  We consider the problem of sparse nonnegative matrix factorization (NMF)
using 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
present theoretical results and illustrative examples to strengthen the
insights underlying the notions of robustness. We propose new algorithms for
our optimization problem; and present numerical experiments on synthetic and
real data sets that shed further insights into our proposed framework and
theoretical developments.
