Metrics for Multivariate Dictionaries

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. In fact, 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, as 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 datasets and 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 both subjects and tasks classification and datasets quality assessment. Besides, a principled connection is made between three close but still disjoint research fields, namely Grassmannian packing, dictionary learning and compressed sensing.