Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices

2025-02-03

AI Summary

Modeling non-Euclidean, anisotropic data on the symmetric positive-definite (SPD) matrix manifold remains challenging due to its intrinsic Riemannian geometry and lack of isotropic statistical models. Method: This paper introduces, for the first time, the Anisotropic Wrapped Gaussian (AWG) distribution—a geometrically aware probabilistic model built upon the exponential map on the SPD manifold. Theoretically, we establish the existence, identifiability, and maximum likelihood estimation framework for AWG. Methodologically, we reinterpret classical classifiers as generative probabilistic models grounded in AWG, yielding a novel manifold-aware classifier. Results: Experiments on synthetic and real-world SPD datasets—including EEG and diffusion MRI—demonstrate substantial improvements in modeling fidelity and classification accuracy, alongside enhanced robustness and flexibility. Our approach establishes a new paradigm for non-Euclidean statistical modeling in manifold learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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 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.

Problem

• Shape Features
• Complex Data Analysis
• Gaussian Models

Innovation

• Gaussian Model for Symmetric Positive Definite Matrices
• Complex Data Handling
• Improved Data Classification