Consistent Collective Matrix Completion under Joint Low Rank Structure

We address the collective matrix completion problem of jointly recovering a collection of matrices with shared structure from partial (and potentially noisy) observations. A collection of matrices form an entity-relationship component, where each matrix is a relation between a pair of entities. We impose a joint low rank structure, wherein each component matrix is low rank and the latent space of the low rank factors corresponding to each entity is shared across the entire collection. In this paper, we first develop a rigorous algebra for the collective-matrix structure, and propose a convex estimate for solving the collective matrix completion problem. We then provide the first theoretical guarantees for consistency of collective matrix completion. We show that for a subset of entity-relationship structures defining a collective matrix (See Assumption 3), with high probability, the proposed estimator exactly recovers the true matrices whenever certain sample complexity requirements (dictated by Theorem 1) are met. We finally corroborate our results through simulated experiments.