论文标题
强大的测量回归
Robust Geodesic Regression
论文作者
论文摘要
本文研究了关于黎曼流形的数据的强大回归。测量回归是线性回归对具有多种价值因变量和一个或多个实现的自变量的设置的概括。现有关于测量回归的工作使用方方误差来找到解决方案，但是与经典的欧几里得案例一样，最小二乘方法对异常值高度敏感。在本文中，我们使用M型估计器，包括$ L_1 $，HUBER和TUKEY BIWEAIGHT估算器来执行强大的测量回归，并描述如何计算后两个的调谐参数。我们还表明，在紧凑的对称空间上，所有M型估计器都是最大的似然估计器，并主张$ L_1 $估计器的总体优势比$ L_2 $和HUBER估计器在高维流形和高维双重估计器上对紧凑型高量含量的估计器的总体优势。数值示例的结果，包括对实际神经成像数据的分析，证明了该方法的有希望的经验特性。
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.
