Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices

Thibault de Surrel, Fabien Lotte, Sylvain Chevallier, Florian Yger

TL;DR: We propose and study the properties and the estimation of wrapped Gaussian distributions on the manifold of SPD matrices

Abstract:

Circular and non-flat data distribution are prevalent across diverse domains of data science, yet their specific geometric structures often remain underutilized in machine learning frameworks.
A principled approach to accounting for the underlying geometry of such data is pivotal, particularly when extending statistical models, like the pervasive Gaussian distribution.
In this work, we tackle those issue by focusing on the manifold of symmetric positive definite matrices, a key focus in information geometry.
We introduced a non-isotropic wrapped Gaussian by leveraging the exponential map, we derive theoretical properties of this distribution and propose a maximum likelihood framework for parameter estimation. Furthermore, we reinterpret established classifiers on SPD through a probabilistic lens and introduce new classifiers based on the wrapped Gaussian model.
Experiments on synthetic and real-world datasets demonstrate the robustness and flexibility of this geometry-aware distribution, underscoring its potential to advance manifold-based data analysis.
This work lays the groundwork for extending classical machine learning and statistical methods to more complex and structured data.

Lay Summary: In this paper, we introduce a new way of modeling data that lie on a non flat space using a probability distribution. We focus on a special type of matrices, that appear in different areas of data science and hope that our modelization will help researchers betters understand the insights of complex data. We study this probability distribution theoretically, deriving some useful properties. We also show how it can be used in practice, in real algorithms on real data. This work paves the way to extending classical machine learning tools to highly complex and structured data.

Link To Code: https://github.com/thibaultdesurrel/wrapped_gaussians_SPD

Primary Area: Probabilistic Methods

Keywords: Gaussian distribution, Wrapped distributions, Symmetric positive definite matrices, estimation, classification, Riemannian geometry, density estimation

Submission Number: 181