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Riemannian Geometry-Based EEG Approaches: A Literature Review

Imad Eddine Tibermacine, Samuele Russo, Ahmed Tibermacine, Abdelaziz Rabehi, Bachir Nail, Kamel Kadri, Christian Napoli

TL;DR

The paper surveys how Riemannian geometry, especially on SPD manifolds, can be integrated with deep learning to improve EEG-based BCIs. It covers feature extraction, classification, manifold learning, tangent-space methods, and transfer learning, illustrating how metrics like the Log-Euclidean and affine-invariant distances, together with tangent-space mappings, enable robust, calibration-efficient decoding. The review highlights methods such as SPDNet, CAPM, HORC, RGIF, MEKT variants, and GNNs on SPD data, showing improvements in MI/SSVEP recognition and cross-subject transfer. The work identifies current limitations—computational complexity, generalization, artifacts, and interpretability—and outlines future directions toward scalable, real-time, multi-modal, and open-platform approaches with potential practical impact in clinical and everyday BCI use.

Abstract

The application of Riemannian geometry in the decoding of brain-computer interfaces (BCIs) has swiftly garnered attention because of its straightforwardness, precision, and resilience, along with its aptitude for transfer learning, which has been demonstrated through significant achievements in global BCI competitions. This paper presents a comprehensive review of recent advancements in the integration of deep learning with Riemannian geometry to enhance EEG signal decoding in BCIs. Our review updates the findings since the last major review in 2017, comparing modern approaches that utilize deep learning to improve the handling of non-Euclidean data structures inherent in EEG signals. We discuss how these approaches not only tackle the traditional challenges of noise sensitivity, non-stationarity, and lengthy calibration times but also introduce novel classification frameworks and signal processing techniques to reduce these limitations significantly. Furthermore, we identify current shortcomings and propose future research directions in manifold learning and riemannian-based classification, focusing on practical implementations and theoretical expansions, such as feature tracking on manifolds, multitask learning, feature extraction, and transfer learning. This review aims to bridge the gap between theoretical research and practical, real-world applications, making sophisticated mathematical approaches accessible and actionable for BCI enhancements.

Riemannian Geometry-Based EEG Approaches: A Literature Review

TL;DR

The paper surveys how Riemannian geometry, especially on SPD manifolds, can be integrated with deep learning to improve EEG-based BCIs. It covers feature extraction, classification, manifold learning, tangent-space methods, and transfer learning, illustrating how metrics like the Log-Euclidean and affine-invariant distances, together with tangent-space mappings, enable robust, calibration-efficient decoding. The review highlights methods such as SPDNet, CAPM, HORC, RGIF, MEKT variants, and GNNs on SPD data, showing improvements in MI/SSVEP recognition and cross-subject transfer. The work identifies current limitations—computational complexity, generalization, artifacts, and interpretability—and outlines future directions toward scalable, real-time, multi-modal, and open-platform approaches with potential practical impact in clinical and everyday BCI use.

Abstract

The application of Riemannian geometry in the decoding of brain-computer interfaces (BCIs) has swiftly garnered attention because of its straightforwardness, precision, and resilience, along with its aptitude for transfer learning, which has been demonstrated through significant achievements in global BCI competitions. This paper presents a comprehensive review of recent advancements in the integration of deep learning with Riemannian geometry to enhance EEG signal decoding in BCIs. Our review updates the findings since the last major review in 2017, comparing modern approaches that utilize deep learning to improve the handling of non-Euclidean data structures inherent in EEG signals. We discuss how these approaches not only tackle the traditional challenges of noise sensitivity, non-stationarity, and lengthy calibration times but also introduce novel classification frameworks and signal processing techniques to reduce these limitations significantly. Furthermore, we identify current shortcomings and propose future research directions in manifold learning and riemannian-based classification, focusing on practical implementations and theoretical expansions, such as feature tracking on manifolds, multitask learning, feature extraction, and transfer learning. This review aims to bridge the gap between theoretical research and practical, real-world applications, making sophisticated mathematical approaches accessible and actionable for BCI enhancements.
Paper Structure (16 sections, 70 equations, 3 figures, 1 table)

This paper contains 16 sections, 70 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Brain-Computer Interface
  3. Riemannian Geometry Concepts
  4. The Covariance Matrix of EEG
  5. Riemannian Metrics on SPD Manifolds
  6. Tangent Space at a Point on a Manifold
  7. Geodesic Distances on Riemannian Manifolds
  8. Riemannian Geometry-Based EEG Approaches
  9. Feature Extraction Methods
  10. Classficiation Approaches
  11. manifold learning
  12. Tangent Space Approaches
  13. Transfer Learning Approaches
  14. Discussion and Challenges
  15. Future Directions in Research
  16. ...and 1 more sections

Figures (3)

  • Figure 1: The SPD Manifold pipelinea76
  • Figure 2: Illustration of the gradient descent algorithms on the Riemannian product manifoldb49.
  • Figure 3: Classification tasks presented in the current literature