TY - JOUR

T1 - A novel update rule of HALS algorithm for nonnegative matrix factorization and Zangwill’s global convergence

AU - Sano, Takehiro

AU - Migita, Tsuyoshi

AU - Takahashi, Norikazu

N1 - Funding Information:
This work was supported by JSPS KAKENHI Grant Number JP21H03510.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022

Y1 - 2022

N2 - Nonnegative Matrix Factorization (NMF) has attracted a great deal of attention as an effective technique for dimensionality reduction of large-scale nonnegative data. Given a nonnegative matrix, NMF aims to obtain two low-rank nonnegative factor matrices by solving a constrained optimization problem. The Hierarchical Alternating Least Squares (HALS) algorithm is a well-known and widely-used iterative method for solving such optimization problems. However, the original update rule used in the HALS algorithm is not well defined. In this paper, we propose a novel well-defined update rule of the HALS algorithm, and prove its global convergence in the sense of Zangwill. Unlike conventional globally-convergent update rules, the proposed one allows variables to take the value of zero and hence can obtain sparse factor matrices. We also present two stopping conditions that guarantee the finite termination of the HALS algorithm. The practical usefulness of the proposed update rule is shown through experiments using real-world datasets.

AB - Nonnegative Matrix Factorization (NMF) has attracted a great deal of attention as an effective technique for dimensionality reduction of large-scale nonnegative data. Given a nonnegative matrix, NMF aims to obtain two low-rank nonnegative factor matrices by solving a constrained optimization problem. The Hierarchical Alternating Least Squares (HALS) algorithm is a well-known and widely-used iterative method for solving such optimization problems. However, the original update rule used in the HALS algorithm is not well defined. In this paper, we propose a novel well-defined update rule of the HALS algorithm, and prove its global convergence in the sense of Zangwill. Unlike conventional globally-convergent update rules, the proposed one allows variables to take the value of zero and hence can obtain sparse factor matrices. We also present two stopping conditions that guarantee the finite termination of the HALS algorithm. The practical usefulness of the proposed update rule is shown through experiments using real-world datasets.

KW - Global convergence

KW - Hierarchical alternating least squares algorithm

KW - Nonnegative matrix factorization

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U2 - 10.1007/s10898-022-01167-7

DO - 10.1007/s10898-022-01167-7

M3 - Article

AN - SCOPUS:85129129345

JO - Journal of Global Optimization

JF - Journal of Global Optimization

SN - 0925-5001

ER -