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.

翻译：低秩矩阵补全旨在计算一个具有最小复杂度的矩阵，以尽可能精确地恢复给定的观测集合。然而，现有的矩阵补全方法均为启发式算法，尽管其扩展性强且常能识别出高质量解，但缺乏最优性保证。我们以最优性为导向重新审视矩阵补全问题。将这类低秩问题重构为基于非凸投影矩阵集合上的凸问题，并实现一种析取分支定界方案，以可验证的最优性求解此类问题。此外，通过将低秩矩阵分解为秩一矩阵之和，并激励每个秩一矩阵中的二阶子式行列式为零，我们推导出一类新颖且通常紧的凸松弛。数值实验表明，与现有方法相比，我们的新型凸松弛将最优性差距降低了两个数量级；对于n≤150且r≤5的n×n秩r矩阵补全问题，所提出的析取分支定界方案可在数小时内达到可验证的最优解。