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.
翻译:从2010年代开始,NMF的可识别性研究取得了很大进展:信号处理和机器学习社区已经发现了许多令人感兴趣的重要成果。NMF的可识别性对实践的许多方面产生了很大影响,例如,对模型的可识别性的理解,也许有点令人惊讶,对于模型的可识别性的理解,在诸如专题采矿和超光谱成像等许多应用中,其解释性直到近些年一直相当有限。从2010年代开始,NMF的可识别性研究取得了很大进展:信号处理和机器学习社区已经发现了许多有趣的重要成果。NMF的可识别性对实践的许多方面产生了很大影响,例如对模型的可识别性设计避免和有性能保障的算法设计等。另一方面,没有一份指导性文件将NMF从可识别性观点引入NMF。我们的目标是填补这一空白,对NMF的可识别性模型的可识别性以及与算法和应用的联系进行全面深入的辅导。这一辅导将有助于研究人员和研究生了解NMF的本质和洞察力,从而也避免了典型的“可塑性设计”工具的成熟期。