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 $\mathcal{P}_d$.

翻译：本文旨在阐明对称正定矩阵黎曼流形 $\mathcal{P}_d$ 上的多种机器学习工具如何通过概率框架实现统一。为此，我们需要在 $\mathcal{P}_d$ 上定义若干高斯分布。我们将展示 $\mathcal{P}_d$ 上常用的分类器如何通过这些高斯分布重新解释为贝叶斯分类器。这些分布还将用于异常检测与降维分析。通过证明这些分布在 $\mathcal{P}_d$ 相关工具中的普遍性，我们为其他机器学习工具向 $\mathcal{P}_d$ 的扩展提供了理论基础。