Projective Wishart Distributions

TL;DR

This paper introduces the concept of projective Wishart distributions, exploring their properties and connections with affine-invariant geometry, and provides explicit density expressions for 2x2 matrices.

Contribution

It defines projective Wishart distributions, links them to affine-invariant geometry, and characterizes their Fréchet mean and explicit densities for small matrices.

Findings

Fréchet mean corresponds to the covariance parameter of Wishart distribution.

Explicit density expressions for 2x2 matrices in terms of affine-invariant distance.

Strong geometric links established with symmetric and Hermitian positive definite matrices.

Abstract

We are interested in the distribution of Wishart samples after forgetting their scaling factors. We call such a distribution a projective Wishart distribution. We show that projective Wishart distributions have strong links with the affine-invariant geometry of symmetric positive definite matrices in the real case or Hermitian positive definite matrices in the complex case. First, the Fr{\'e}chet mean of a projective Wishart distribution is the covariance parameter, up to a scaling factor, of the corresponding Wishart distribution. Second, in the case of 2 by 2 matrices, the densities have simple expressions in term of the affine-invariant distance.