Geometric distance between positive definite matrices of different dimensions

TL;DR

This paper introduces a method to define a geometric distance between positive definite matrices of different dimensions using the Riemannian distance on the cone of positive definite matrices, with applications to ellipsoids and covariance matrices.

Contribution

It proposes a novel approach to measure distances between positive definite matrices of different sizes using Riemannian geometry, extending existing concepts.

Findings

Defines a natural distance between matrices of different dimensions.

Connects the distance to ellipsoids and covariance matrices.

Provides a geometric framework for cross-dimensional comparisons.

Abstract

We show how the Riemannian distance on $\mathbb{S}^n_{++}$, the cone of $n\times n$ real symmetric or complex Hermitian positive definite matrices, may be used to naturally define a distance between two such matrices of different dimensions. Given that $\mathbb{S}^n_{++}$ also parameterizes $n$-dimensional ellipsoids, and inner products on $\mathbb{R}^n$, $n \times n$ covariance matrices of nondegenerate probability distributions, this gives us a natural way to define a geometric distance between a pair of such objects of different dimensions.