Consistent Collective Matrix Completion under Joint Low Rank Structure

TL;DR

This paper introduces a new approach for jointly recovering multiple matrices with shared low-rank structures from partial observations, providing theoretical guarantees and scalable algorithms that outperform standard methods.

Contribution

It develops a rigorous algebraic framework, proposes a convex estimator with proven consistency, and offers scalable algorithms for collective matrix completion with optimal sample complexity.

Findings

Exact recovery with high probability under certain sample complexity

Sample complexity is near-optimal up to logarithmic factors

Scalable approximate algorithms effectively solve the proposed convex program

Abstract

We address the collective matrix completion problem of jointly recovering a collection of matrices with shared structure from partial (and potentially noisy) observations. To ensure well--posedness of the problem, 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. We first develop a rigorous algebra for representing and manipulating collective--matrix structure, and identify sufficient conditions for consistent estimation of collective matrices. We then propose a tractable convex estimator for solving the collective matrix completion problem, and provide the first non--trivial theoretical guarantees for consistency of collective matrix completion. We show that under reasonable assumptions stated in Section 3.1, with high probability, the proposed estimator exactly recovers the true matrices whenever sample complexity requirements dictated by Theorem 1 are met. The sample complexity requirement derived in the paper are optimum up to logarithmic factors, and significantly improve upon the requirements obtained by trivial extensions of standard matrix completion. Finally, we propose a scalable approximate algorithm to solve the proposed convex program, and corroborate our results through simulated experiments.