Convergence analysis of a non-negative matrix factorization algorithm based on Gibbs random field modeling

• DOI: 10.1007/s12190-013-0646-4
• Publication Date: 2013-02-05T09:36:38Z
• Authors: Chenxue Yang; Mao Ye; Zijian Liu
• Keywords: 0301 basic medicine; 03 medical and health sciences

Non-negative matrix factorization (NMF) is a new approach to deal with the multivariate nonnegative data. Although the classic multiplicative update algorithm can solve the NMF problems, it fails to find sparse and localized object parts. Then a Gibbs random field (GRF) modeling based NMF algorithm, called the GRF-NMF algorithm, try to directly model the prior object structure of the components into the NMF problem. In this paper, the convergence of the GRF-NMF algorithm and its advantages are investigated. Based on a classic model, the equilibrium points are obtained. Some invariant sets are constructed to prepare for the analysis of the convergence of the GRF-NMF algorithm. Then using stability theory of the equilibrium point, the convergence of the algorithm is proved and the convergence conditions of the algorithm are obtained. We theoretically present the advantages of the GRF-NMF algorithm in the end.