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

2023-05-20

AI Summary

Existing low-rank matrix completion methods are efficient and scalable but lack optimality guarantees. This paper proposes the first verifiably optimal solution framework. To address the nonconvex rank constraint, we formulate a convex reconstruction model based on projection matrices and introduce determinant-zero constraints on all second-order minors to achieve nearly exact convex relaxation. We then design a tailored branch-and-bound algorithm for the resulting nonconvex structure, integrating low-rank matrix factorization with convex optimization techniques. On problem instances with dimensions $n,m leq 2500$ and rank $r leq 5$, our method computes certifiably optimal or near-optimal solutions within hours. It reduces the average optimality gap by two orders of magnitude and lowers test error by 2%–50% compared to state-of-the-art heuristics. This breakthrough overcomes the long-standing limitation of heuristic approaches—namely, their inability to verify global optimality.

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 possess any optimality guarantees. 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 $n imes m$ rank-$r$ matrix completion problems to certifiable optimality or near optimality in hours for $max {m, n} leq 2500$ and $r leq 5$. Moreover, this improvement in the training error translates into an average $2%$--$50%$ improvement in the test set error.

Problem

• Certifiably optimal low-rank matrix completion
• Overcoming heuristic methods lacking optimality guarantees
• Improving test set error by 2% to 50%

Innovation

• Disjunctive branch-and-bound for optimal matrix completion
• Convex relaxations via rank-one matrix decomposition
• Certifiable optimality for low-rank matrices