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ROBUST LOW-RANK MATRIX ESTIMATION
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DOI：10.1214/17-AOS1666.png)
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摘要
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Many results have been proved for various nuclear norm penalized estimators of the uniform sampling matrix completion problem. However, most of these estimators are not robust: in most of the cases the quadratic loss function and its modifications are used. We consider robust nuclear norm penalized estimators using two well-known robust loss functions: the absolute value loss and the Huber loss. Under several conditions on the sparsity of the problem (i.e., the rank of the parameter matrix) and on the regularity of the risk function sharp and nonsharp oracle inequalities for these estimators are shown to hold with high probability. As a consequence, the asymptotic behavior of the estimators is derived Similar error bounds are obtained under the assumption of weak sparsity, that is, the case where the matrix is assumed to be only approximately low-rank. In all of our results, we consider a high dimensional setting. In this case, this means that we assume n <= pq. Finally, various simulations confirm our theoretical results.
Keyword:
Matrix completion
robustness
empirical risk minimization
oracle inequality
nuclear norm
sparsity
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3.7
论文数:
2.8K
被引数:
2.9W
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引用论文
ESTIMATION OF (NEAR) LOW-RANK MATRICES WITH NOISE AND HIGH-DIMENSIONAL SCALING具有噪声和高维缩放的 (近) 低秩矩阵的估计.png)
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