Some Geometric Properties of Matrix Means with respect to Different Distance Function

TL;DR

This paper investigates geometric properties of matrix means under various metrics, demonstrating in-betweenness and sphere inclusion properties for specific matrix means with respect to Bures-Wasserstein, Hellinger, and Log-Determinant distances.

Contribution

It establishes new geometric properties of matrix means, including in-betweenness and sphere inclusion, under different distance functions, extending prior understanding of matrix mean behavior.

Findings

Matrix power means satisfy in-betweenness in Hellinger metric.

Weighted Heron means and geodesic curves lie inside spheres defined by Log-Determinant distance.

Geometric and arithmetic means' geodesic curves are contained within specific metric spheres.

Abstract

In this paper we study the monotonicity, in-betweenness and in-sphere properties of matrix means with respect to Bures-Wasserstein, Hellinger and Log-Determinant metrics. More precisely, we show that the matrix power means (Kubo-Ando and non-Kubo-Ando extensions) satisfy the in-betweenness property in the Hellinger metric. We also show that for two positive definite matrices $A$ and $B$, the curve of weighted Heron means, the geodesic curve of the arithmetic and the geometric mean lie inside the sphere centered at the geometric mean with the radius equal to half of the Log-Determinant distance between $A$ and $B$.