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Nonnegative matrix factorization (NMF), which is the approximation of a datamatrix as the product of two nonnegative matrices, is a key issue in machinelearning and data analysis. One approach to NMF is to formulate the problem asa nonconvex optimization problem of minimizing the distance between a datamatrix and the product of two nonnegative matrices with nonnegativityconstraints and then solve the problem using an iterative algorithm. Thealgorithms commonly used are the multiplicative update algorithm and thealternating least-squares algorithm. Although both algorithms converge quickly,they may not converge to a stationary point to the problem that is equal to thesolution to a nonmonotone variational inequality for the gradient of thedistance function. This paper presents an iterative algorithm for solving theproblem that is based on the Krasnosel'ski\u\i-Mann fixed point algorithm.Convergence analysis showed that, under certain assumptions, any accumulationpoint of the sequence generated by the proposed algorithm belongs to thesolution set of the variational inequality. Application of the {\tt 'mult'} and{\tt'als'} algorithms in MATLAB and the proposed algorithm to various NMFproblems showed that the proposed algorithm had fast convergence and waseffective.

Fixed Point Algorithm for Solving Nonmonotone Variational Inequalities in Nonnegative Matrix Factorization