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Geometry-Aware Deep Congruence Networks for Manifold Learning in Cross-Subject Motor Imagery

Sanjeev Manivannan, Chandrashekar Lakshminarayan

TL;DR

This work tackles zero-shot cross-subject motor-imagery decoding in EEG by leveraging the geometry of SPD covariance matrices. It introduces geometry-aware preprocessing modules (DCR and RiFU) and two deep congruence network classifiers (SPD-DCNet and RiFUNet) that operate directly on SPD data to learn subject-invariant representations. On the BCI-IV 2a dataset under LOSO, the approach yields a 3–4% improvement over strong classical baselines, with SPD-DCNet achieving the highest mean cross-subject accuracy. The framework demonstrates calibration-free, subject-generalizable EEG decoding by integrating Riemannian geometry with learnable SPD transforms and Fisher-based regularizers.

Abstract

Cross-subject motor-imagery decoding remains a major challenge in EEG-based brain-computer interfaces due to strong subject variability and the curved geometry of covariance matrices on the symmetric positive definite (SPD) manifold. We address the zero-shot cross-subject setting, where no target-subject labels or adaptation are allowed, by introducing novel geometry-aware preprocessing modules and deep congruence networks that operate directly on SPD covariance matrices. Our preprocessing modules, DCR and RiFU, extend Riemannian Alignment by improving action separation while reducing subject-specific distortions. We further propose two manifold classifiers, SPD-DCNet and RiFUNet, which use hierarchical congruence transforms to learn discriminative, subject-invariant covariance representations. On the BCI-IV 2a benchmark, our framework improves cross-subject accuracy by 3-4% over the strongest classical baselines, demonstrating the value of geometry-aware transformations for robust EEG decoding.

Geometry-Aware Deep Congruence Networks for Manifold Learning in Cross-Subject Motor Imagery

TL;DR

This work tackles zero-shot cross-subject motor-imagery decoding in EEG by leveraging the geometry of SPD covariance matrices. It introduces geometry-aware preprocessing modules (DCR and RiFU) and two deep congruence network classifiers (SPD-DCNet and RiFUNet) that operate directly on SPD data to learn subject-invariant representations. On the BCI-IV 2a dataset under LOSO, the approach yields a 3–4% improvement over strong classical baselines, with SPD-DCNet achieving the highest mean cross-subject accuracy. The framework demonstrates calibration-free, subject-generalizable EEG decoding by integrating Riemannian geometry with learnable SPD transforms and Fisher-based regularizers.

Abstract

Cross-subject motor-imagery decoding remains a major challenge in EEG-based brain-computer interfaces due to strong subject variability and the curved geometry of covariance matrices on the symmetric positive definite (SPD) manifold. We address the zero-shot cross-subject setting, where no target-subject labels or adaptation are allowed, by introducing novel geometry-aware preprocessing modules and deep congruence networks that operate directly on SPD covariance matrices. Our preprocessing modules, DCR and RiFU, extend Riemannian Alignment by improving action separation while reducing subject-specific distortions. We further propose two manifold classifiers, SPD-DCNet and RiFUNet, which use hierarchical congruence transforms to learn discriminative, subject-invariant covariance representations. On the BCI-IV 2a benchmark, our framework improves cross-subject accuracy by 3-4% over the strongest classical baselines, demonstrating the value of geometry-aware transformations for robust EEG decoding.

Paper Structure

This paper contains 27 sections, 2 theorems, 35 equations, 2 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Dataset and Notation
  3. Preliminaries
  4. SPD Manifold and Geometry
  5. Congruence Transforms on SPD Matrices
  6. Classical Preprocessing Methods
  7. Riemannian Alignment (RA)
  8. Common Spatial Patterns (CSP)
  9. LDA Projection
  10. Riemannian Procrustes Analysis (RPA)
  11. Classical Classifiers
  12. Tangent Space Logistic Regression (TSLR)
  13. Minimum Distance to Mean (MDM)
  14. LDA Classifier
  15. Methods
  16. ...and 12 more sections

Key Result

Lemma 3.1

For any $C \in \mathcal{S}_{++}^C$ and full-rank $W$, the matrix $C' = W C W^\top$ remains SPD.

Figures (2)

  • Figure 1: (i) DCR Pre-Aligner: Learns dispersion scaling and rotation in the tangent space after RA to produce an aligned SPD. (ii) RiFU Pre-Aligner: Uses an RA-initialized Riemannian U-Net to reconstruct an aligned SPD with multi-scale geometric structure.
  • Figure 2: i) RiFUNet classifier with a Riemannian U-Net feature extractor followed by a TSLR action head. (ii) SPD-DCNet classifier composed of deep congruence layers with a linear (no-activation) action head. Together, these architectures illustrate Deep Congruence Networks for cross-subject action decoding.

Theorems & Definitions (2)

  • Lemma 3.1: SPD Preservation
  • Lemma 3.2: Affine Invariance