Exponential-wrapped distributions on symmetric spaces

TL;DR

This paper introduces a unified framework for constructing exponential-wrapped probability distributions on affine locally symmetric spaces, facilitating statistical modeling on complex manifolds with applications demonstrated in classification tasks.

Contribution

It generalizes the construction of exponential-wrapped distributions to a broad class of symmetric spaces, providing explicit Jacobian formulas and practical insights for data science applications.

Findings

Distributions are explicitly constructed on Grassmannians and pseudo-hyperboloids.

The approach improves classification performance on simulated data.

Provides a unifying theoretical framework for manifold-based distributions.

Abstract

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.