Enhancing Computational Efficiency of Motor Imagery BCI Classification with Block-Toeplitz Augmented Covariance Matrices and Siegel Metric

TL;DR

An enhancement to the augmented covariance method (ACM) is introduced, exploiting more thoroughly its mathematical properties, in order to improve motor imagery classification and improves consequently the computational efficiency over ACM, making it even more suitable for real time experiments.

Abstract

Electroencephalographic signals are represented as multidimensional datasets. We introduce an enhancement to the augmented covariance method (ACM), exploiting more thoroughly its mathematical properties, in order to improve motor imagery classification.Standard ACM emerges as a combination of phase space reconstruction of dynamical systems and of Riemannian geometry. Indeed, it is based on the construction of a Symmetric Positive Definite matrix to improve classification. But this matrix also has a Block-Toeplitz structure that was previously ignored. This work treats such matrices in the real manifold to which they belong: the set of Block-Toeplitz SPD matrices. After some manipulation, this set is can be seen as the product of an SPD manifold and a Siegel Disk Space.The proposed methodology was tested using the MOABB framework with a within-session evaluation procedure. It achieves a similar classification performance to ACM, which is typically better than -- or at worse comparable to -- state-of-the-art methods. But, it also improves consequently the computational efficiency over ACM, making it even more suitable for real time experiments.

Figures (2)

• Figure 1: Schematic illustration of the BT-ACM + TG + SVM methodology. The presented example uses only 3 electrodes (in red on the top left plot). The measurement process of the original dynamic system is thus a 3-dimensional time series. The process then begins with the extraction of epoch signal representing left and right hand tasks. We then use the phase space reconstruction process to obtain a dynamic system equivalent to the original one (selection of hyper-parameters are made via grid search using the nested approach). In this figure, we see an embedding corresponding to $p=3$ and $\tau=10$). The BT-ACM matrix is computed as the autocovariance of this high-dimensional time series. Subsequently the main blocks are converted in Verblunsky coefficients. Then, each component is mapped to the tangent space using the appropriate Riemannian manifold computations and vectorized. The final step is the application of an SVM-based classification algorithm.
• Figure 2: Results for Right vs Left hand classification, using WS evaluation. Plot (a) provides a combined meta analysis (over all datasets) of the different pipelines. It shows the significance that the algorithm on the y-axis is better than the one on the x-axis. The color represents the significance level of the difference of accuracy, in terms of t-values. We only show significant interactions ($p < 0.05$). Plots (b) summarizes the computational time and carbon footprint of ACM+TS+SVM vs BT-ACM+TS+SBM. Plots (c), (d) and (e) show the meta analysis of BT-ACM+TS+SVM against respectively ACM+TS+SVM (Grid), TS+EN, DeepConvNet. We show the standardized mean differences of p-values computed as one-tailed Wilcoxon signed-rank test for the hypothesis given as title of the plot. The gray bar denotes the $95\%$ interval. * stands for $p < 0.05$, ** for $p < 0.01$, and *** for $p < 0.001$.