Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Riemannian median of positive-definite matrices

Yutaro Nakagawa

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 \[ δ(M, Λ) \leq \frac{1}{n}\sum_{k=1}^{n}δ(A_{k}, Λ) \leq \sqrt{\frac{1}{n} \sum_{k=1}^{n} δ(A_{k}, Λ)^{2}}, \] where $M=M(\mathbb{A})$, $Λ$ is the Karcher mean of $\mathbb{A}$, and $δ$ is the Riemannian distance induced by the Riemannian 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 in a special case, and construct a counter example showing that the monotonicity of the Riemannian median does not hold in general.

The Riemannian median of positive-definite matrices

Abstract

We propose a definition of the Riemannian median of a tuple of positive-definite matrices . 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 where , is the Karcher mean of , and is the Riemannian distance induced by the Riemannian trace metric. This inequality is an analogue of , where , and 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.
Paper Structure (15 sections, 25 theorems, 82 equations)

This paper contains 15 sections, 25 theorems, 82 equations.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. CAT$(0)$ space
  4. Karcher mean for positive-definite matrices
  5. The exponential map and the logarithmic map
  6. Jensen's inequality for a probability measure on a global NPC space
  7. DEFINITION OF THE RIEMANNIAN MEDIAN
  8. SOME PROPERTIES OF THE RIEMANNIAN MEDIAN
  9. Inequality of the distance between the Riemannian median and the Karcher mean
  10. Commutative case
  11. Effect of outliers
  12. Congruence invariance and joint homogeneity
  13. Self-duality
  14. Monotonicity
  15. Weighted Riemannian median

Key Result

Lemma 2.3

Let $(X, d)$ be a CAT$(0)$ space and $p, q, r$ be three points in $X$. Then, there are points $\overline{p}, \overline{q}, \overline{r} \in \mathbb{R}^{2}$ such that $d(p, q)=d_{\mathbb{R}^{2}}(\overline{p}, \overline{q})$, $d(q, r)=d_{\mathbb{R}^{2}}(\overline{q}, \overline{r})$ and $d(r, p)=d_{\ma

Theorems & Definitions (49)

  • Definition 2.1: Bri99
  • Definition 2.2: Bri99
  • Lemma 2.3: Bri99
  • Definition 2.4: Bri99
  • Proposition 2.5: Bri99
  • Proposition 2.6: Bri99
  • Definition 2.7: Bha062, Lim122
  • Proposition 2.8: Bha062
  • Proposition 2.9: Bha062
  • Lemma 2.10: Bha062
  • ...and 39 more