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Efficient method for symmetric nonnegative matrix factorization with an approximate augmented Lagrangian scheme

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 454卷

作者： Zhu, Hong Niu, Chenchen Liang, Yongjin Jiangsu Univ Sch Math Sci 301 Xuefu Rd Zhenjiang 212013 Jiangsu Peoples R China Wuxi Yanqiao High Sch Jiangsu 214171 Peoples R China

In this paper, we propose an efficient method for solving symmetric nonnegative matrix factorization following an approximate augmented Lagrangian scheme. The augmented Lagrangian subproblem was solved column by column under the block coordinate descent (BCD) framework. In particular, we extend the recursive formula for rank-k nonnegative least squares problems by Chu et al. (2021) to subproblems generated by BCD framework in our method. Thereafter we derive the closed-form solution for k = 4. Experiments for clustering show that the proposed method is noticeably efficient and achieves competitive performance compared with state-of-the-art methods.

关键词： symmetric nonnegative matrix factorization Augmented Lagrangian method BCD framework ARkNLS Clustering

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RANDOMIZED ALGORITHMS FOR symmetric nonnegative matrix factorization

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SIAM JOURNAL ON matrix ANALYSIS AND APPLICATIONS 2025年第1期46卷 584-625页

作者： Hayashi, Koby Aksoy, Sinan g. Ballard, Grey Park, Haesun Georgia Inst Technol Sch Computat Sci & Engn Atlanta GA 30332 USA Pacific Northwest Natl Lab Seattle WA 98109 USA Wake Forest Univ Dept Comp Sci Winston Salem NC 27109 USA

symmetric nonnegative matrix factorization (SymNMF) is a technique in data analysis and machine learning that approximates a symmetric matrix with a product of a nonnegative, low-rank matrix and its transpose. To design faster and more scalable algorithms for SymNMF, we develop two randomized algorithms for its computation. The first algorithm uses randomized matrix sketching to compute an initial low-rank approximation to the input matrix and proceeds to rapidly compute a SymNMF of the approximation. The second algorithm uses randomized leverage score sampling to approximately solve constrained least squares problems. Many successful methods for SymNMF rely on (approximately) solving sequences of constrained least squares problems. We prove theoretically that leverage score sampling can approximately solve nonnegative least squares problems to a chosen accuracy with high probability. Additionally, we prove sampling complexity results for previously proposed hybrid sampling techniques which deterministically include high leverage score rows. This hybrid scheme is crucial for obtaining speedups in practice. Finally, we demonstrate that both methods work well in practice by applying them to graph clustering tasks on large real world data sets. These experiments show that our methods approximately maintain solution quality and achieve significant speedups for both large dense and large sparse problems.

关键词： nonnegative matrix factorization symmetric nonnegative matrix factorization leverage scores matrix sketching randomized numerical linear algebra (RandNLA)

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One-hot constrained symmetric nonnegative matrix factorization for image clustering

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PATTERN RECOGNITION 2025年 162卷

作者： Li, Jie Li, Chaoqian Kunming Univ Sch Math 2 Puxin Rd Kunming 650214 Yunnan Peoples R China Yunnan Univ Sch Math & Stat 2 Cuihu North Rd Kunming 650091 Yunnan Peoples R China

Semi-supervised symmetric Non-Negative matrix factorization (SNMF) has proven to bean effective clustering method. However, most existing semi-supervised SNMF approaches rely on sophisticated techniques to incorporate supervised information, which results in increased hyper-parameter tuning and model complexity. To achieve better performance with lower complexity, we propose a novel semi-supervised SNMF method called One-hot Constrained SNMF (OCSNMF). This method introduces a parameter-free embedding strategy for partial label information, representing the clustering assignments of labeled data points as one-hot vectors in the SNMF decomposition. We present an iterative algorithm to solve the optimization problem of the proposed OCSNMF, along with analyses of convergence and complexity. Experimental results on six image datasets demonstrate the superiority of OCSNMF compared to several state-of-the-art methods. The code can be obtained from: https://***/ljisxz/OCSNMF.

关键词： Semi-supervised symmetric nonnegative matrix factorization Clustering One-hot encoding

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symmetric nonnegative matrix factorization Based on Box-Constrained Half-Quadratic Optimization

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IEEE ACCESS 2020年 8卷 170976-170990页

作者： Chen, Bo-Wei Natl Sun Yat Sen Univ Dept Elect Engn Kaohsiung 80424 Taiwan Pervas Artificial Intelligence Res PAIR Labs Hsinchu 30010 Taiwan

nonnegative matrix factorization (NMF) based on half-quadratic (HQ) functions was proven effective and robust when dealing with data contaminated by continuous occlusion according to the half-quadratic optimization theory. Nonetheless, state-of-the-art HQ NMF still cannot handle symmetric data matrices, and this caused problems when applications require processing symmetric matrices, e.g., similarity matrices. In view of such, this study presents HQ symmetric NMF along with HQ optimization algorithms that can solve symmetric matrix decomposition and avoid oscillations during alternating updates. Moreover, variants of HQ symmetric NMF are proposed and examined in this study - The direct method and the asymmetry penalty constrained method. Both show their distinct advantages. Experiments on open datasets were conducted for comparison with other methods. Experimental results showed that the proposed HQ symmetric NMF was more robust against continuous occlusion than the baselines. Both root-mean-squared errors and coefficients of determination were better than those of the baselines, thereby demonstrating effectiveness of the proposed method.

关键词： symmetric matrices Optimization Covariance matrices matrix decomposition Oscillators Robustness Mathematical model Half-quadratic optimization symmetric nonnegative matrix factorization similarity matrix

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symmetric nonnegative matrix factorization: Algorithms and Applications to Probabilistic Clustering

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IEEE TRANSACTIONS ON NEURAL NETWORKS 2011年第12期22卷 2117-2131页

作者： He, Zhaoshui Xie, Shengli Zdunek, Rafal Zhou, Guoxu Cichocki, Andrzej Guangdong Univ Technol Fac Automat Guangzhou 510641 Guangdong Peoples R China RIKEN Brain Sci Inst Lab Adv Brain Signal Proc Wako Saitama 3510198 Japan Wroclaw Univ Technol Inst Telecommun Teleinformat & Acoust PL-50370 Wroclaw Poland Polish Acad Sci Syst Res Inst PL-00901 Warsaw Poland

nonnegative matrix factorization (NMF) is an unsupervised learning method useful in various applications including image processing and semantic analysis of documents. This paper focuses on symmetric NMF (SNMF), which is a special case of NMF decomposition. Three parallel multiplicative update algorithms using level 3 basic linear algebra subprograms directly are developed for this problem. First, by minimizing the Euclidean distance, a multiplicative update algorithm is proposed, and its convergence under mild conditions is proved. Based on it, we further propose another two fast parallel methods: alpha-SNMF and beta-SNMF algorithms. All of them are easy to implement. These algorithms are applied to probabilistic clustering. We demonstrate their effectiveness for facial image clustering, document categorization, and pattern clustering in gene expression.

关键词： Basic linear algebra subprograms completely positive coordinate update multiplicative update nonnegative matrix factorization parallel update probabilistic clustering symmetric nonnegative matrix factorization

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symmetric nonnegative matrix factorization with elastic-net regularized block-wise weighted representation for clustering

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PATTERN ANALYSIS AND APPLICATIONS 2022年第4期25卷 807-817页

作者： Rodriguez-Dominguez, Ulises Dalmau, Oscar Math Res Ctr Guanajuato Mexico

In unsupervised learning, symmetric nonnegative matrix factorization (NMF) has proven its efficacy for various clustering tasks in recent years, considering both linearly and nonlinearly separable data. On the other hand, block-wise weighted sparse representation-based classification (BW-SRC), a recently proposed sparse representation technique improved sparse coding features for supervised classification. In this work, we take advantage of both techniques to device a new unsupervised image clustering algorithm. A disadvantage of symmetric NMF is its computational burden associated to the quadratic growth of its similarity matrix. We reduce this computation by first working with sub-sampled data and then by working with the full data samples in a second stage elastic-net coefficient estimation problem with previously learned block-wise weights. This decreases both the computational time and the memory requirements when solving for symmetric NMF, but at the same time allows to cluster the full data samples in a robust way. We either outperform or achieve highly competitive results with previous matrix factorization clustering methods on seven benchmark image datasets.

关键词： symmetric nonnegative matrix factorization Elastic-net Regularization Weighted features Clustering

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symmetric nonnegative matrix factorization: A systematic review

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NEUROCOMPUTING 2023年第1期557卷

作者： Chen, Wen-Sheng Xie, Kexin Liu, Rui Pan, Binbin Shenzhen Univ Coll Math & Stat Shenzhen 518060 Peoples R China Guangdong Key Lab Intelligent Informat Proc Shenzhen 518060 Peoples R China

In recent years, symmetric non-negative matrix factorization (SNMF), a variant of non-negative matrix factorization (NMF), has emerged as a promising tool for data analysis. This paper mainly focuses on the theoretical idea, the basic model, the optimization method, and the variants of SNMF. The SNMF-related approaches can be generally classified into two main categories, Classic SNMFs and Extended SNMFs. The classic SNMFs contain Orthogonal SNMF, Sparse SNMF, Manifold structure based SNMF and Pairwise constraint based SNMF, besides, extended SNMFs include Self-supervised SNMF, MV-WSNMF and Multi-view SNMF. According to different classes of SNMFs, this review elaborates on the key concepts, characteristics, and current issues of these algorithms. The clustering performance of SNMF and its variants on three object image datasets is empirically compared. In addition, the effects of some algorithms for solving SNMF have been compared and the performance of similarity matrix construction methods is also compared. Finally, some open problems with SNMF are discussed.

关键词： symmetric nonnegative matrix factorization Feature extraction Supervised learning

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Inexact Block Coordinate Descent Methods for symmetric nonnegative matrix factorization

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2017年第22期65卷 5995-6008页

作者： Shi, Qingjiang Sun, Haoran Lu, Songtao Hong, Mingyi Razaviyayn, Meisam Iowa State Univ Dept Ind & Mfg Syst Engn Ames IA 50011 USA Nanjing Univ Aeronaut & Astronaut Coll Elect Informat Engn Nanjing 210016 Jiangsu Peoples R China Univ Southern Calif Daniel J Epstein Dept Ind & Syst Engn Los Angeles CA 90089 USA

symmetric nonnegativematrix factorization (SNMF) is equivalent to computing a symmetric nonnegative low rank approximation of a data similarity matrix. It inherits the good data interpretability of the well-known nonnegative matrix factorization technique and has better ability of clustering nonlinearly separable data. In this paper, we focus on the algorithmic aspect of the SNMF problem and propose simple inexact block coordinate decentmethods to address the problem, leading to both serial and parallel algorithms. The proposed algorithms have guaranteed convergence to stationary solutions and can efficiently handle large-scale and/or sparse SNMF problems. Extensive simulations verify the effectiveness of the proposed algorithms compared to recent state-of-the-art algorithms.

关键词： symmetric nonnegative matrix factorization block coordinate decent block successive upper-bounding minimization parallel algorithm stationary solution

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An augmented Lagrangian alternating direction method for overlapping community detection based on symmetric nonnegative matrix factorization

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INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS 2020年第2期11卷 403-415页

作者： Hu, Liying Guo, Gongde Fujian Normal Univ Sch Math & Informat Fuzhou Peoples R China Fujian Normal Univ Digital Fujian Internet Of Things Lab Environm Mo Fuzhou Peoples R China

In this paper, we present an augmented Lagrangian alternating direction algorithm for symmetric nonnegative matrix factorization. The convergence of the algorithm is also proved in detail and strictly. Then we present a modified overlapping community detection method which is based on the presented symmetric nonnegative matrix factorization algorithm. We apply the modified community detection method to several real world networks. The obtained results show the capability of our method in detecting overlapping communities, hubs and outliers. We find that our experimental results have better quality than several competing methods for identifying communities.

关键词： Augmented Lagrangian function Alternating direction method symmetric nonnegative matrix factorization Overlapping community detection

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MHSNMF: multi-view hessian regularization based symmetric nonnegative matrix factorization for microbiome data analysis

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BMC BIOINFORMATICS 2020年第Sup6期21卷 234-234页

作者： Ma, Yuanyuan Zhao, Junmin Ma, Yingjun Anyang Normal Univ Sch Comp & Informat Engn Anyang Peoples R China Henan Univ Urban Construct Sch Comp & Data Sci Pingdingshan Peoples R China Cent China Normal Sch Comp Wuhan Peoples R China

BackgroundWith the rapid development of high-throughput technique, multiple heterogeneous omics data have been accumulated vastly (e.g., genomics, proteomics and metabolomics data). Integrating information from multiple sources or views is challenging to obtain a profound insight into the complicated relations among micro-organisms, nutrients and host environment. In this paper we propose a multi-view Hessian regularization based symmetric nonnegative matrix factorization algorithm (MHSNMF) for clustering heterogeneous microbiome data. Compared with many existing approaches, the advantages of MHSNMF lie in: (1) MHSNMF combines multiple Hessian regularization to leverage the high-order information from the same cohort of instances with multiple representations;(2) MHSNMF utilities the advantages of SNMF and naturally handles the complex relationship among microbiome samples;(3) uses the consensus matrix obtained by MHSNMF, we also design a novel approach to predict the classification of new microbiome *** conduct extensive experiments on two real-word datasets (Three-source dataset and Human Microbiome Plan dataset), the experimental results show that the proposed MHSNMF algorithm outperforms other baseline and state-of-the-art methods. Compared with other methods, MHSNMF achieves the best performance (accuracy: 95.28%, normalized mutual information: 91.79%) on microbiome data. It suggests the potential application of MHSNMF in microbiome data *** show that the proposed MHSNMF algorithm can effectively combine the phylogenetic, transporter, and metabolic profiles into a unified paradigm to analyze the relationships among different microbiome samples. Furthermore, the proposed prediction method based on MHSNMF has been shown to be effective in judging the types of new microbiome samples.

关键词： symmetric nonnegative matrix factorization Hessian regularization Multi-view clustering Human microbiome

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