A New Canonical Log-Euclidean Kernel for Symmetric Positive Definite Matrices for EEG Analysis (Oct 2024)

Objective: Working with the Riemannian manifold of symmetric positive-definite (SPD) matrices has become popular in electroencephalography (EEG) analysis. Frequently selected for its speed property is the manifold geometry provided by the log-euclidean Riemannian metric. However, the kernels used in the log-euclidean framework are not canonically based on the underlying geometry. Therefore, we introduce a new canonical log-euclidean (CLE) kernel. Methods: We used the log-euclidean metric tensor on the SPD manifold to derive the CLE kernel. We compared it with existing kernels, namely the affine-invariant, log-euclidean, and Gaussian log-euclidean kernel. For comparison, we tested the kernels on two paradigms: classification and dimensionality reduction. Each paradigm was evaluated on five open-access brain-computer interface datasets with motor-imagery tasks across multiple sessions. Performance was measured as balanced classification accuracy using a leave-one-session-out cross-validation. Dimensionality reduction performance was measured using AUClogRNX. Results: The CLE kernel itself is simple and easily turned into code, which is provided in addition to all the analytical solutions to relevant equations in the log-euclidean framework. The CLE kernel significantly outperformed existing log-euclidean kernels in classification tasks and was several times faster than the affine-invariant kernel for most datasets. Conclusion: We found that adhering to the geometrical structure significantly improves the accuracy over two commonly used log-euclidean kernels while keeping the speed advantages of the log-euclidean framework. Significance: The CLE provides a good choice as a kernel in time-critical applications and fills a gap in the kernel methods of the log-euclidean framework.