Global convergence of a modified HALS algorithm for nonnegative matrix factorization

Takumi Kimura; Norikazu Takahashi

doi:10.1109/CAMSAP.2015.7383726

Global convergence of a modified HALS algorithm for nonnegative matrix factorization

Takumi Kimura, Norikazu Takahashi

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

9 Citations (Scopus)

Abstract

Hierarchical alternating least squares (HALS) algorithms are efficient computational methods for nonnegative matrix factorization (NMF). Given an initial solution, HALS algorithms update the solution block by block iteratively so that the error decreases monotonically. However, update rules in HALS algorithms are not well-defined. In addition, due to this problem, the convergence of the sequence of solutions to a stationary point cannot be proved theoretically. In this paper, we consider the HALS algorithm for the Frobenius norm-based NMF, and prove that a modified version has the global convergence property in the sense of Zangwill.

Original language	English
Title of host publication	2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	21-24
Number of pages	4
ISBN (Electronic)	9781479919635
DOIs	https://doi.org/10.1109/CAMSAP.2015.7383726
Publication status	Published - 2015
Event	6th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015 - Cancun, Mexico Duration: Dec 13 2015 → Dec 16 2015

Publication series

Name	2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015

Other

Other	6th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015
Country/Territory	Mexico
City	Cancun
Period	12/13/15 → 12/16/15

ASJC Scopus subject areas

Signal Processing
Computational Mathematics

Access to Document

10.1109/CAMSAP.2015.7383726

Cite this

Kimura, T., & Takahashi, N. (2015). Global convergence of a modified HALS algorithm for nonnegative matrix factorization. In 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015 (pp. 21-24). [7383726] (2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CAMSAP.2015.7383726

Global convergence of a modified HALS algorithm for nonnegative matrix factorization. / Kimura, Takumi; Takahashi, Norikazu.
2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015. Institute of Electrical and Electronics Engineers Inc., 2015. p. 21-24 7383726 (2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Kimura, T & Takahashi, N 2015, Global convergence of a modified HALS algorithm for nonnegative matrix factorization. in 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015., 7383726, 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015, Institute of Electrical and Electronics Engineers Inc., pp. 21-24, 6th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015, Cancun, Mexico, 12/13/15. https://doi.org/10.1109/CAMSAP.2015.7383726

Kimura T, Takahashi N. Global convergence of a modified HALS algorithm for nonnegative matrix factorization. In 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015. Institute of Electrical and Electronics Engineers Inc. 2015. p. 21-24. 7383726. (2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015). doi: 10.1109/CAMSAP.2015.7383726

Kimura, Takumi ; Takahashi, Norikazu. / Global convergence of a modified HALS algorithm for nonnegative matrix factorization. 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015. Institute of Electrical and Electronics Engineers Inc., 2015. pp. 21-24 (2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, CAMSAP 2015).

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abstract = "Hierarchical alternating least squares (HALS) algorithms are efficient computational methods for nonnegative matrix factorization (NMF). Given an initial solution, HALS algorithms update the solution block by block iteratively so that the error decreases monotonically. However, update rules in HALS algorithms are not well-defined. In addition, due to this problem, the convergence of the sequence of solutions to a stationary point cannot be proved theoretically. In this paper, we consider the HALS algorithm for the Frobenius norm-based NMF, and prove that a modified version has the global convergence property in the sense of Zangwill.",

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Global convergence of a modified HALS algorithm for nonnegative matrix factorization

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