# 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.

Contact: cyril.furtlehner à inria.fr