Affine-Invariant Midrange Statistics

TL;DR

This paper introduces an affine-invariant approach to the matrix midrange problem on positive definite matrices, proposing a scalable midpoint computation based on the Thompson metric, and extends it to multiple matrices.

Contribution

It formulates the affine-invariant matrix midrange problem on the cone of positive definite matrices and investigates a computationally efficient midpoint as an average within this framework.

Findings

Proposes an affine-invariant midrange problem formulation.

Develops a scalable midpoint computation based on the Thompson metric.

Extends the approach to N-point problems for matrices.

Abstract

We formulate and discuss the affine-invariant matrix midrange problem on the cone of $n\times n$ positive definite Hermitian matrices $\mathbb{P}(n)$, which is based on the Thompson metric. A particular computationally efficient midpoint of this metric is investigated as a highly scalable candidate for an average of two positive definite matrices within this context, before studying the $N$-point problem in the vector and matrix settings.