# A unified global convergence analysis of multiplicative update rules for nonnegative matrix factorization

Norikazu Takahashi, Jiro Katayama, Masato Seki, Junnichi Takeuchi

Research output: Contribution to journalArticle

1 Citation (Scopus)

### Abstract

Multiplicative update rules are a well-known computational method for nonnegative matrix factorization. Depending on the error measure between two matrices, various types of multiplicative update rules have been proposed so far. However, their convergence properties are not fully understood. This paper provides a sufficient condition for a general multiplicative update rule to have the global convergence property in the sense that any sequence of solutions has at least one convergent subsequence and the limit of any convergent subsequence is a stationary point of the optimization problem. Using this condition, it is proved that many of the existing multiplicative update rules have the global convergence property if they are modified slightly so that all variables take positive values. This paper also proposes new multiplicative update rules based on Kullback–Leibler, Gamma, and Rényi divergences. It is shown that these three rules have the global convergence property if the same modification as above is made.

Original language English 221-250 30 Computational Optimization and Applications 71 1 https://doi.org/10.1007/s10589-018-9997-y Published - Sep 1 2018

### Fingerprint

Non-negative Matrix Factorization
Global Analysis
Factorization
Convergence Analysis
Global Convergence
Multiplicative
Update
Convergence Properties
Computational methods
Subsequence
Stationary point
Computational Methods
Divergence
Optimization Problem
Sufficient Conditions

### All Science Journal Classification (ASJC) codes

• Control and Optimization
• Computational Mathematics
• Applied Mathematics

### Cite this

A unified global convergence analysis of multiplicative update rules for nonnegative matrix factorization. / Takahashi, Norikazu; Katayama, Jiro; Seki, Masato; Takeuchi, Junnichi.

In: Computational Optimization and Applications, Vol. 71, No. 1, 01.09.2018, p. 221-250.

Research output: Contribution to journalArticle

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AB - Multiplicative update rules are a well-known computational method for nonnegative matrix factorization. Depending on the error measure between two matrices, various types of multiplicative update rules have been proposed so far. However, their convergence properties are not fully understood. This paper provides a sufficient condition for a general multiplicative update rule to have the global convergence property in the sense that any sequence of solutions has at least one convergent subsequence and the limit of any convergent subsequence is a stationary point of the optimization problem. Using this condition, it is proved that many of the existing multiplicative update rules have the global convergence property if they are modified slightly so that all variables take positive values. This paper also proposes new multiplicative update rules based on Kullback–Leibler, Gamma, and Rényi divergences. It is shown that these three rules have the global convergence property if the same modification as above is made.

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