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

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