Annals of Mathematical Sciences and Applications

Volume 7 (2022)

Number 2

Index search method for solving nonnegative matrix factorization

Pages: 281 – 300



Yi-Shin Cheng (Department of Applied Mathematics, National University of Kaohsiung, Taiwan)

Ching-Sung Liu (Department of Applied Mathematics, National University of Kaohsiung, Taiwan)


Nonnegative matrix factorization (NMF) has been widely used for dimensionality reduction in recent years, while playing an important role in many fields such as image processing and data analysis. NMF is a classic non-convex optimization problem, and the alternating nonnegative least squares (ANLS) framework is a popular method for solving the problem. In general, ANLS divides the NMF problem into two convex optimization problems, called nonnegative least squares (NNLS) problems. In this paper, we first introduce an active set method for NNLS, called indexed search method (ISM). Meanwhile, our goal is to propose a robust algorithm that combines ANLS and ISM for solving NMF. Finally, numerical experiments are provided to support the theoretical results.


nonnegative matrix factorization, index search method, KKT conditions

2010 Mathematics Subject Classification

65F15, 65F60

C.-S. Liu was supported in part by the National Science and Technology Council in Taiwan.

Received 23 August 2022

Accepted 25 August 2022

Published 12 September 2022