This paper studies the problem of deterministic rank-one matrix completion. It is known that the simplest semidefinite programming relaxation, involving minimization of the nuclear norm, does not in general return the solution for this problem. In this paper, we show that in every instance where the problem has a unique solution, one can provably recover the original matrix through the level 2 Lasserre relaxation with minimization of the trace norm. We further show that the solution of the proposed semidefinite program is Lipschitz stable with respect to perturbations of the observed entries, unlike more basic algorithms such as nonlinear propagation or ridge regression. Our proof is based on recursively building a certificate of optimality corresponding to a dual sum-of-squares (SoS) polynomial. This SoS polynomial is built from the polynomial ideal generated by the completion constraints and the monomials provided by the minimization of the trace. The proposed relaxation fits in the framework of the Lasserre hierarchy, albeit with the key addition of the trace objective function. Finally, we show how to represent and manipulate the moment tensor in favorable complexity by means of a hierarchical low-rank factorization.

本文研究了确定性秩矩阵完成的问题。众所周知，涉及到核规范最小化的最简单的半定程序松弛通常不会返回该问题的解决方案。在本文中，我们表明，在每种情况下，如果问题有独特的解决方案，则可以通过2级Lasserre松弛并最小化迹线范数来证明可恢复原始矩阵。我们进一步表明，与更基本的算法（例如非线性传播或岭回归）不同，所提出的半定程序的解相对于所观察条目的扰动是Lipschitz稳定的。我们的证明基于递归构建与对偶平方和（SoS）多项式相对应的最优性证书。该SoS多项式是根据由完成约束生成的多项式理想和由迹线最小化提供的单项式建立的。所建议的放宽符合Lasserre层次结构的框架，尽管添加了跟踪目标函数。最后，我们展示如何通过分层的低秩分解来以有利的复杂度表示和操纵矩张量。