双不变测地线回归：存在性、唯一性与收敛性
Bi-invariant Geodesic Regression: Existence, Uniqueness, and Convergence
- Zuse Institute Berlin（齐泽研究所柏林）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究李群上双不变测地线回归估计量的局部存在唯一性，并证明所提迭代算法线性收敛，给出邻域显式界限。
AI中文摘要：
双不变测地线回归将线性回归推广到李群。其主要特点是尊重群的对称性，使得所得估计量独立于诸如参考系等任意选择。然而，该估计量的局部存在性和唯一性至今尚未得到证明。此外，用于计算该估计量的所提算法的收敛性质也未知。在本工作中，我们研究这些问题。我们证明，在局部范围内，存在唯一估计量，并给出了该邻域大小的显式界限。我们还表明，所提出的迭代算法线性收敛于该估计量。
英文摘要：
Bi-invariant geodesic regression generalizes linear regression to Lie groups. Its main feature is that it respects the symmetries of the group so that the resulting estimator is independent of arbitrary choices such as a reference frame. However, the local existence and uniqueness of the underlying estimator have not been shown until now. Furthermore, the convergence properties of the proposed algorithm for computing the estimator are not known. In this work, we investigate these questions. We prove that, locally, a unique estimator exists and give explicit bounds on the size of this neighborhood. We also show that the proposed iterative algorithm converges linearly to this estimator.
