Fusion over the Grassmannian for High-Rank Matrix Completion

Jeremy Scott Johnson, Huanran Li, Daniel L. Pimentel-Alarcón

Primary Area: general machine learning (i.e., none of the above)

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Keywords: Manifold Learning, Subspace Clustering, Matrix Completion

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Abstract: This paper presents a new paradigm to cluster and complete data lying in a union of subspaces using points in the Grassmannian as proxies. Our main theoretical contribution exploits the geometry of this Riemannian manifold to obtain local convergence guarantees. Furthermore, 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 selection, and sketching techniques that can be used in practice. We complement our analysis with synthetic and real-data experiments, which show that our approach performs comparable to the state-of-the-art in the {\em easy} cases (high sampling rates), and significantly better in the {\em difficult} cases (low sampling rates), thus shortening the gap towards the fundamental sampling limit of HRMC.

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Submission Number: 1373