Rank-preserving geometric means of positive semi-definite matrices

TL;DR

This paper introduces a new rank-preserving geometric mean for positive semi-definite matrices, extending classical matrix means to low-rank approximations, with properties suitable for high-dimensional applications.

Contribution

It proposes a novel rank-preserving geometric mean for positive semi-definite matrices, generalizing existing matrix means to fixed-rank cases.

Findings

The mean satisfies all expected geometric properties.

It is applicable to low-rank matrix approximations.

Supports operations in high-dimensional spaces.

Abstract

The generalization of the geometric mean of positive scalars to positive definite matrices has attracted considerable attention since the seminal work of Ando. The paper generalizes this framework of matrix means by proposing the definition of a rank-preserving mean for two or an arbitrary number of positive semi-definite matrices of fixed rank. The proposed mean is shown to be geometric in that it satisfies all the expected properties of a rank-preserving geometric mean. The work is motivated by operations on low-rank approximations of positive definite matrices in high-dimensional spaces.