Geometric statistics with subspace structure preservation for SPD matrices

TL;DR

This paper introduces a geometric framework for processing symmetric positive definite (SPD) matrices that preserves subspace structures, utilizing Thompson geometry and a novel inductive mean for improved analysis.

Contribution

It proposes a new geometric approach based on Thompson geometry for SPD matrices, including properties of a specific geodesic space and a novel inductive mean method.

Findings

Preserves subspace structures in SPD data.

Provides efficient computation of extreme generalized eigenvalues.

Introduces a new inductive mean for SPD matrices.

Abstract

We present a geometric framework for the processing of SPD-valued data that preserves subspace structures and is based on the efficient computation of extreme generalized eigenvalues. This is achieved through the use of the Thompson geometry of the semidefinite cone. We explore a particular geodesic space structure in detail and establish several properties associated with it. Finally, we review a novel inductive mean of SPD matrices based on this geometry.