Robust Extrinsic Regression Analysis for Manifold Valued Data

TL;DR

This paper introduces a robust extrinsic regression framework for manifold-valued data, extending the geometric median concept and demonstrating improved performance over existing methods through simulations.

Contribution

It proposes a novel extrinsic median for manifolds and integrates it into local polynomial regression for robust analysis of manifold data.

Findings

The proposed extrinsic median is robust against noise and outliers.

The new regression method outperforms existing approaches in simulations.

The paper provides an efficient algorithm for implementation.

Abstract

Recently, there has been a growing need in analyzing data on manifolds owing to their important role in diverse fields of science and engineering. In the literature of manifold-valued data analysis up till now, however, only a few works have been carried out concerning the robustness of estimation against noises, outliers, and other sources of perturbations. In this regard, we introduce a novel extrinsic framework for analyzing manifold valued data in a robust manner. First, by extending the notion of the geometric median, we propose a new robust location parameter on manifolds, so-called the extrinsic median. A robust extrinsic regression method is also developed by incorporating the conditional extrinsic median into the classical local polynomial regression method. We present the Weiszfeld's algorithm for implementing the proposed methods. The promising performance of our approach against existing methods is illustrated through simulation studies.