流形数据处理方法提供了一般误差范围。进行了矩形柱正交矩阵的紧凑Stiefel流形的数值实验。作为用例，我们计算Hermite插值曲线以进行正交矩阵分解，例如奇异值分解和QR分解。