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# Statistical Optimization of Non-Negative Matrix Factorization

*Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics*, PMLR 15:128-136, 2011.

#### Abstract

Non-Negative Matrix Factorization (NMF) is a dimensionality reduction method that has been shown to be very useful for a variety of tasks in machine learning and data mining. One of the fastest algorithms for NMF is the Block Principal Pivoting method (BPP) of (Kim & Park, 2008b), which follows a block coordinate descent approach. The optimization in each iteration involves solving a large number of expensive least squares problems. Taking the view that the design matrix was generated by a stochastic process, and using the asymptotic normality of the least squares estimator, we propose a method for improving the performance of BPP. Our method starts with a small subset of the columns and rows of the original matrix and uses frequentist hypothesis tests to adaptively increase the size of the problem. This achieves two objectives: 1) during the initial phase of the algorithm we solve far fewer, much smaller sized least squares problems and 2) all hypothesis tests failing while using all the data represents a principled, automatic stopping criterion. Our experiments on three real world datasets show that our algorithm significantly improves the performance of the original BPP algorithm.