A probabilistic view on Riemannian machine learning models for SPD matrices

TL;DR

This paper unifies various machine learning methods on the Riemannian manifold of SPD matrices within a probabilistic framework using Gaussian distributions, enabling reinterpretation, outlier detection, and extension of tools.

Contribution

It introduces a probabilistic approach on the SPD manifold, connecting classifiers, outlier detection, and dimension reduction through Gaussian distributions.

Findings

Reinterprets popular classifiers as Bayes classifiers on SPD matrices.

Demonstrates Gaussian distributions' role in outlier detection.

Facilitates extension of machine learning tools to the SPD manifold.

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 $\mathcal{P}_d$.