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Nonnegative matrix factorization (NMF) is a prominent technique for datadimensionality reduction that has been widely used for text mining, computervision, pattern discovery, and bioinformatics. In this paper, a frameworkcalled ARkNLS (Alternating Rank-k Nonnegativity constrained Least Squares) isproposed for computing NMF. First, a recursive formula for the solution of therank-k nonnegativity-constrained least squares (NLS) is established. Thisrecursive formula can be used to derive the closed-form solution for the Rank-kNLS problem for any integer k $\ge$ 1. As a result, each subproblem for analternating rank-k nonnegative least squares framework can be obtained based onthis closed form solution. Assuming that all matrices involved in rank-k NLS inthe context of NMF computation are of full rank, two of the currently best NMFalgorithms HALS (hierarchical alternating least squares) and ANLS-BPP(Alternating NLS based on Block Principal Pivoting) can be considered asspecial cases of ARkNLS with k = 1 and k = r for rank r NMF, respectively. Thispaper is then focused on the framework with k = 3, which leads to a newalgorithm for NMF via the closed-form solution of the rank-3 NLS problem.Furthermore, a new strategy that efficiently overcomes the potentialsingularity problem in rank-3 NLS within the context of NMF computation is alsopresented. Extensive numerical comparisons using real and synthetic data setsdemonstrate that the proposed algorithm provides state-of-the-art performancein terms of computational accuracy and cpu time.

An Alternating Rank-K Nonnegative Least Squares Framework (ARkNLS) for Nonnegative Matrix Factorization