Compressive Bayesian non-negative matrix factorization for mutational signatures analysis

Non-negative matrix factorization (NMF) is widely used in many applications for dimensionality reduction. Inferring an appropriate number of factors for NMF is a challenging problem, and several approaches based on information criteria or sparsity-inducing priors have been proposed. However, inference in these models is often complicated and computationally challenging. In this paper, we introduce a novel methodology for overfitted Bayesian NMF models using ``compressive hyperpriors'' that force unneeded factors down to negligible values while only imposing mild shrinkage on needed factors. The method is based on using simple semi-conjugate priors to facilitate inference, while setting the strength of the hyperprior in a data-dependent way to achieve this compressive property. We apply our method to mutational signatures analysis in cancer genomics, where we find that it outperforms state-of-the-art alternatives. In particular, we illustrate how our compressive hyperprior enables the use of biologically informed priors on the signatures, yielding significantly improved accuracy. We provide theoretical results characterizing the posterior and its concentration, and we demonstrate the method in simulations and on real data from cancer applications.