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 provide an instance-wise certificate of optimality. 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 $n \times m$ rank-$k$ matrix completion problems to certifiable optimality or near optimality in hours for $\max \{m, n\} \leq 2500$ and $k \leq 5$. Moreover, this reduction in the training error translates into an average $2\%$--$50\%$ reduction in the test set error compared with alternating minimization-based methods.

翻译：低秩矩阵补全旨在计算一个复杂度最小的矩阵，以尽可能准确地恢复给定观测集。然而，现有的矩阵补全方法均为启发式算法，虽然具有高度可扩展性且通常能识别高质量解，但无法提供实例层面的最优性证明。本文以最优性为导向重新审视矩阵补全问题。我们将低秩矩阵补全问题重新表述为在非凸投影矩阵集上的凸优化问题，并实现了一种析取分支定界方案，可求解至可证明的最优性。此外，通过将低秩矩阵分解为多个秩一矩阵之和，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近乎精确的凸松弛方法。数值实验表明，与现有尝试相比，我们的新凸松弛方法将最优性间隙降低了两个数量级；对于 $\max \{m, n\} \leq 2500$ 且 $k \leq 5$ 的 $n \times m$ 秩-$k$ 矩阵补全问题，我们的析取分支定界方案可在数小时内求解至可证明的最优解或近似最优解。此外，与基于交替最小化的方法相比，训练误差的降低可转化为测试集误差平均减少 $2\%$--$50\%$。