Sparse NMF with Archetypal Regularization: Computational and Robustness Properties

TL;DR

This paper introduces a robust sparse NMF framework with archetypal regularization, providing theoretical guarantees and new algorithms, demonstrated through experiments on synthetic and real datasets.

Contribution

It generalizes robustness notions for sparse NMF with archetypal regularization, offering minimal-assumption guarantees and novel algorithms.

Findings

Robustness guarantees hold under minimal data assumptions.

Algorithms effectively recover archetypes in synthetic and real data.

The framework enhances interpretability and stability of NMF solutions.

Abstract

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.