The Riemannian median of positive-definite matrices

TL;DR

This paper introduces a definition of the Riemannian median for positive-definite matrices, establishes inequalities relating it to the Karcher mean, and explores its properties and limitations within Riemannian geometry.

Contribution

It defines the Riemannian median for positive-definite matrices and derives inequalities analogous to classical median-mean relations, expanding understanding of median concepts in Riemannian geometry.

Findings

Established inequality relating Riemannian median and Karcher mean

Analyzed properties like invariance, homogeneity, and self-duality

Provided counterexample showing monotonicity does not always hold

Abstract

We propose a definition of the Riemannian median $M(\mathbb{A})$ of a tuple of positive-definite matrices $\mathbb{A}:=(A_{1}, \cdots, A_{n})$. We will define it as a positive-definite matrix using Landers and Rogge's work \cite{Lan81} partially, not as a set unlike Yang's work \cite{Yan10}. Then, in the set of positive-definite matrices with the Riemannian trace metric, we show \[ \delta(M, \Lambda) \leq \frac{1}{n}\sum_{k=1}^{n}\delta(A_{k}, \Lambda) \leq \sqrt{\frac{1}{n} \sum_{k=1}^{n} \delta(A_{k}, \Lambda)^{2}}, \] where $M=M(\mathbb{A})$, $\Lambda$ is the Karcher mean of $\mathbb{A}$, and $\delta$ is the Riemannian distance induced by the Riemannian trace metric. This inequality is an analogue of $|\mu-m| \leq \sigma$, where $\mu$, $m$ and $\sigma$ are the mean, the median and the standard deviation of real-valued data points. Moreover, we investigate the commutative case, how outliers have an effect on the Riemannian median, the congruence invariance, the joint homogeneity, the self-duality and the monotonicity in a special case, and construct a counter example showing that the monotonicity of the Riemannian median does not hold in general.