Fusion over the Grassmann Manifold for Incomplete-Data Clustering

Jeremy Scott Johnson, Huanran Li, Chun Gan, Zheng Lu, Matthew Malloy, Daniel L. Pimentel-Alarcón

Keywords: high-rank matrix completion, subspace clustering, manifold learning

TL;DR: We introduce a novel paradigm that optimizes on the Grassmannian to complete and cluster incomplete data in a union of subspaces.

Abstract: This paper presents a new paradigm to cluster incomplete vectors using subspaces as proxies to exploit the geometry of the Grassmannian. We leverage this new perspective to develop an algorithm to cluster and complete data in a union of subspaces via a fusion penalty formulation. 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. In developing our model, we present local convergence guarantees. We describe clustering, completion, model selection, and sketching techniques that can be used in practice, and complement our analysis with synthetic and real-data experiments.

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