Sparse NMF with Archetypal Regularization: Computational and Robustness Properties
This work addresses robustness in sparse NMF for data analysis, offering incremental improvements in theoretical guarantees and algorithms.
The paper tackles sparse nonnegative matrix factorization with archetypal regularization to represent data points as nonnegative linear combinations of sparse factors, introducing strong and weak robustness guarantees under minimal assumptions and providing theoretical results and algorithms validated on synthetic and real data.
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.