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Randomized algorithms for Orthogonal Nonnegative Matrix Factorization

Journal of the Operations Research Society of China 2023年第2期11卷 327-345页

作者： Yong-Yong Chen Fang-Fang Xu College of Mathematics and Systems Science Shandong University of Science and TechnologyQingdao 266590ShandongChina

Orthogonal nonnegative matrix factorization(ONMF)is widely used in blind image separation problem,document classification,and human face *** model of ONMF can be efficiently solved by the alternating direction method of multipliers and hierarchical alternating least squares *** the given matrix is huge,the cost of computation and communication is too ***,ONMF becomes challenging in the large-scale *** random projection is an efficient method of dimensionality *** this paper,we apply the random projection to ONMF and propose two randomized *** experiments show that our proposed algorithms perform well on both simulated and real data.

关键词： Orthogonal nonnegative matrix factorization Random projection method Dimensionality reduction Augmented lagrangian method hierarchical alternating least squares algorithm

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

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JOURNAL OF GLOBAL OPTIMIZATION 2022年第3期84卷 755-781页

作者： Sano, Takehiro Migita, Tsuyoshi Takahashi, Norikazu Okayama Univ Grad Sch Nat Sci & Technol Kita Ku 3-1-1 Tsushima Naka Okayama 7008530 Japan

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.

关键词： Nonnegative matrix factorization hierarchical alternating least squares algorithm Global convergence

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Distributed HALS algorithm for NMF based on Simple Average Consensus algorithm 8

Distributed HALS Algorithm for NMF based on Simple Average C...

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IEEE International Conference on Progress in Informatics and Computing (IEEE PIC)

作者： Hayashi, Keiju Migita, Tsuyoshi Takahashi, Norikazu Okayama Univ Okayama Japan

ISBN: (纸本)9781665426558

Nonnegative Matrix Factorization (NMF) is an efficient dimensionality reduction method for nonnegative data. Recently, a distributed algorithm has been proposed for multiple agents in a network to execute the hierarchical alternating least squares algorithm, which is well known as a fast computation method for NMF. However, the average consensus algorithm used there requires each agent to store the entire history of the values of its variables until the complete average consensus is reached, which increases the memory usage and computational cost. In this paper, we propose to replace the complicated average consensus algorithm with a simple one, and show through simulations that this replacement does not degrade the quality of the result if the values of the hyper-parameters are properly chosen.

关键词： nonnegative matrix factorization hierarchical alternating least squares algorithm distributed algorithm average consensus multiagent system

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An algorithm for Randomized Nonnegative Matrix Factorization and Its Global Convergence

An Algorithm for Randomized Nonnegative Matrix Factorization...

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IEEE Symposium Series on Computational Intelligence (IEEE SSCI)

作者： Masuda, Takao Migita, Tsuyoshi Takahashi, Norikazu Okayama Univ Okayama Japan

ISBN: (纸本)9781728190488

Nonnegative Matrix Factorization (NMF) is to decompose a given nonnegative matrix into two nonnegative factor matrices. Recently, randomized NMF has been proposed as an approach to fast NMF of large nonnegative matrices. The main idea of this approach is to perform NMF after reducing the dimensionality of the given nonnegative matrix by multiplying it by a random matrix. Since randomized NMF is formulated as a constrained optimization problem which is slightly different from the one for original NMF, it is necessary to develop suitable algorithms for solving it. However, the conventional algorithm has a serious drawback that the constraints are not satisfied. In addition, the convergence of the algorithm has not been analyzed. In this paper, in order to overcome these drawbacks, we propose to modify the optimization problem and design an algorithm based on the hierarchical alternating least squares method to solve the modified optimization problem. We also prove the global convergence of the designed algorithm.

关键词： nonnegative matrix factorization hierarchical alternating least squares algorithm randomized NMF global convergence

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Distributed geometric nonnegative matrix factorization and hierarchical alternating least squares-based nonnegative tensor factorization with the MapReduce paradigm

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CONCURRENCY AND COMPUTATION-PRACTICE & EXPERIENCE 2018年第17期30卷

作者： Zdunek, Rafal Fonal, Krzysztof Wroclaw Univ Sci & Technol Dept Elect Wybrzeze Wyspianskiego 27 PL-50370 Wroclaw Poland

Nonnegative matrix factorization and its multilinear extension known as nonnegative tensor factorization are commonly used methods in machine learning and data analysis for feature extraction and dimensionality reduction for nonnegative high-dimensional data. Dimensionality reduction for massive amounts of data usually involves distributed computation across multi-node computer architectures. In this study, we propose various computational strategies for parallel and distributed computation of the latent factors in both factorization models, all of which are based on partitioning the computational tasks according to the MapReduce paradigm. We extend the previously reported distributed hierarchical alternating least squares algorithm to the multi-way array factorization model, where we assume that the observed multi-way data can be partitioned into chunks along one mode. Moreover, we propose a new geometry-based distributed computational strategy for solving nonnegative matrix factorization problems. Numerical experiments performed using various large-scale data sets demonstrated that these algorithms are efficient and robust to noisy data.

关键词： distributed nonnegative matrix factorization distributed nonnegative tensor factorization geometric nonnegative matrix factorization hierarchical alternating least squares algorithm MapReduce paradigm

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Randomized Nonnegative Tensor Factorization for Feature Extraction from High-dimensional Signals 25

Randomized Nonnegative Tensor Factorization for Feature Extr...

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25th International Conference on Systems, Signals and Image Processing (IWSSIP)

作者： Zdunek, Rafal Fonal, Krzysztof Wroclaw Univ Sci & Technol Fac Elect Wybrzeze Wyspianskiego 27 PL-50370 Wroclaw Poland

ISBN: (纸本)9781538669792

Tensor decomposition methods are well-known tools for multilinear feature extraction from multi-way arrays with many important applications in signal processing and machine learning. Nonnegative Tensor Factorization (NTF) is a particular case of such methods, mostly addressed for processing nonnegative multi-way arrays, such as hyperspectral observations or a set of images. One of the most efficient algorithms for NTF is the hierarchical alternating least squares (HALS) algorithm that belongs to a family of coordinate gradient descent updates. Despite its very good numerical properties, its computational complexity is quite large for large-scale datasets. In this study, we propose the randomized extension of the HALS, which considerably decreases its computational complexity with respect to the standard HALS. The numerical experiments, performed for various large-scale observations, confirm that the proposed algorithm is much faster than the standard one at the cost of slightly decreased performance.

关键词： Nonnegative Tensor Factorization hierarchical alternating least squares algorithm Randomized Approximation algorithm Multilinear Feature Extraction Dimensionality Reduction

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