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A probabilistic view on Riemannian machine learning models for SPD matrices

Thibault de Surrel, Florian Yger, Fabien Lotte, Sylvain Chevallier

TL;DR

This work addresses unifying machine learning methods on the SPD matrix manifold $\mathcal{P}_d$ by building a probabilistic framework based on Gaussian-like distributions with respect to the Affine-Invariant Riemannian Metric. It shows that classifiers such as Minimum Distance to Mean and Tangent Space LDA/QDA can be viewed as Bayes classifiers under isotropic or wrapped Gaussian models, providing a common interpretation for classification, outlier detection, and dimension reduction on SPD data. The framework naturally encompasses the Riemannian potato for outlier detection and dimension-reduction tools like Riemannian t-SNE and PCA, and it points to extensions to other Gaussian families and kernel-based or deep learning approaches on $\mathcal{P}_d$. Overall, the paper offers a versatile probabilistic foundation for extending SPD-based ML tools to broader manifolds and motivates future work on kernel design and deep models in this setting.

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 $\mathcal{P}_d$.

A probabilistic view on Riemannian machine learning models for SPD matrices

TL;DR

This work addresses unifying machine learning methods on the SPD matrix manifold by building a probabilistic framework based on Gaussian-like distributions with respect to the Affine-Invariant Riemannian Metric. It shows that classifiers such as Minimum Distance to Mean and Tangent Space LDA/QDA can be viewed as Bayes classifiers under isotropic or wrapped Gaussian models, providing a common interpretation for classification, outlier detection, and dimension reduction on SPD data. The framework naturally encompasses the Riemannian potato for outlier detection and dimension-reduction tools like Riemannian t-SNE and PCA, and it points to extensions to other Gaussian families and kernel-based or deep learning approaches on . Overall, the paper offers a versatile probabilistic foundation for extending SPD-based ML tools to broader manifolds and motivates future work on kernel design and deep models in this setting.

Abstract

The goal of this paper is to show how different machine learning tools on the Riemannian manifold of Symmetric Positive Definite (SPD) matrices can be united under a probabilistic framework. For this, we will need several Gaussian distributions defined on . We will show how popular classifiers on 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 , we allow for other machine learning tools to be extended to .
Paper Structure (15 sections, 3 theorems, 11 equations, 1 figure)

This paper contains 15 sections, 3 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Different probability distributions on the manifold of SPD matrices
  3. The isotropic Gaussian distribution
  4. Wrapped distributions
  5. Other probability distributions on $\mathcal{P}_d$
  6. Application to classification
  7. Minimum Distance to Mean (MDM)
  8. Tangent Space LDA/QDA
  9. Other probabilistic classifiers for SPD matrices
  10. Application to outlier detection: Riemannian potato
  11. Application to dimension reduction
  12. Riemannian t-SNE
  13. Riemannian PCA
  14. Conclusion
  15. Acknowledgments.

Key Result

proposition thmcounterproposition

Let us suppose that each class is modeled by an isotropic Gaussian distribution centered at $\bar{X}^k$ with a shared $\sigma$, i.e. $\alpha_k = G(\bar{X}^k, \sigma^2)$, then, the MDM converges to the BC when the number of data tends to infinity.

Figures (1)

  • Figure 1: Results of the Riemannian t-SNE and PCA algorithms applied to the covariance matrices of EEG signals from subject 8 of the BNCI2014001 BNCI2014004 dataset. The points are colored according to the class of the task performed by the subject. Both algorithms reduce the data to $2 \times 2$ SPD matrices, which can be visualized in 3D. Indeed, a $2 \times 2$ SPD matrices is of the form $abbc$, which can be represented in 3D by the point $(a, b, c)$.

Theorems & Definitions (4)

  • definition thmcounterdefinition: Bayes Classifier (BC) bishop2007
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proposition thmcounterproposition