Low-rank matrix completion consists of computing a matrix of minimal complexity that recovers a given set of observations as accurately as possible, and has numerous applications such as product recommendation. Unfortunately, existing methods for solving low-rank 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, by reformulating low-rank problems as convex problems over the non-convex set of projection matrices and implementing 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, via a Shor relaxation, that each two-by-two minor in each rank-one matrix has determinant zero. In numerical experiments, our new convex relaxations decrease the optimality gap by two orders of magnitude compared to existing attempts. Moreover, we showcase the performance of our disjunctive branch-and-bound scheme and demonstrate that it solves matrix completion problems over 150x150 matrices to certifiable optimality in hours, constituting an order of magnitude improvement on the state-of-the-art for certifiably optimal methods.

翻译：低秩矩阵补全包括计算一个最小复杂度的矩阵，以尽可能准确地恢复给定的观测集合，并在产品推荐等领域有众多应用。遗憾的是，现有求解低秩矩阵补全的方法均为启发式算法，虽然具有高度可扩展性且常能找到高质量解，但缺乏任何最优性保证。我们以最优性为导向重新审视矩阵补全，将低秩问题重新表述为投影矩阵非凸集上的凸问题，并实现一种可证明最优解的析取分支定界方案。进一步，我们通过将低秩矩阵分解为秩一矩阵之和，并利用Shor松弛激励每个秩一矩阵中每个二阶子矩阵的行列式为零，推导出一类新颖且通常紧致的凸松弛。在数值实验中，我们的新型凸松弛将最优性差距比现有尝试降低两个数量级。此外，我们展示了析取分支定界方案的性能，并证明其能在数小时内将150x150矩阵的补全问题求解至可证明最优，相较现有可证明最优方法实现了一个数量级的改进。