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**Stable rank-one matrix completion is solved by the level \(2\) Lasserre relaxation.**
*(English)*
Zbl 1478.90076

Summary: 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.

### MSC:

90C22 | Semidefinite programming |

90C25 | Convex programming |

65K05 | Numerical mathematical programming methods |

90C59 | Approximation methods and heuristics in mathematical programming |

### Keywords:

matrix completion; convex optimization; semidefinite programming; semidefinite programming hierarchies; duality in optimization; sum-of-squares polynomials
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\textit{A. Cosse} and \textit{L. Demanet}, Found. Comput. Math. 21, No. 4, 891--940 (2021; Zbl 1478.90076)

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