The rise of nonnegative matrix factorization: Algorithms and applications

Although nonnegative matrix factorization (NMF) is widely used, some matrix factorization methods result in misleading results and waste of computing resources due to lack of timely optimization and case-by-case consideration. Therefore, an up-to-date and comprehensive review on its algorithms and applications is needed to promote improvement and applications for NMF. Here, we start with introducing background and gathering the principles and formulae of NMF algorithms. There have been dozens of new algorithms since its birth in the 1990s. Generally, several or even more algorithms are adopted in a single software package written in R, Python, C/C++, etc. Besides, the applications of NMF are analyzed. NMF is not only most widely used in modern subjects or techniques such as computer science, telecommunications, imaging science, and remote sensing but also increasingly used in traditional subjects such as physics, chemistry, biology, medicine, and psychology, being accepted by around 130 fields (disciplines) in about 20 years. Finally, the features and performance of different categories of NMF are summarized and evaluated. The summarized advantages and disadvantages and proposed suggestions for improvements are expected to enlighten the future efforts to polish the mathematical principles and procedures of NMF to realize higher accuracy and productivity in practical use.