Gauss-Seidel HALS Algorithm for Nonnegative Matrix Factorization with Sparseness and Smoothness Constraints

Takumi KIMURA; Norikazu TAKAHASHI

doi:10.1587/transfun.E100.A.2925

Gauss-Seidel HALS Algorithm for Nonnegative Matrix Factorization with Sparseness and Smoothness Constraints

Takumi KIMURA, Norikazu TAKAHASHI

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Nonnegative Matrix Factorization (NMF) with sparseness and smoothness constraints has attracted increasing attention. When these properties are considered, NMF is usually formulated as an optimization problem in which a linear combination of an approximation error term and some regularization terms must be minimized under the constraint that the factor matrices are nonnegative. In this paper, we focus our attention on the error measure based on the Euclidean distance and propose a new iterative method for solving those optimization problems. The proposed method is based on the Hierarchical Alternating Least Squares (HALS) algorithm developed by Cichocki et al. We first present an example to show that the original HALS algorithm can increase the objective value. We then propose a new algorithm called the Gauss-Seidel HALS algorithm that decreases the objective value monotonically. We also prove that it has the global convergence property in the sense of Zangwill. We finally verify the effectiveness of the proposed algorithm through numerical experiments using synthetic and real data.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E100-A No.12 pp.2925-2935

Publication Date: 2017/12/01

Publicized

Online ISSN: 1745-1337

DOI: 10.1587/transfun.E100.A.2925

Type of Manuscript: PAPER

Category: Digital Signal Processing

Authors

Takumi KIMURA
Okayama University
Norikazu TAKAHASHI
Okayama University

Keyword

nonnegative matrix factorization, hierarchical alternating least squares algorithm, Euclidean distance, global convergence, sparseness, smoothness

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Takumi KIMURA, Norikazu TAKAHASHI, "Gauss-Seidel HALS Algorithm for Nonnegative Matrix Factorization with Sparseness and Smoothness Constraints" in IEICE TRANSACTIONS on Fundamentals, vol. E100-A, no. 12, pp. 2925-2935, December 2017, doi: 10.1587/transfun.E100.A.2925.
Abstract: Nonnegative Matrix Factorization (NMF) with sparseness and smoothness constraints has attracted increasing attention. When these properties are considered, NMF is usually formulated as an optimization problem in which a linear combination of an approximation error term and some regularization terms must be minimized under the constraint that the factor matrices are nonnegative. In this paper, we focus our attention on the error measure based on the Euclidean distance and propose a new iterative method for solving those optimization problems. The proposed method is based on the Hierarchical Alternating Least Squares (HALS) algorithm developed by Cichocki et al. We first present an example to show that the original HALS algorithm can increase the objective value. We then propose a new algorithm called the Gauss-Seidel HALS algorithm that decreases the objective value monotonically. We also prove that it has the global convergence property in the sense of Zangwill. We finally verify the effectiveness of the proposed algorithm through numerical experiments using synthetic and real data.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1587/transfun.E100.A.2925/_p

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@ARTICLE{e100-a_12_2925,
author={Takumi KIMURA, Norikazu TAKAHASHI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Gauss-Seidel HALS Algorithm for Nonnegative Matrix Factorization with Sparseness and Smoothness Constraints},
year={2017},
volume={E100-A},
number={12},
pages={2925-2935},
abstract={Nonnegative Matrix Factorization (NMF) with sparseness and smoothness constraints has attracted increasing attention. When these properties are considered, NMF is usually formulated as an optimization problem in which a linear combination of an approximation error term and some regularization terms must be minimized under the constraint that the factor matrices are nonnegative. In this paper, we focus our attention on the error measure based on the Euclidean distance and propose a new iterative method for solving those optimization problems. The proposed method is based on the Hierarchical Alternating Least Squares (HALS) algorithm developed by Cichocki et al. We first present an example to show that the original HALS algorithm can increase the objective value. We then propose a new algorithm called the Gauss-Seidel HALS algorithm that decreases the objective value monotonically. We also prove that it has the global convergence property in the sense of Zangwill. We finally verify the effectiveness of the proposed algorithm through numerical experiments using synthetic and real data.},
keywords={},
doi={10.1587/transfun.E100.A.2925},
ISSN={1745-1337},
month={December},}

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TY - JOUR
TI - Gauss-Seidel HALS Algorithm for Nonnegative Matrix Factorization with Sparseness and Smoothness Constraints
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2925
EP - 2935
AU - Takumi KIMURA
AU - Norikazu TAKAHASHI
PY - 2017
DO - 10.1587/transfun.E100.A.2925
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E100-A
IS - 12
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - December 2017
AB - Nonnegative Matrix Factorization (NMF) with sparseness and smoothness constraints has attracted increasing attention. When these properties are considered, NMF is usually formulated as an optimization problem in which a linear combination of an approximation error term and some regularization terms must be minimized under the constraint that the factor matrices are nonnegative. In this paper, we focus our attention on the error measure based on the Euclidean distance and propose a new iterative method for solving those optimization problems. The proposed method is based on the Hierarchical Alternating Least Squares (HALS) algorithm developed by Cichocki et al. We first present an example to show that the original HALS algorithm can increase the objective value. We then propose a new algorithm called the Gauss-Seidel HALS algorithm that decreases the objective value monotonically. We also prove that it has the global convergence property in the sense of Zangwill. We finally verify the effectiveness of the proposed algorithm through numerical experiments using synthetic and real data.
ER -