将SPD矩阵的机器学习工具统一到概率框架下，实现分类、异常检测与降维
A probabilistic view on Riemannian machine learning models for SPD matrices
- 基于流形上的高斯分布构建概率模型
- 主流分类器可重解释为贝叶斯分类器
- 适用于医学影像等需处理对称正定矩阵的场景
本文旨在展示如何将定义在对称正定（SPD）矩阵流形 $\mathcal{P}_d$ 上的不同机器学习工具统一到一个概率框架中。为此，需要在 $\mathcal{P}_d$ 上定义多种高斯分布，并证明流行的分类器可重新解释为使用这些分布的贝叶斯分类器。这些分布还可用于异常检测和降维。通过揭示这些分布广泛存在于 $\mathcal{P}_d$ 的各类工具中，本文为其他机器学习方法向 $\mathcal{P}_d$ 的扩展提供了理论基础。
原文摘要 · Abstract (English)
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 $\mathcal{P}_d$.
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