  We consider the problem of sparse nonnegative matrix factorization (NMF) with
archetypal regularization. The goal is to represent a collection of data points
as nonnegative linear combinations of a few nonnegative sparse factors with
appealing geometric properties, arising from the use of archetypal
regularization. We generalize the notion of robustness studied in Javadi and
Montanari (2019) (without sparsity) to the notions of (a) strong robustness
that implies each estimated archetype is close to the underlying archetypes and
(b) weak robustness that implies there exists at least one recovered archetype
that is close to the underlying archetypes. Our theoretical results on
robustness guarantees hold under minimal assumptions on the underlying data,
and applies to settings where the underlying archetypes need not be sparse. We
propose new algorithms for our optimization problem; and present numerical
experiments on synthetic and real datasets that shed further insights into our
proposed framework and theoretical developments.
