Optimal Low-Rank Matrix Completion: Semidefinite Relaxations and Eigenvector Disjunctions

• URL: http://arxiv.org/abs/2305.12292v2
• Date: Fri, 26 Jan 2024 17:34:25 GMT
• Title: Optimal Low-Rank Matrix Completion: Semidefinite Relaxations and Eigenvector Disjunctions
• Authors: Dimitris Bertsimas, Ryan Cory-Wright, Sean Lo, and Jean Pauphilet
• Abstract summary: Low-rank matrix completion consists of a matrix of minimal complexity that recovers a given set of observations as accurately as possible. New convex relaxations decrease the optimal by magnitude compared to existing methods.
• Score: 6.537257913467247
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 these low-rank 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 tight 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 nxn rank-r matrix completion problems to certifiable optimality in hours for n<=150 and r<=5.

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