Technical Summary: A Sliced-Wasserstein Framework on Correlation Matrices for EEG Decoding

Problem Statement

Electroencephalography (EEG) decoding relies heavily on covariance descriptors for their robustness to noise. However, these representations are sensitive to channel-wise scaling, which acts as a nuisance factor in scientific settings where absolute signal magnitudes vary significantly across subjects and sessions. While full-rank correlation matrices offer a scale-invariant alternative, they form a quotient manifold of Symmetric Positive Definite (SPD) matrices that does not admit standard SPD geometry. Consequently, they have received limited attention in deep geometric learning.

Furthermore, comparing distributions of correlation matrices extracted from EEG recordings typically requires Optimal Transport (OT) tools. While the Wasserstein distance provides a principled framework for distribution comparison, computing it on Riemannian manifolds involves solving large-scale linear programs with prohibitive computational complexity ($O(n^3 \log n)$). Existing Sliced-Wasserstein (SW) extensions to non-Euclidean spaces (e.g., spherical, hyperbolic, or SPD manifolds) do not yet provide a principled construction tailored specifically to the correlation manifold under recently introduced intrinsic geometries.

Methodology

1. Pullback Euclidean Metric Sliced Wasserstein (PEMSW)

The authors propose a general framework for defining SW discrepancies on manifolds endowed with Pullback Euclidean Metrics (PEMs).

• Core Concept: A PEM is defined via a global diffeomorphism $\phi: \mathcal{M} \to \mathcal{E}$, where $\mathcal{E}$ is a finite-dimensional inner-product space. This embedding maps geodesics on the manifold $\mathcal{M}$ to straight lines in the embedding space $\mathcal{E}$.
• Geodesic Slicing: Instead of complex geodesic slicing on the manifold, the framework performs standard linear slicing in the embedding space $\mathcal{E}$. For a reference point $x_0$ and a direction $a$ in the unit sphere of $\mathcal{E}$, the slicing coordinate is a linear projection of the embedded point $\phi(x)$.
• Equivalence: Theorem 3.4 establishes that the PEMS distance on the manifold is exactly equivalent to the Euclidean SW distance between the embedded measures ($\tilde{\mu} = \phi_\# \mu$ and $\tilde{\nu} = \phi_\# \nu$). This allows for efficient computation using standard Euclidean SW algorithms while preserving the manifold's intrinsic geometry.
• Properties: The framework inherits metric properties, weak convergence guarantees, and provides upper/lower bounds relative to the true Wasserstein distance on the manifold.

2. Correlation Sliced-Wasserstein (CorSW)

The authors instantiate PEMS on the manifold of full-rank correlation matrices ($\mathcal{C}^{++}_n$) using two recently proposed intrinsic Riemannian metrics:

• Off-Log Metric (OLM): Uses the diffeomorphism $\text{Logo}$ mapping $\mathcal{C}^{++}_n$ to the space of hollow symmetric matrices ($\text{Hol}(n)$). The slicing coordinate is the Frobenius inner product between the direction and the $\text{Logo}$-transformed matrix.
• Log-Scaled Metric (LSM): Uses the diffeomorphism $\text{Log}^\star$ mapping $\mathcal{C}^{++}_n$ to the space of symmetric matrices with zero row sums ($\text{Row}_0(n)$).
• Unified View: The paper demonstrates that the existing SPDSW (Sliced-Wasserstein on SPD manifolds) is a special case of PEMS when the manifold is SPD and the embedding is the matrix logarithm.

3. Domain Generalization Framework

To address distribution shifts in EEG decoding (e.g., across subjects or sessions), the authors develop a domain generalization (DG) framework using CorSW:

• Objective: Align the empirical distribution of correlation matrices from each source domain to a common, domain-agnostic reference law.
• Implementation: A reference distribution is generated in the embedding space (as a Gaussian) and mapped back to the correlation manifold. The loss minimizes the CorSW distance between the source domain's correlation matrices and this reference.
• Efficiency: The method requires no iterative OT solvers; it relies on closed-form embeddings and Monte Carlo sampling of slicing directions, adding minimal overhead to training.

Key Contributions

1. PEMSW Framework: A unified construction of SW discrepancies on manifolds with PEMs, extending classical SW to non-Euclidean spaces while preserving geometric structure.
2. CorSW Instantiation: Two specific CorSW discrepancies derived for correlation matrices under OLM and LSM geometries, providing closed-form slicing coordinates.
3. Theoretical Unification: A proof that existing SPDSW distances are special cases of PEMS, clarifying the geometric foundations of sliced discrepancies.
4. Empirical Validation: A DG framework for EEG decoding that demonstrates improved generalization across three benchmark datasets (BCIC-IV-2a, MAMEM-SSVEP-II, BCI-ERN) with low training overhead and no inference cost.

Experimental Results

The proposed method was evaluated on three EEG decoding paradigms: Motor Imagery (MI), Steady-State Visual Evoked Potentials (SSVEP), and Error-Related Negativity (ERN).

• Performance: Across all datasets, models augmented with CorSW (e.g., CorAtt-OLM+CorSW-OLM) consistently outperformed baseline Euclidean models, standard Riemannian models, and other domain generalization strategies (such as Instance Weighting and various whitening techniques).
  • On the ERN task, CorSW improved the mean accuracy of the baseline CorAtt-OLM by +4.66%.
  • On the SSVEP task, it achieved a +2.76% gain.
  • On the MI task, it provided a +2.45% gain.
• Robustness: In exhaustive cross-session domain generalization protocols (leave-one-session-out), CorSW achieved the highest mean scores among all compared DG methods.
• Stability: The addition of CorSW reduced run-to-run variability (standard deviation) significantly, indicating improved training stability under session-wise shifts.
• Efficiency: Training time increased by only ~5.2 ms per iteration compared to the baseline, with no additional cost during inference.
• Interpretability: Gradient-based visualizations confirmed that the learned representations focused on neurophysiologically meaningful patterns (e.g., Oz electrode for SSVEP, FCz for ERN), aligning with established neuroscience findings.

Significance and Claims

The paper claims that PEMSW provides a general and extensible approach for distribution comparison on pullback manifolds. By reducing geodesic slicing to linear slicing in an embedding space, it offers a principled and efficient geometry-aware discrepancy.

Specifically for EEG, the authors argue that CorSW effectively addresses the challenge of scale-invariance and distribution shifts. Unlike previous methods that either ignore the correlation manifold structure or rely on computationally expensive OT, CorSW leverages intrinsic geometries (OLM/LSM) to align distributions efficiently. The results suggest that aligning source-domain distributions directly on the correlation manifold is more effective than applying Euclidean alignment losses to correlation-based representations.

The authors note limitations, primarily that the framework is restricted to manifolds with global diffeomorphic embeddings (PEMs) and does not yet extend to general Riemannian manifolds without such structures. Ethically, they emphasize that while the method uses existing benchmarks, practical applications involving sensitive EEG data must adhere to strict data governance and privacy protocols.