Sparse Nonnegative Matrix Factorization Based on a Hyperbolic Tangent Approximation of L₀-Norm and Neurodynamic Optimization

Sparse nonnegative matrix factorization (SNMF) attracts much attention in the past two decades because its sparse and part-based representations are desirable in many machine learning applications. Due to the combinatorial nature of the sparsity constraint in form of l₀-norm, the problem is hard to solve. In this paper, a hyperbolic tangent function is introduced to approximate the l₀-norm. A discrete-time neurodynamic approach is developed for solving the proposed formulation. The stability and the convergence behavior are shown for the state vectors. Experiment results are discussed to demonstrate the superiority of the approach. The results show that this approach outperforms other sparse NMF approaches with the smallest relative reconstruction error and the required level of sparsity.