Robust low-rank matrix estimation.

*(English)*Zbl 1412.62068The authors consider so called matrix completion problems, that means there is a high-dimensional matrix with \(p\) rows and \(q\) columns but with only \(n<pq\) observed (noisy) entries and the challenge is now to predict the missing entries. This uniform sampling matrix completion problem is often discussed in statistical papers but the used estimators which are optimal WRT quadratic loss are not robust. In this paper the authors consider so called robust nuclear norm (trace norm) penalized estimators which are optimal WRT an absolute value loss function or WRT Huber’s loss function (with given tuning parameter). Under some assumptions on the sparsity of the problem and on the regularity of the risk functions they present so called oracle inequalities for these estimators. “An oracle inequality relates the performance of a real estimator with that of an ideal estimator which relies on perfect information supplied by an oracle, and which is not available in practice.” (from [E. J. Candès, Acta Numerica 15, 257–325 (2006; Zbl 1141.62001)]). Moreover the asymptotic behaviour of the estimators is investigated and simulation studies are added.

Reviewer: Wolfgang Näther (Freiberg)

##### MSC:

62H12 | Estimation in multivariate analysis |

62J05 | Linear regression; mixed models |

62F30 | Parametric inference under constraints |

62F35 | Robustness and adaptive procedures (parametric inference) |

##### Keywords:

matrix completion; robustness; empirical risk minimization; oracle inequality; nuclear norm; sparsity##### Software:

CVX##### References:

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