Generalized Matrix Factorization

Unmeasured or latent variables are often the cause of correlations between multivariate measurements and are studied in a variety of fields such as psychology, ecology, and medicine. For Gaussian measurements, there are classical tools such as factor analysis or principal component analysis with a well-established theory and fast algorithms. Generalized Linear Latent Variable models (GLLVM) generalize such factor models to non-Gaussian responses. However, current algorithms for estimating model parameters in GLLVMs require intensive computation and do not scale to large datasets with thousands of observational units or responses. In this article, we propose a new approach for fitting GLLVMs to such high-volume, high-dimensional datasets. We approximate the likelihood using penalized quasi-likelihood and use a Newton method and Fisher scoring to learn the model parameters. Our method greatly reduces the computation time and can be easily parallelized, enabling factorization at unprecedented scale using commodity hardware. We illustrate application of our method on a dataset of 48,000 observational units with over 2,000 observed species in each unit, finding that most of the variability can be explained with a handful of factors.