Constructing matrix geometric means

TL;DR

This paper explores methods to construct new matrix geometric means from existing ones, highlighting a simpler mean for four matrices and limitations for larger sets.

Contribution

It introduces a new, computationally simpler matrix mean for four matrices and discusses the limitations of current methods for larger sets.

Findings

A new matrix mean for four matrices that is easier to compute.

Existing strategies do not simplify computation for more than four matrices.

Limitations of current methods for constructing matrix means for larger sets.

Abstract

In this paper, we analyze the process of "assembling" new matrix geometric means from existing ones, through function composition or limit processes. We show that for n=4 a new matrix mean exists which is simpler to compute than the existing ones. Moreover, we show that for n>4 the existing proving strategies cannot provide a mean computationally simpler than the existing ones.