  We investigate a general matrix factorization for deviance-based data losses,
extending the ubiquitous singular value decomposition beyond squared error
loss. While similar approaches have been explored before, our method leverages
classical statistical methodology from generalized linear models (GLMs) and
provides an efficient algorithm that is flexible enough to allow for structural
zeros via entry weights. Moreover, by adapting results from GLM theory, we
provide support for these decompositions by (i) showing strong consistency
under the GLM setup, (ii) checking the adequacy of a chosen exponential family
via a generalized Hosmer-Lemeshow test, and (iii) determining the rank of the
decomposition via a maximum eigenvalue gap method. To further support our
findings, we conduct simulation studies to assess robustness to decomposition
assumptions and extensive case studies using benchmark datasets from image face
recognition, natural language processing, network analysis, and biomedical
studies. Our theoretical and empirical results indicate that the proposed
decomposition is more flexible, general, and robust, and can thus provide
improved performance when compared to similar methods. To facilitate
applications, an R package with efficient model fitting and family and rank
determination is also provided.
