  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 establishing
the compressive property, and we demonstrate the method in simulations and on
real data from a breast cancer application.
