Robust Geodesic Regression

Abstract

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 show that, on compact symmetric spaces, all M-type estimators are maximum likelihood estimators, and argue in favor of a general preference for the \(L_1\) estimator over the \(L_2\) and Huber estimators on high-dimensional spaces. A derivation of the Riemannian normal distribution on \(S^n\) and \(\mathbb {H}^n\) is also included. Results from numerical examples, including analysis of real neuroimaging data, demonstrate the promising empirical properties of the proposed approach.