Robust Multivariate Regression on Riemannian Manifolds

Abstract

This paper considers expanding the scope of geodesic regression by proposing a robust estimation method. Most existing geodesic regressions have been developed based on the least-squares approach. However, these methods are sensitive to outliers, as in the classic Euclidean case. In this paper, we propose a robust regression approach on Riemannian manifolds through a novel combination of Euclidean robust approaches and geodesic regression when errors follow heavy-tailed distributions or include outliers. We also discuss how to sample from various distributions on Riemannian manifolds, and further conduct numerical experiments for empirical performance evaluation of the proposed method. Note that MATLAB codes used to implement the methods and to carry out some experiments are available at https://github.com/sangyulism/robust-multivariate-regression-on-Riemannian-manifolds in order that one can reproduce the same results.