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 |
|---|---|
| Pages (from-to) | 221-250 |
| Number of pages | 30 |
| Journal | Computational Optimization and Applications |
| Volume | 71 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - Sep 1 2018 |
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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 journal › Article
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TY - JOUR
T1 - A unified global convergence analysis of multiplicative update rules for nonnegative matrix factorization
AU - Takahashi, Norikazu
AU - Katayama, Jiro
AU - Seki, Masato
AU - Takeuchi, Junnichi
PY - 2018/9/1
Y1 - 2018/9/1
N2 - 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.
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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UR - http://www.scopus.com/inward/citedby.url?scp=85044089143&partnerID=8YFLogxK
U2 - 10.1007/s10589-018-9997-y
DO - 10.1007/s10589-018-9997-y
M3 - Article
AN - SCOPUS:85044089143
VL - 71
SP - 221
EP - 250
JO - Computational Optimization and Applications
JF - Computational Optimization and Applications
SN - 0926-6003
IS - 1
ER -