The Riemannian median of positive-definite matrices

Abstract

Using Landers and Rogge's work \cite{Lan81} partially, we define the Riemannian median $M(\mathbb{A})$ of a tuple of positive-definite matrices $\mathbb{A}=(A_{1}, \cdots, A_{n})$ as a positive-definite matrix, not as a set unlike Yang's work \cite{Yan10}. Then, in the Riemannian manifold of positive-definite matrices with the 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 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.