Convergence of a Fast Hierarchical Alternating Least Squares Algorithm for Nonnegative Matrix Factorization

The hierarchical alternating least squares (HALS) algorithms are powerful tools for nonnegative matrix factorization (NMF), among which the Fast-HALS, proposed in [A. Cichocki and A.-H. Phan, 2009], is one of the most efficient. This paper investigates the convergence of Fast-HALS. First, a more gen...

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Published in:IEEE transactions on knowledge and data engineering Vol. 36; no. 1; pp. 77 - 89
Main Authors: Hou, Liangshao, Chu, Delin, Liao, Li-Zhi
Format: Journal Article
Language:English
Published: New York IEEE 01.01.2024
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithm design and analysis
Algorithms
Approximation algorithms
Columns (structural)
Convergence
Factorization
Kurdyka-Łojasiewicz property
Least squares
Least squares approximations
nonnegative matrix factorization
the fast hierarchical alternating least squares algorithm
ISSN:1041-4347, 1558-2191
Online Access:Get full text
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Summary:The hierarchical alternating least squares (HALS) algorithms are powerful tools for nonnegative matrix factorization (NMF), among which the Fast-HALS, proposed in [A. Cichocki and A.-H. Phan, 2009], is one of the most efficient. This paper investigates the convergence of Fast-HALS. First, a more general weak convergence (converged subsequences exist and converge to the stationary point set) is established without any assumption, while most existing results assume all the columns of iterates are strictly away from the origin. Then, a simplified strong convergence (the entire sequence converges to a stationary point) proof is provided. The existing strong convergence is attributed to the block prox-linear (BPL) method, which is a more general framework including Fast-HALS as a special case. So, the convergence proof under BPL is quite complex. Our simplified proof explores the structure of Fast-HALS and can be regarded as a complement to the results under BPL. In addition, some numerical verifications are presented.
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ISSN:1041-4347
1558-2191
DOI:10.1109/TKDE.2023.3279369