Disjunctive Branch-and-Bound for Certifiably Optimal Low-Rank Matrix Completion

TL;DR

This paper introduces a novel branch-and-bound approach with convex relaxations for certifiably optimal low-rank matrix completion, significantly reducing optimality gaps and improving test error over existing heuristics.

Contribution

It reformulates low-rank matrix completion as convex problems over projection matrices and develops a disjunctive branch-and-bound scheme for certifiable optimality.

Findings

Reduces optimality gap by two orders of magnitude.

Solves large-scale problems to certifiable optimality or near optimality within hours.

Achieves 2% to 50% reduction in test error compared to existing heuristics.

Abstract

Low-rank matrix completion consists of computing a matrix of minimal complexity that recovers a given set of observations as accurately as possible. Unfortunately, existing methods for matrix completion are heuristics that, while highly scalable and often identifying high-quality solutions, do not provide an instance-wise certificate of optimality. We reexamine matrix completion with an optimality-oriented eye. We reformulate low-rank matrix completion problems as convex problems over the non-convex set of projection matrices and implement a disjunctive branch-and-bound scheme that solves them to certifiable optimality. Further, we derive a novel and often near-exact class of convex relaxations by decomposing a low-rank matrix as a sum of rank-one matrices and incentivizing that two-by-two minors in each rank-one matrix have determinant zero. In numerical experiments, our new convex relaxations decrease the optimality gap by two orders of magnitude compared to existing attempts, and our disjunctive branch-and-bound scheme solves $n \times m$ rank-$k$ matrix completion problems to certifiable optimality or near optimality in hours for $\max \{m, n\} \leq 2500$ and $k \leq 5$. Moreover, this reduction in the training error translates into an average $2\%$--$50\%$ reduction in the test set error compared with alternating minimization-based methods.