A probabilistic view on Riemannian machine learning models for SPD matrices
This work provides a foundational framework for researchers in machine learning and statistics working with SPD matrices, though it appears incremental in unifying existing tools.
The paper tackles the problem of unifying machine learning tools on the Riemannian manifold of symmetric positive definite matrices by developing a probabilistic framework using Gaussian distributions, enabling reinterpretation of classifiers as Bayes classifiers and extending to outlier detection and dimension reduction.
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 $\mathcal{P}_d$.