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 \[ δ(M, Λ) \leq \frac{1}{n} \sum_{k=1}^{n} δ(A_{k}, Λ) \leq \sqrt{\frac{1}{n} \sum_{k=1}^{n} δ(A_{k}, Λ)^{2}}, \] where $M=M(\mathbb{A})$, $Λ$ is the Karcher mean of $\mathbb{A}$, and $δ$ is the Riemannian distance induced by the trace metric. This inequality is an analogue of $|μ-m| \leq σ$, where $μ$, $m$ and $σ$ 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.