Parallelized preconditioned model building algorithm for matrix factorization

Matrix factorization is a common task underlying several machine learning applications such as recommender systems, topic modeling, or compressed sensing. Given a large and possibly sparse matrix A, we seek two smaller matrices W and H such that their product is as close to A as possible. The objective is minimizing the sum of square errors in the approximation. Typically such problems involve hundreds of thousands of unknowns, so an optimizer must be exceptionally efficient. In this study, a new algorithm, Preconditioned Model Building is adapted to factorize matrices composed of movie ratings in the MovieLens data sets with 1, 10, and 20 million entries. We present experiments that compare the sequential MATLAB implementation of the PMB algorithm with other algorithms in the minFunc package. We also employ a lock-free sparse matrix factorization algorithm and provide a scalable shared-memory parallel implementation. We show that (a) the optimization performance of the PMB algorithm is comparable to the best algorithms in common use, and (b) the computational performance can be significantly increased with parallelization