Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices

TL;DR

This paper introduces a wrapped Gaussian distribution on the manifold of symmetric positive definite matrices, enabling geometry-aware statistical modeling and classification for complex structured data.

Contribution

It presents a novel non-isotropic wrapped Gaussian on SPD manifolds, with theoretical properties, parameter estimation, and new classifiers based on this distribution.

Findings

Demonstrates robustness on synthetic and real datasets

Provides a probabilistic reinterpretation of existing classifiers

Lays groundwork for advanced manifold-based machine learning

Abstract

Circular and non-flat data distributions 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 (SPD) matrices, a key focus in information geometry. We introduce 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.