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Subspace metrics for multivariate dictionaries and application to EEG

Overcomplete representations and dictionary learning algorithms are attracting a growing interest in the machine learning community. This paper addresses the emerging problem of comparing multivariate overcomplete dictionaries. Despite a recurrent need to rely on a distance for learning or assessing multivariate overcomplete dictionaries, no metrics in their underlying spaces have yet been proposed. Henceforth we propose to study overcomplete representations from the perspective of matrix manifolds. We consider distances between multivariate dictionaries as distances between their spans which reveal to be elements of a Grassmannian manifold. We introduce set-metrics defined on Grassmannian spaces and study their properties both theoretically and numerically. Thanks to the introduced metrics, experimental convergences of dictionary learning algorithms are assessed on synthetic datasets. Set-metrics are embedded in a clustering algorithm for a qualitative analysis of real EEG signals for Brain-Computer Interfaces (BCI). The obtained clusters of subjects are associated with subject performances. This is a major methodological advance to understand the BCI-inefficiency phenomenon and to predict the ability of a user to interact with a BCI.