Recurrent neural network for approximate nonnegative matrix factorization

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A recurrent neural network solving the approximate nonnegative matrix factorization (NMF) problem is presented in this paper. The proposed network is based on the Lagrangian approach, and exploits a partial dual method in order to limit the number of dual variables. Sparsity constraints on basis or activation matrices are included by adding a weighted sum of constraint functions to the least squares reconstruction error. However, the corresponding Lagrange multipliers are computed by the network dynamics itself, avoiding empirical tuning or a validation process. It is proved that local solutions of the NMF optimization problem correspond to as many stable steady-state points of the network dynamics. The validity of the proposed approach is verified through several simulation examples concerning both synthetic and real-world datasets for feature extraction and clustering applications.