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

Cross-subject motor-imagery decoding remains a major challenge in EEG-based brain-computer interfaces. To mitigate strong inter-subject variability, recent work has emphasized manifold-based approaches operating on covariance representations. Yet dispersion scaling and orientation alignment remain largely unaddressed in existing methods. In this paper, we address both issues through congruence transforms and introduce three complementary geometry-aware models: (i) Discriminative Congruence Transform (DCT), (ii) Deep Linear DCT (DLDCT), and (iii) Deep DCT-UNet (DDCT-UNet). These models are evaluated both as pre-alignment modules for downstream classifiers and as end-to-end discriminative systems trained via cross-entropy backpropagation with a custom logistic-regression head. Across challenging motor-imagery benchmarks, the proposed framework improves transductive cross-subject accuracy by 2-3%, demonstrating the value of geometry-aware congruence learning.

💡 Research Summary

This paper tackles one of the most persistent challenges in EEG‑based motor‑imagery (MI) brain‑computer interfaces (BCIs): the large inter‑subject variability that hampers cross‑subject decoding. Recent work has shown that representing each trial by its covariance matrix, which lies on the manifold of symmetric positive‑definite (SPD) matrices, yields a more robust feature space than raw time‑series. However, existing manifold‑based pipelines typically rely on Riemannian Alignment (RA) – a subject‑wise whitening that centers each subject’s covariances at the identity. While RA removes the mean shift, it does not address two crucial geometric aspects: (1) dispersion (the eigen‑value spectrum) and (2) orientation (the eigen‑vectors). Consequently, class‑specific ellipsoids may remain anisotropically scaled or rotated, inflating within‑class scatter and degrading discriminability.

The authors propose a family of geometry‑aware transformations that extend RA by learning both a global dispersion scaling factor and a global orthogonal rotation, jointly optimized with a Fisher‑type discriminant loss. The simplest model, Discriminative Congruence Transform (DCT), proceeds as follows: after RA, each covariance is mapped to the tangent space at the identity via the matrix logarithm. A scalar γ > 0 uniformly scales the tangent representation (dispersion scaling). Then an orthogonal matrix R = exp(A − Aᵀ) – where A is an unconstrained matrix and the exponential guarantees R ∈ SO(d) – rotates the scaled tangents (orientation alignment). The transformed tangents L_O = Rᵀ γ L′ R are fed into a Fisher objective that maximizes the ratio of between‑class scatter to within‑class scatter, defined on the Frobenius norm in the tangent space. Additional regularizers keep γ close to 1, R close to the identity, and penalize off‑diagonal energy of the dataset‑averaged rotated matrix, ensuring that the learned transform does not drift away from a well‑conditioned geometry.

While DCT already yields a 2–3 % absolute accuracy gain over plain RA on two benchmark MI datasets (BCI‑IV‑2a and PhysioNet), the authors recognize that a single linear transform in the tangent space may be insufficient for more complex inter‑subject distortions. To this end they introduce two deep extensions that operate directly on the SPD manifold:

1. Deep Linear DCT (DLDCT) – a stack of linear congruence layers Φ_ℓ(C) = W_ℓᵀ C W_ℓ + εI, where each W_ℓ can change dimensionality (d_ℓ → d_{ℓ+1}). By composing several such layers, the model gradually reshapes covariances while preserving SPD structure at every step. Weight matrices are initialized from the eigenvectors of the Euclidean mean of RA‑aligned covariances, replicated to match target dimensions, which yields well‑conditioned operators and reduces reliance on the εI stabilization term. The Fisher loss is still evaluated in the tangent space at the identity, using the log of the input and output covariances, allowing stable back‑propagation.

2. Deep DCT‑UNet (DDCT‑UNet) – a U‑Net‑style encoder‑decoder architecture where each resolution level contains SPD congruence layers. Skip connections fuse multi‑scale representations, and a variance‑regularization term is added to prevent the deep stack from drifting toward subject‑specific identity matrices. This design introduces non‑linearity (via activation functions and multi‑scale fusion) while still respecting the manifold geometry.

Both deep models can be used in two ways: (a) as pre‑aligners, where the transformed covariances are fed to conventional Riemannian classifiers such as Minimum‑Distance‑to‑Mean (MDM), Tangent‑Space Logistic Regression (TSLR), or Tangent‑Space Analysis with LDA (TSA‑LDA); and (b) as end‑to‑end (E2E) discriminative systems, where the same Fisher‑type loss is back‑propagated through the deep stack and a final logistic‑regression head, jointly learning alignment and classification parameters.

The experimental protocol follows a transductive leave‑one‑subject‑out (LOSO) setting: all source subjects are used for training, while the held‑out target subject provides only unlabeled trials at test time (no calibration). Across both datasets, the three proposed models consistently outperform the RA baseline by 2–3 % absolute accuracy, regardless of the downstream classifier. Notably, the deep variants (DLDCT and DDCT‑UNet) achieve larger between‑class scatter and smaller within‑class scatter than DCT, confirming that deeper geometry‑aware transformations can capture more nuanced inter‑subject distortions.

In summary, the paper makes three key contributions: (1) a principled, geometry‑aware extension of Riemannian alignment that jointly learns dispersion scaling and orientation alignment via a discriminative Fisher objective; (2) two deep architectures that preserve SPD structure while providing hierarchical, non‑linear transformations; and (3) a thorough empirical validation showing that these methods improve cross‑subject MI decoding without requiring any labeled target data. The work opens avenues for further research on manifold‑constrained deep learning, including extensions to other neuroimaging modalities, adaptive depth selection, and real‑time BCI deployment.