Fusion over the Grassmannian for High-Rank Matrix Completion

This paper presents a new paradigm to cluster and complete data lying in a union of subspaces using points on the Grassmannian as proxies. Our approach does not require prior knowledge of the number of subspaces, is naturally suited to handle noise, and only requires an upper bound on the subspaces' dimensions. We detail clustering, completion, model section, and sketching techniques that can be used in practice. We complement our discussion with synthetic and real-data experiments, which show that our approach performs comparable to the state-of-the art in the easy cases (high sampling rates), and significantly better in the difficult cases (low sampling rates), thus shortening the gap towards the fundamental sampling limit of HRMC.