Bicriteria Sparse Nonnegative Matrix Factorization via Two-Timescale Duplex Neurodynamic Optimization
Hangjun Che, Jun Wang, Andrzej Cichocki
Abstract
In this article, sparse nonnegative matrix factorization (SNMF) is formulated as a mixed-integer bicriteria optimization problem for minimizing matrix factorization errors and maximizing factorized matrix sparsity based on an exact binary representation of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{0}$ </tex-math></inline-formula> matrix norm. The binary constraints of the problem are then equivalently replaced with bilinear constraints to convert the problem to a biconvex problem. The reformulated biconvex problem is finally solved by using a two-timescale duplex neurodynamic approach consisting of two recurrent neural networks (RNNs) operating collaboratively at two timescales. A Gaussian score (GS) is defined as to integrate the bicriteria of factorization errors and sparsity of resulting matrices. The performance of the proposed neurodynamic approach is substantiated in terms of low factorization errors, high sparsity, and high GS on four benchmark datasets.