Efficient computation of robust weighted low-rank matrix approximations using the L₁ norm

Abstract

The calculation of a low-rank approximation to a matrix is fundamental to many algorithms in computer vision and other fields. One of the primary tools used for calculating such low-rank approximations is the Singular Value Decomposition, but this method is not applicable in the case where there are outliers or missing elements in the data. Unfortunately, this is often the case in practice. We present a method for low-rank matrix approximation which is a generalization of the Wiberg algorithm. Our method calculates the rank-constrained factorization, which minimizes the L1 norm and does so in the presence of missing data. This is achieved by exploiting the differentiability of linear programs, and results in an algorithm can be efficiently implemented using existing optimization software. We show the results of experiments on synthetic and real data.