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

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,...

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Published in:	Journal of applied mathematics & computing Vol. 42; no. 1-2; pp. 491 - 508
Main Authors:	Yang, Chenxue, Ye, Mao, Liu, Zijian
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer-Verlag 01.07.2013 Springer Nature B.V
Subjects:	Algorithms Analysis Approximation Computation Computational Mathematics and Numerical Analysis Computer science Construction Convergence Decomposition Equilibrium Factorization Invariants Mathematical and Computational Engineering Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Matrix Neighborhoods Original Research Stability Studies Theory of Computation 68Q25 68W40 Gibbs random field (GRF) Stability Non-negative matrix factorizations Existence Convergence Equilibrium point
ISSN:	1598-5865, 1865-2085
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Abstract	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.
AbstractList	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. 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.[PUBLICATION ABSTRACT]
Author	Ye, Mao Liu, Zijian Yang, Chenxue
Author_xml	– sequence: 1 givenname: Chenxue surname: Yang fullname: Yang, Chenxue organization: School of Computer Science and Engineering, University of Electronic Science and Technology of China – sequence: 2 givenname: Mao surname: Ye fullname: Ye, Mao organization: School of Computer Science and Engineering, University of Electronic Science and Technology of China – sequence: 3 givenname: Zijian surname: Liu fullname: Liu, Zijian email: hbliuzijian@126.com organization: School of Science, Chongqing Jiaotong University
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Cites_doi	10.1007/s11063-009-9126-0 10.1002/env.3170050203 10.1142/S0129065707001275 10.1038/44565 10.1101/gr.903503 10.1073/pnas.0308531101 10.1016/j.laa.2005.06.025 10.1093/bioinformatics/bti653 10.1007/978-1-4612-9892-2 10.1016/j.ipm.2004.11.005 10.1109/TNN.2007.895831 10.1016/j.csda.2006.11.006 10.1109/9.471210 10.1109/ICCVW.2009.5457714 10.21437/Eurospeech.2003-343 10.1007/11785231_58
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Snippet	Non-negative matrix factorization (NMF) is a new approach to deal with the multivariate nonnegative data. Although the classic multiplicative update algorithm...
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SubjectTerms	Algorithms Analysis Approximation Computation Computational Mathematics and Numerical Analysis Computer science Construction Convergence Decomposition Equilibrium Factorization Invariants Mathematical and Computational Engineering Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Matrix Neighborhoods Original Research Stability Studies Theory of Computation
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Title	Convergence analysis of a non-negative matrix factorization algorithm based on Gibbs random field modeling
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