DC-NMF: nonnegative matrix factorization based on divide-and-conquer for fast clustering and topic modeling

The importance of unsupervised clustering and topic modeling is well recognized with ever-increasing volumes of text data available from numerous sources. Nonnegative matrix factorization (NMF) has proven to be a successful method for cluster and topic discovery in unlabeled data sets. In this paper, we propose a fast algorithm for computing NMF using a divide-and-conquer strategy, called DC-NMF. Given an input matrix where the columns represent data items, we build a binary tree structure of the data items using a recently-proposed efficient algorithm for computing rank-2 NMF, and then gather information from the tree to initialize the rank-k NMF, which needs only a few iterations to reach a desired solution. We also investigate various criteria for selecting the node to split when growing the tree. We demonstrate the scalability of our algorithm for computing general rank-k NMF as well as its effectiveness in clustering and topic modeling for large-scale text data sets, by comparing it to other frequently utilized state-of-the-art algorithms. The value of the proposed approach lies in the highly efficient and accurate method for initializing rank-k NMF and the scalability achieved from the divide-and-conquer approach of the algorithm and properties of rank-2 NMF. In summary, we present efficient tools for analyzing large-scale data sets, and techniques that can be generalized to many other data analytics problem domains along with an open-source software library called SmallK.