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

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.

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.

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.

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.

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.