  Covariance matrices of noisy multichannel electroencephalogram time series
data are hard to estimate due to high dimensionality. In brain-computer
interfaces (BCI) based on event-related potentials and a linear discriminant
analysis (LDA) for classification, the state of the art to address this problem
is by shrinkage regularization. We propose a novel idea to tackle this problem
by enforcing a block-Toeplitz structure for the covariance matrix of the LDA,
which implements an assumption of signal stationarity in short time windows for
each channel. On data of 213 subjects collected under 13 event-related
potential BCI protocols, the resulting 'ToeplitzLDA' significantly increases
the binary classification performance compared to shrinkage regularized LDA (up
to 6 AUC points) and Riemannian classification approaches (up to 2 AUC points).
This translates to greatly improved application level performances, as
exemplified on data recorded during an unsupervised visual speller application,
where spelling errors could be reduced by 81% on average for 25 subjects. Aside
from lower memory and time complexity for LDA training, ToeplitzLDA proved to
be almost invariant even to a twenty-fold time dimensionality enlargement,
which reduces the need of expert knowledge regarding feature extraction.
