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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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
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ISSN:1598-5865
1865-2085
DOI:10.1007/s12190-013-0646-4