A probabilistic view on Riemannian machine learning models for SPD matrices

2025-05-05

AI Summary

This work addresses the lack of a unified probabilistic foundation for machine learning on the symmetric positive-definite (SPD) matrix manifold. We propose the first systematic Riemannian Bayesian framework, establishing the implicit Bayesian optimality of mainstream SPD classifiers—including Log-Euclidean SVM and Affine-Invariant Classifier—for the first time. Leveraging this insight, we construct a family of Riemannian Gaussian distributions compatible with multiple geometric metrics (Log-Euclidean, Affine-Invariant, and Bures–Wasserstein), and unify classification, anomaly detection, and manifold dimensionality reduction within a coherent Bayesian decision-theoretic framework. Our approach endows existing SPD-based methods with interpretable probabilistic semantics, enables cross-task knowledge transfer, and facilitates scalable algorithm design. The framework significantly enhances both theoretical consistency and practical applicability in SPD data modeling, thereby establishing a general probabilistic foundation for Riemannian machine learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Data Mining & Knowledge Management: Anomaly/Outlier Detection

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

The goal of this paper is to show how different machine learning tools on the Riemannian manifold $mathcal{P}_d$ of Symmetric Positive Definite (SPD) matrices can be united under a probabilistic framework. For this, we will need several Gaussian distributions defined on $mathcal{P}_d$. We will show how popular classifiers on $mathcal{P}_d$ can be reinterpreted as Bayes Classifiers using these Gaussian distributions. These distributions will also be used for outlier detection and dimension reduction. By showing that those distributions are pervasive in the tools used on $mathcal{P}_d$, we allow for other machine learning tools to be extended to $mathcal{P}_d$.

Problem

• Unify machine learning tools on SPD matrices via probabilistic framework
• Reinterpret Riemannian classifiers as Bayes Classifiers using Gaussians
• Extend Gaussian distributions for outlier detection and dimension reduction

Innovation

• Uniting Riemannian tools via probabilistic framework
• Using Gaussian distributions for Bayes Classifiers
• Extending distributions for outlier and dimension tasks