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

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Abstract

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.

Original languageEnglish
JournalJournal of Global Optimization
DOIs
Publication statusAccepted/In press - 2022

Keywords

  • Global convergence
  • Hierarchical alternating least squares algorithm
  • Nonnegative matrix factorization

ASJC Scopus subject areas

  • Computer Science Applications
  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics

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