Low Rank Matrix Completion with Exponential Family Noise
published: Aug. 20, 2015, recorded: July 2015, views: 1885
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The matrix completion problem consists in reconstructing a matrix from a sample of entries observed with noise. A popular class of estimator, known as nuclear norm penalized estimators, are based on minimizing the sum of a data fitting term and a nuclear norm penalization. Here, we investigate the case where the noise distribution belongs to the exponential family, is sub-exponential and consider a general sampling scheme. We first consider an estimator defined as the minimizer of the sum of a log-likelihood term and a nuclear norm penalization and prove an upper bound on the Frobenius prediction risk. The rate obtained improves on previous works on exponential family completion. When the sampling distribution is known, we propose a second estimator and prove an oracle inequality on the Kullback-Leibler divergence risk, which translates immediatly into an upper bound on the Frobenius risk. Finally, we show that all the rates obtained are minimax optimal up to a logarithmic factor.
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