Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications

TL;DR

This paper provides a comprehensive tutorial on the identifiability of nonnegative matrix factorization (NMF), explaining its importance for interpretability, algorithms, and applications in signal and data analytics, especially in recent research developments.

Contribution

It fills a gap by systematically reviewing NMF identifiability, connecting theoretical insights with practical algorithms and applications, aiding researchers and practitioners.

Findings

Recent progress in NMF identifiability research since 2010s

Connections between identifiability and algorithm performance

Guidelines for choosing suitable NMF models for applications

Abstract

Nonnegative matrix factorization (NMF) has become a workhorse for signal and data analytics, triggered by its model parsimony and interpretability. Perhaps a bit surprisingly, the understanding to its model identifiability---the major reason behind the interpretability in many applications such as topic mining and hyperspectral imaging---had been rather limited until recent years. Beginning from the 2010s, the identifiability research of NMF has progressed considerably: Many interesting and important results have been discovered by the signal processing (SP) and machine learning (ML) communities. NMF identifiability has a great impact on many aspects in practice, such as ill-posed formulation avoidance and performance-guaranteed algorithm design. On the other hand, there is no tutorial paper that introduces NMF from an identifiability viewpoint. In this paper, we aim at filling this gap by offering a comprehensive and deep tutorial on model identifiability of NMF as well as the connections to algorithms and applications. This tutorial will help researchers and graduate students grasp the essence and insights of NMF, thereby avoiding typical `pitfalls' that are often times due to unidentifiable NMF formulations. This paper will also help practitioners pick/design suitable factorization tools for their own problems.