Robust Geodesic Regression

• Published in: International journal of computer vision 2022-02, Vol.130 (2), p.478-503
• Main Authors: Shin, Ha-Young, Oh, Hee-Seok
• Format: Article
• Language: English
• Subjects: Alzheimer's disease; Artificial Intelligence; Computer Imaging; Computer Science; Dependent variables; Empirical analysis; Euclidean space; Image Processing and Computer Vision; Independent variables; Least squares method; Manifolds (mathematics); Maximum likelihood estimators; Medical imaging; Normal distribution; Outliers (statistics); Pattern Recognition; Pattern Recognition and Graphics; Real variables; Regression; Regression analysis; Riemann manifold; Robustness (mathematics); Statistical analysis; Vision
• ISSN: 0920-5691; 1573-1405
• DOI: 10.1007/s11263-021-01561-w

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 H n is also included. Results from numerical examples, including analysis of real neuroimaging data, demonstrate the promising empirical properties of the proposed approach.