Enhancing Motor Imagery BCI classification with Block-Toeplitz Augmented Covariance Matrices and Siegel metric

International audience ; Standard Augmented Covariance Method (ACM) emerges as a combination of phase space reconstruction of dynamical systems and of Riemannian geometry. This approach creates a matrix that contains not only an average spatial representation of the signal but also a representation of its evolution in time. As the ACM matrix also turns out to be an SPD matrix, it can be classified using the same Riemannian framework that was so successful for pure spatial covariance matrix. However, it also possesses a structural property of being Block-Toeplitz, which was not exploited. Recently, an approach has been proposed to better deal with such Block-Toeplitz SPD matrices in the fields of audio processing or radar signal analysis.The idea of this research is thus to endow the smooth manifold of Block-Toeplitz SPD matrices with a Riemannian metric, thus allowing the ACM matrix to be treated within its true manifold membership. It is actually possible to treat the Block-Toeplitz SPD matrix manifold as the product of an SPD manifold and a Siegel Disk Space, after applying an appropriate conversion of the blocks of the ACM matrix into the Verblusky coefficients.The new -- Siegel metric based -- pipeline was tested and validated against several state-of-the-art algorithms (Machine Learning (ML) and Deep Learning (DL)) on several datasets for motor imagery (MI) classification using several subjects and several tasks with the MOABB framework, and a within-session evaluation procedure. The resulting algorithm achieves performance that is significantly better than state-of-the-art BCI algorithms, except for TS+EL to which it is comparable.However, this novel algorithm achieve this at substantial reduction in computational costs and carbon footprint compared with standard ACM.