Probabilistic low-rank matrix completion on finite alphabets

The task of reconstructing a matrix given a sample of observed entries is known as the \emph{matrix completion problem}. Such a consideration arises in a wide variety of problems, including recommender systems, collaborative filtering, dimensionality reduction, image processing, quantum physics or multi-class classification to name a few. Most works have focused on recovering an unknown real-valued low-rank matrix from randomly sub-sampling its entries. Here, we investigate the case where the observations take a finite numbers of values, corresponding for examples to ratings in recommender systems or labels in multi-class classification. We also consider a general sampling scheme (non-necessarily uniform) over the matrix entries.