Weighted low-rank approximations

We study the common problem of approximating a target matrix with a matrix of lower rank. We provide a simple and efficient (EM) algorithm for solving weighted low-rank approximation problems, which, unlike their unweighted version, do not admit a closed-form solution in general. We analyze, in addition, the nature of locally optimal solutions that arise in this context, demonstrate the utility of accommodating the weights in reconstructing the underlying low-rank representation, and extend the formulation to non-Gaussian noise models such as logistic models. Finally, we apply the methods developed to a collaborative filtering task.