Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices
This work addresses the challenge of incorporating geometric structures into machine learning for domains like information geometry, though it is incremental as it extends existing Gaussian models to SPD manifolds.
The authors tackled the problem of modeling non-flat data distributions on symmetric positive definite (SPD) matrices by introducing a non-isotropic wrapped Gaussian distribution using the exponential map, and experiments showed its robustness and flexibility on synthetic and real-world datasets.
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.