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

CoRR（2023）

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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Key words

semidefinite relaxations,eigenvector disjunctions,matrix,low-rank

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