Alice Le Brigant

Alice Le Brigant, Jules Deschamps, Antoine Collas et al.

We introduce the information geometry module of the Python package Geomstats. The module first implements Fisher-Rao Riemannian manifolds of widely used parametric families of probability distributions, such as normal, gamma, beta, Dirichlet distributions, and more. The module further gives the Fisher-Rao Riemannian geometry of any parametric family of distributions of interest, given a parameterized probability density function as input. The implemented Riemannian geometry tools allow users to compare, average, interpolate between distributions inside a given family. Importantly, such capabilities open the door to statistics and machine learning on probability distributions. We present the object-oriented implementation of the module along with illustrative examples and show how it can be used to perform learning on manifolds of parametric probability distributions.

Nina Vesseron, Elsa Cazelles, Alice Le Brigant et al.

This paper focuses on Geodesic Principal Component Analysis (GPCA) on a collection of probability distributions using the Otto-Wasserstein geometry. The goal is to identify geodesic curves in the space of probability measures that best capture the modes of variation of the underlying dataset. We first address the case of a collection of Gaussian distributions, and show how to lift the computations in the space of invertible linear maps. For the more general setting of absolutely continuous probability measures, we leverage a novel approach to parameterizing geodesics in Wasserstein space with neural networks. Finally, we compare to classical tangent PCA through various examples and provide illustrations on real-world datasets.

Alice Le Brigant, Nicolas Guigui, Sana Rebbah et al.

In this paper, we use tools of information geometry to compare, average and classify histograms. Beta distributions are fitted to the histograms and the corresponding Fisher information geometry is used for comparison. We show that this geometry is negatively curved, which guarantees uniqueness of the notion of mean, and makes it suitable to classify histograms through the popular K-means algorithm. We illustrate the use of these geometric tools in supervised and unsupervised classification procedures of two medical data-sets, cardiac shape deformations for the detection of pulmonary hypertension and brain cortical thickness for the diagnosis of Alzheimer's disease.