Brain computer interfaces (BCIs) have been attracting a great interest in recent years. The common spatial patterns (CSP) technique is a well-established approach to the spatial filtering of the electroencephalogram (EEG) data in BCI applications. Even though CSP was originally proposed from a heuristic viewpoint, it can be also built on very strong foundations using information theory. This paper reviews the relationship between CSP and several information-theoretic approaches, including the Kullback-Leibler divergence, the Beta divergence and the Alpha-Beta log-det (AB-LD)divergence. We also revise other approaches based on the idea of selecting those features that are maximally informative about the class labels. The performance of all the methods will be also compared via experiments.

In brain-computer interfaces (BCIs), the typical models of the EEG observations usually lead to a poor estimation of the trial covariance matrices, given the high non-stationarity of the EEG sources. We propose the application of two techniques that significantly improve the accuracy of these estimations and can be combined with a wide range of motor imagery BCI (MI-BCI) methods. The first one scales the observations in such a way that implicitly normalizes the common temporal strength of the source activities. When the scaling applies independently to the trials of the observations, the procedure justifies and improves the classical preprocessing for the EEG data. In addition, when the scaling is instantaneous and independent for each sample, the procedure particularizes to Tyler's method in statistics for obtaining a distributionfree estimate of scattering. In this case, the proposal provides an original interpretation of this existing method as a technique that pursuits an implicit instantaneous power-normalization of the underlying source processes. The second technique applies to the classifier and improves its performance through a convenient regularization of the features covariance matrix. Experimental tests reveal that a combination of the proposed techniques with the state-of-the-art algorithms for motor-imagery classification provides a significant improvement in the classification results. Index Terms-Common spatial pattern, brain-computer interfaces, motor-imagery classification, covariance matrix estimation. I. INTRODUCTION B RAIN-COMPUTER interfaces (BCI) have a great potential for enabling the communication between machine and humans by means of the analysis of the electroencephalographic activity. Nowadays, almost all the Motor Imagery BCI (MI-BCI) systems summarize most of the relevant information about the measurements in two kinds of covariance matrices: the covariance matrices of the filtered observations (employed for dimensionality reduction) and the covariance matrices of the features (which are required for classification). In the dimensionality reduction Manuscript

Abstract:The Alpha-Beta Log-Det divergences for positive definite matrices are flexible divergences that are parameterized by two real constants and are able to specialize several relevant classical cases like the squared Riemannian metric, the Steins loss, the S-divergence, etc. A novel classification criterion based on these divergences is optimized to address the problem of classification of the motor imagery movements. This research paper is divided into three main sections in order to address the above mentioned problem: (1) Firstly, it is proven that a suitable scaling of the class conditional covariance matrices can be used to link the Common Spatial Pattern (CSP) solution with a predefined number of spatial filters for each class and its representation as a divergence optimization problem by making their different filter selection policies compatible; (2) A closed form formula for the gradient of the Alpha-Beta Log-Det divergences is derived that allows to perform optimization as well as easily use it in many practical applications; (3) Finally, in similarity with the work of Samek et al. 2014, which proposed the robust spatial filtering of the motor imagery movements based on the beta-divergence, the optimization of the Alpha-Beta Log-Det divergences is applied to this problem. The resulting subspace algorithm provides a unified framework for testing the performance and robustness of the several divergences in different scenarios.

The common spatial pattern (CSP) method is a dimensionality reduction technique widely used in brain-computer interface (BCI) systems. In the two-class CSP problem, training data are linearly projected onto directions maximizing or minimizing the variance ratio between the two classes. The present contribution proves that kurtosis maximization performs CSP in an unsupervised manner, i.e., with no need for labeled data, when the classes follow Gaussian or elliptically symmetric distributions. Numerical analyses on synthetic and real data validate these findings in various experimental conditions, and demonstrate the interest of the proposed unsupervised approach.