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GaussSeidel HALS Algorithm for Nonnegative Matrix Factorization with Sparseness and Smoothness Constraints
Takumi KIMURA Norikazu TAKAHASHI
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E100A
No.12
pp.29252935 Publication Date: 2017/12/01
Online ISSN: 17451337 Type of Manuscript: PAPER Category: Digital Signal Processing Keyword: nonnegative matrix factorization, hierarchical alternating least squares algorithm, Euclidean distance, global convergence, sparseness, smoothness,
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Summary:
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 GaussSeidel 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.

