An efficient method for non-negative low-rank completion

Abstract

In this article, we propose a new method for low-rank completion of a large sparse matrix, subject to non-negativity constraint. As a challenging prototype of this problem, we have in mind the well-known Netflix problem. Our method is based on the derivation of a constrained gradient system and its numerical integration. The methods we propose are based on the constrained minimization of a functional associated to the low-rank completion matrix having minimal distance to the given matrix. In the main 2-level method, the low-rank matrix is expressed in the form of the non-negative factorization X = eUVT, where the factors U and V are assumed to be normalized with unit Frobenius norm. In the inner level—for a given e—we minimize the functional; in the outer level, we tune the parameter e until we reach a solution. Numerical experiments on well-known large test matrices show the effectiveness of the method when compared with other algorithms available in the literature.