  Nonnegative matrix factorization (NMF) has an established reputation as a
useful data analysis technique in numerous applications. However, its usage in
practical situations is undergoing challenges in recent years. The fundamental
factor to this is the increasingly growing size of the datasets available and
needed in the information sciences. To address this, in this work we propose to
use structured random compression, that is, random projections that exploit the
data structure, for two NMF variants: classical and separable. In separable NMF
(SNMF) the left factors are a subset of the columns of the input matrix. We
present suitable formulations for each problem, dealing with different
representative algorithms within each one. We show that the resulting
compressed techniques are faster than their uncompressed variants, vastly
reduce memory demands, and do not encompass any significant deterioration in
performance. The proposed structured random projections for SNMF allow to deal
with arbitrarily shaped large matrices, beyond the standard limit of
tall-and-skinny matrices, granting access to very efficient computations in
this general setting. We accompany the algorithmic presentation with
theoretical foundations and numerous and diverse examples, showing the
suitability of the proposed approaches.
