Robust Estimation of Location in Matrix Manifolds Using the Projected Frobenius Median

TL;DR

This paper introduces a robust, computationally efficient method for location estimation on matrix manifolds using the projected Frobenius median, applicable to various complex geometric spaces.

Contribution

It develops a novel median-based estimation technique for matrix manifolds, with proven robustness, uniqueness, and asymptotic properties, extending median concepts to complex geometric spaces.

Findings

Method is computationally attractive and robust.

Unique solution under non-colinearity condition.

Effective in real-world earthquake data analysis.

Abstract

We propose a robust method for location estimation in various matrix manifolds based on the projected Frobenius median, which is closely related to the spatial median. This method applies broadly to matrix manifolds, including Stiefel and Grassmann manifolds, Kendall shape spaces as well as to projective Stiefel manifolds, a type of quotient space of a Stiefel manifold. Our approach involves computation of the Frobenius median in an ambient Euclidean space followed by projection onto the relevant matrix manifold. Our estimation method is computationally attractive, has a unique solution provided the sample data are not colinear in the ambient Euclidean space, has desirable robustness features and has appropriate equivariance properties under natural groups of transformations. We establish asymptotic normality under mild conditions and derive the influence function for matrix manifolds of interest. Simulation studies on the rank-1 complex Grassmann manifold and the projective Stiefel manifold further show the applicability and robustness of our method. We also apply our method to a real-world earthquake moment tensor dataset.