Global convergence of modified multiplicative updates for nonnegative matrix factorization

Norikazu Takahashi, Ryota Hibi

Research output: Contribution to journalArticlepeer-review

27 Citations (Scopus)


Nonnegative matrix factorization (NMF) is the problem of approximating a given nonnegative matrix by the product of two nonnegative matrices. The multiplicative updates proposed by Lee and Seung are widely used as efficient computational methods for NMF. However, the global convergence of these updates is not formally guaranteed because they are not defined for all pairs of nonnegative matrices. In this paper, we consider slightly modified versions of the original multiplicative updates and study their global convergence properties. The only difference between the modified updates and the original ones is that the former do not allow variables to take values less than a user-specified positive constant. Using Zangwill's global convergence theorem, we prove that any sequence of solutions generated by either of those modified updates has at least one convergent subsequence and the limit of any convergent subsequence is a stationary point of the corresponding optimization problem. Furthermore, we propose algorithms based on the modified updates that always stop within a finite number of iterations.

Original languageEnglish
Pages (from-to)417-440
Number of pages24
JournalComputational Optimization and Applications
Issue number2
Publication statusPublished - Mar 2014


  • Finite termination
  • Global convergence
  • Multiplicative update
  • Nonnegative matrix factorization

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics


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