Title: Non-negative Matrix Factorization via Archetypal Analysis,
Abstract: Given a collection of data points, non-negative matrix factorization (NMF)
suggests to express them as convex combinations of a small set of `archetypes'
with non-negative entries. This decomposition is unique only if the true
archetypes are non-negative and sufficiently sparse (or the weights are
sufficiently sparse), a regime that is captured by the separability condition
and its generalizations.
In this paper, we study an approach to NMF that can be traced back to the
work of Cutler and Breiman (1994) and does not require the data to be
separable, while providing a generally unique decomposition. We optimize the
trade-off between two objectives: we minimize the distance of the data points
from the convex envelope of the archetypes (which can be interpreted as an
empirical risk), while minimizing the distance of the archetypes from the
convex envelope of the data (which can be interpreted as a data-dependent
regularization). The archetypal analysis method of (Cutler, Breiman, 1994) is
recovered as the limiting case in which the last term is given infinite weight.
We introduce a `uniqueness condition' on the data which is necessary for
exactly recovering the archetypes from noiseless data. We prove that, under
uniqueness (plus additional regularity conditions on the geometry of the
archetypes), our estimator is robust. While our approach requires solving a
non-convex optimization problem, we find that standard optimization methods
succeed in finding good solutions both for real and synthetic data.