The calculation of a low-rank approximation of a matrix is a fundamental operation in many computer vision applications. The workhorse of this class of problems has long been the Singular Value Decomposition. However, in the presence of missing data and outliers this method is not applicable, and unfortunately, this is often the case in practice. In this paper we present a method for calculating the low-rank factorization of a matrix which minimizes the L1 norm in the presence of missing data. Our approach represents a generalization the Wiberg algorithm of one of the more convincing methods for factorization under the L2 norm. By utilizing the differentiability of linear programs, we can extend the underlying ideas behind this approach to include this class of L1 problems as well. We show that the proposed algorithm can be efficiently implemented using existing optimization software. We also provide preliminary experiments on synthetic as well as real world data with very convincing results.