Nonnegative Matrix Factorizations for Clustering: A Survey

Recently there has been significant development in the use of non-negative matrix factorization (NMF) methods for various clustering tasks. NMF factorizes an input nonnegative matrix into two nonnegative matrices of lower rank. Although NMF can be used for conventional data analysis, the recent overwhelming interest in NMF is due to the newly discovered ability of NMF to solve challenging data mining and machine learning problems. In particular, NMF with the sum of squared error cost function is equivalent to a relaxed K-means clustering, the most widely used unsupervised learning algorithm. In addition, NMF with the I-divergence cost function is equivalent to probabilistic latent semantic indexing, another unsupervised learning method popularly used in text analysis. Many other data mining and machine learning problems can be reformulated as an NMF problem. This chapter aims to provide a comprehensive review of non-negative matrix factorization methods for clustering. In particular, we outline the theoretical foundations on NMF for clustering, provide an overview of different variants on NMF formulations, and examine several practical issues in NMF algorithms. We also summarize recent advances on using NMF-based methods for solving many other clustering problems including coclustering, semi-supervised clustering, and consensus clustering and discuss some future research directions.