This paper studies robust regression for data on Riemannian manifolds. Geodesic regression is the generalization of linear regression to a setting with a manifold-valued dependent variable and one or more real-valued independent variables. The existing work on geodesic regression uses the sum-of-squared errors to find the solution, but as in the classical Euclidean case, the least-squares method is highly sensitive to outliers. In this paper, we use M-type estimators, including the $L_1$, Huber and Tukey biweight estimators, to perform robust geodesic regression, and describe how to calculate the tuning parameters for the latter two. We also show that, on compact symmetric spaces, all M-type estimators are maximum likelihood estimators, and argue for the overall superiority of the $L_1$ estimator over the $L_2$ and Huber estimators on high-dimensional manifolds and over the Tukey biweight estimator on compact high-dimensional manifolds. Results from numerical examples, including analysis of real neuroimaging data, demonstrate the promising empirical properties of the proposed approach.

翻译：本文研究里曼尼方形数据的稳健回归。 大地回归是将线性回归普遍化为具有多重价值的依附变量和一个或多个实际价值独立变量的设置。 有关大地回归的现有工作使用定量误差总和来找到解决方案, 但正如古典欧几里德案例一样, 最小方形方法对外部值高度敏感。 在本文中, 我们使用M型测算仪, 包括$L_ 1美元、 Huber 和 Tukey 双重量估测器, 来进行稳健的地标回归, 并描述如何计算后两个变量的调值参数。 我们还显示, 在紧凑的对称空间, 所有M型估测器都是最大可能性的估测器, 并论证$L_ 1 美元比$L_ 2 和Huber 测算器的总体优势。 在本文中, 我们使用M型测算器测算器, 包括 $L_ 2 美元 和 Huber 测算器测算器, 双称 双称测算器, 用于紧凑高维度的多元数数数数数矩阵。 我们用的数字示例示例示例示例示例示例示例示例, 显示, 分析, 分析, 分析 分析 分析 分析 分析