Adaptive Multinomial Matrix Completion
This work addresses matrix completion for quantized data in applications like recommender systems and multi-class classification, offering an adaptive estimator with proven optimality, though it is incremental as it builds on existing nuclear norm methods.
The paper tackles matrix completion with highly quantized observations, such as ratings or labels, by proposing a constrained nuclear norm penalized maximum likelihood estimator that is adaptive and does not require prior knowledge of rank or nuclear norm bounds. It provides theoretical guarantees of minimax optimality and supports claims with Monte-Carlo experiments on simulated and real data.
The task of estimating a matrix given a sample of observed entries is known as the \emph{matrix completion problem}. Most works on matrix completion have focused on recovering an unknown real-valued low-rank matrix from a random sample of its entries. Here, we investigate the case of highly quantized observations when the measurements can take only a small number of values. These quantized outputs are generated according to a probability distribution parametrized by the unknown matrix of interest. This model corresponds, for example, to ratings in recommender systems or labels in multi-class classification. We consider a general, non-uniform, sampling scheme and give theoretical guarantees on the performance of a constrained, nuclear norm penalized maximum likelihood estimator. One important advantage of this estimator is that it does not require knowledge of the rank or an upper bound on the nuclear norm of the unknown matrix and, thus, it is adaptive. We provide lower bounds showing that our estimator is minimax optimal. An efficient algorithm based on lifted coordinate gradient descent is proposed to compute the estimator. A limited Monte-Carlo experiment, using both simulated and real data is provided to support our claims.