Inductive Geometric Matrix Midranges

TL;DR

This paper introduces a geometric clustering method for SPD matrices using the Thompson metric and a new inductive midrange centroid, offering a balance between Euclidean simplicity and Riemannian accuracy.

Contribution

It proposes a novel inductive midrange centroid computation for SPD matrices and integrates it with Thompson metric-based clustering algorithms.

Findings

The inductive midrange centroid is effective and numerically validated.

Thompson metric-based clustering improves data interpretation.

Method is incorporated into X-means and K-means++ algorithms.

Abstract

Covariance data as represented by symmetric positive definite (SPD) matrices are ubiquitous throughout technical study as efficient descriptors of interdependent systems. Euclidean analysis of SPD matrices, while computationally fast, can lead to skewed and even unphysical interpretations of data. Riemannian methods preserve the geometric structure of SPD data at the cost of expensive eigenvalue computations. In this paper, we propose a geometric method for unsupervised clustering of SPD data based on the Thompson metric. This technique relies upon a novel "inductive midrange" centroid computation for SPD data, whose properties are examined and numerically confirmed. We demonstrate the incorporation of the Thompson metric and inductive midrange into X-means and K-means++ clustering algorithms.