October 20th

14:30 , R2014 Digiteo Shannon (660) (see location):

Jean Lafond

Title : Low Rank Matrix Completion with Exponential Family Noise

Abstract :

The matrix completion problem consists in reconstructing a matrix from a
sample of entries, possibly 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 and is sub-exponential. Our framework alllows for 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 matrix completion for
exponential family. When the sampling distribution is known, we propose
another estimator and prove an oracle inequality w.r.t. the Kullback-Leibler
prediction risk, which translates immediatly into an upper bound on the
Frobenius prediction risk. Finally, we show that all the rates obtained are
minimax optimal up to a logarithmic factor.

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