Nuclear norm penalization and optimal rates for noisy low rank matrix completion
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This paper deals with the trace regression model where n entries or linear combinations of entries of an unknown m_1× m_2 matrix A_0 corrupted by noise are observed. We propose a new nuclear norm penalized estimator of A_0 and establish a general sharp oracle inequality for this estimator for arbitrary values of n,m_1,m_2 under the condition of isometry in expectation. Then this method is applied to the matrix completion problem. In this case, the estimator admits a simple explicit form and we prove that it satisfies oracle inequalities with faster rates of convergence than in the previous works. They are valid, in particular, in the high-dimensional setting m_1m_2≫ n. We show that the obtained rates are optimal up to logarithmic factors in a minimax sense and also derive, for any fixed matrix A_0, a non-minimax lower bound on the rate of convergence of our estimator, which coincides with the upper bound up to a constant factor. Finally, we show that our procedure provides an exact recovery of the rank of A_0 with probability close to 1. We also discuss the statistical learning setting where there is no underlying model determined by A_0 and the aim is to find the best trace regression model approximating the data.
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