  Overcomplete representations and dictionary learning algorithms kept
attracting a growing interest in the machine learning community. This paper
addresses the emerging problem of comparing multivariate overcomplete
representations. Despite a recurrent need to rely on a distance for learning or
assessing multivariate overcomplete representations, no metrics in their
underlying spaces have yet been proposed. Henceforth we propose to study
overcomplete representations from the perspective of frame theory and matrix
manifolds. We consider distances between multivariate dictionaries as distances
between their spans which reveal to be elements of a Grassmannian manifold. We
introduce Wasserstein-like set-metrics defined on Grassmannian spaces and study
their properties both theoretically and numerically. Indeed a deep experimental
study based on tailored synthetic datasetsand real EEG signals for
Brain-Computer Interfaces (BCI) have been conducted. In particular, the
introduced metrics have been embedded in clustering algorithm and applied to
BCI Competition IV-2a for dataset quality assessment. Besides, a principled
connection is made between three close but still disjoint research fields,
namely, Grassmannian packing, dictionary learning and compressed sensing.
