Robust Matrix Completion

This paper considers the problem of recovery of a low-rank matrix in the situation when most of its entries are not observed and a fraction of observed entries are corrupted. The observations are noisy realizations of the sum of a low rank matrix, which we wish to recover, with a second matrix having a complementary sparse structure such as element-wise or column-wise sparsity. We analyze a class of estimators obtained by solving a constrained convex optimization problem that combines the nuclear norm and a convex relaxation for a sparse constraint. Our results are obtained for the simultaneous presence of random and deterministic patterns in the sampling scheme. We provide guarantees for recovery of low-rank and sparse components from partial and corrupted observations in the presence of noise and show that the obtained rates of convergence are minimax optimal.

Probabilistic low-rank matrix completion on finite alphabets

The task of reconstructing a matrix given a sample of observedentries is known as the matrix completion problem. It arises ina wide range of problems, including recommender systems, collaborativefiltering, dimensionality reduction, image processing, quantum physics or multi-class classificationto name a few. Most works have focused on recovering an unknown real-valued low-rankmatrix from randomly sub-sampling its entries.Here, we investigate the case where the observations take a finite number of values, corresponding for examples to ratings in recommender systems or labels in multi-class classification.We also consider a general sampling scheme (not necessarily uniform) over the matrix entries.The performance of a nuclear-norm penalized estimator is analyzed theoretically.More precisely, we derive bounds for the Kullback-Leibler divergence between the true and estimated distributions.In practice, we have also proposed an efficient algorithm based on lifted coordinate gradient descent in order to tacklepotentially high dimensional settings.