Emmanuel Chevallier | Searcharxiv

arXiv2024

Projective Wishart Distributions

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.

math.ST↗

arXiv2020

Exponential-wrapped distributions on symmetric spaces

In many applications, the curvature of the space supporting the data makes the statistical modelling challenging. In this paper we discuss the construction and use of probability distributions wrapped around manifolds using exponential maps. These distributions have already been used on specific manifolds. We describe their construction in the unifying framework of affine locally symmetric spaces. Affine locally symmetric spaces are a broad class of manifolds containing many manifolds encountered in data sciences. We show that on these spaces, exponential-wrapped distributions enjoy interesting properties for practical use. We provide the generic expression of the Jacobian appearing in these distributions and compute it on two particular examples: Grassmannians and pseudo-hyperboloids. We illustrate the interest of such distributions in a classification experiment on simulated data.

stat.ME↗