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

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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: 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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Multiview Clustering via Hypergraph Induced Semi-Supervised symmetric nonnegative matrix factorization

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY 2023年第10期33卷 5510-5524页

作者： Peng, Siyuan Yin, Jingxing Yang, Zhijing Chen, Badong Lin, Zhiping Guangdong Univ Technol Sch Informat Engn Guangzhou 510006 Peoples R China Guangdong Prov Key Lab Intellectual Property & Big Guangzhou 510006 Peoples R China Xi An Jiao Tong Univ Inst Artificial Intelligence & Robot Xian 710049 Peoples R China Nanyang Technol Univ Sch Elect & Elect Engn Singapore 639798 Singapore

nonnegative matrix factorization (NMF) based multiview technique has been commonly used in multiview data clustering tasks. However, previous NMF based multiview clustering approaches fail to take advantage of a small amount of supervisory information to effectively improve the clustering performance, and are easily affected by the additional post-processing method in clustering tasks. To cope with these issues, a novel framework named multiview clustering via hypergraph induced semi-supervised symmetric NMF (MVCHSS) is proposed in this paper for multiview data clustering applications. Specifically, the proposed method has the following features: 1) a new multiview based hypergraph pairwise constraints propagation (MHPCP) algorithm is developed in MVCHSS to construct a set of informative similarity matrices, revealing the high-order relationships effectively and fully utilizing the limited pairwise constraint supervisory information among samples of each view data;2) the obtained similarity matrices with much supervisory information are not only enforced into the symmetric NMF (SNMF) model, but also incorporated into the graph regularization for each view data;3) the optimization problem of MVCHSS is formulated for multiview data clustering tasks to acquire a more discriminative clustering indicator matrix (or called consensus assignment matrix) without additional post-processing method. Moreover, the proof of convergence and the computational complexity for MVCHSS are presented. Extensive experiments on five multiview datasets demonstrate that the proposed MVCHSS framework outperforms several state-of-the-art multiview clustering methods.

关键词： Semi-supervised learning symmetric nonnegative matrix factorization multiview clustering pairwise constraints propagation

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A Progressive Hierarchical Alternating Least Squares Method for symmetric nonnegative matrix factorization

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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2023年第5期45卷 5355-5369页

作者： Hou, Liangshao Chu, Delin Liao, Li-Zhi Hong Kong Baptist Univ Dept Math Hong Kong Peoples R China Natl Univ Singapore Dept Math Singapore 119076 Singapore

In this article, we study the symmetric nonnegative matrix factorization (SNMF) which is a powerful tool in data mining for dimension reduction and clustering. The main contributions of the present work include: (i) a new descent direction for the rank-one SNMF is derived and a strategy for choosing the step size along this descent direction is established;(ii) a progressive hierarchical alternating least squares (PHALS) method for SNMF is developed, which is parameter-free and updates the variables column by column. Moreover, every column is updated by solving a rank-one SNMF subproblem;and (iii) the convergence to the Karush-Kuhn-Tucker (KKT) point set (or the stationary point set) is proved for PHALS. Several synthetical and real data sets are tested to demonstrate the effectiveness and efficiency of the proposed method. Our PHALS provides better performance in terms of the computational accuracy, the optimality gap, and the CPU time, compared with a number of state-of-the-art SNMF methods.

关键词： symmetric matrices Convergence matrix decomposition Dimensionality reduction Data mining Minimization Systematics symmetric nonnegative matrix factorization progressive hierarchical alternating least squares Karush-Kuhn-Tucker points clustering

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A Provable Splitting Approach for symmetric nonnegative matrix factorization

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IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 2023年第3期35卷 2206-2219页

作者： Li, Xiao Zhu, Zhihui Li, Qiuwei Liu, Kai Chinese Univ Hong Kong Sch Data Sci Shenzhen 518172 Guangdong Peoples R China Shenzhen Inst Artificial Intelligence & Robot Soc Shenzhen Guangdong Peoples R China Univ Denver Dept Elect & Comp Engn Denver CO 80208 USA Alibaba Grp Damo Acad Decis Intelligence Lab Bellevue WA 98004 USA Clemson Univ Comp Sci Div Clemson SC 29634 USA

The symmetric nonnegative matrix factorization (NMF), a special but important class of the general NMF, has found numerous applications in data analysis such as various clustering tasks. Unfortunately, designing fast algorithms for the symmetric NMF is not as easy as for its nonsymmetric counterpart, since the latter admits the splitting property that allows state-of-the-art alternating-type algorithms. To overcome this issue, we first split the decision variable and transform the symmetric NMF to a penalized nonsymmetric one, paving the way for designing efficient alternating-type algorithms. We then show that solving the penalized nonsymmetric reformulation returns a solution to the original symmetric NMF. Moreover, we design a family of alternating-type algorithms and show that they all admit strong convergence guarantee: the generated sequence of iterates is convergent and converges at least sublinearly to a critical point of the original symmetric NMF. Finally, we conduct experiments on both synthetic data and real image clustering to support our theoretical results and demonstrate the performance of the alternating-type algorithms.

关键词： Convergence symmetric matrices Signal processing algorithms Clustering algorithms Minimization Transforms Optimization symmetric nonnegative matrix factorization convergence image clustering alternating minimization

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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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Self-Supervised symmetric nonnegative matrix factorization

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY 2022年第7期32卷 4526-4537页

作者： Jia, Yuheng Liu, Hui Hou, Junhui Kwong, Sam Zhang, Qingfu Southeast Univ Sch Comp Sci & Engn Minist Educ Nanjing 210096 Peoples R China Southeast Univ Key Lab Comp Network & Informat Integrat Minist Educ Nanjing 210096 Peoples R China Caritas Inst Higher Educ Sch Comp & Informat Sci Hong Kong Peoples R China City Univ Hong Kong CityU Dept Comp Sci Hong Kong Peoples R China City Univ Hong Kong Shenzhen Res Inst Shenzhen 51800 Peoples R China

symmetric nonnegative matrix factorization (SNMF) has demonstrated to be a powerful method for data clustering. However, SNMF is mathematically formulated as a non-convex optimization problem, making it sensitive to the initialization of variables. Inspired by ensemble clustering that aims to seek a better clustering result from a set of clustering results, we propose self-supervised SNMF ((SNMF)-N-3), which is capable of boosting clustering performance progressively by taking advantage of the sensitivity to initialization characteristic of SNMF, without relying on any additional information. Specifically, we first perform SNMF repeatedly with a random positive matrix for initialization each time, leading to multiple decomposed matrices. Then, we rank the quality of the resulting matrices with adaptively learned weights, from which a new similarity matrix that is expected to be more discriminative is reconstructed for SNMF again. These two steps are iterated until the stopping criterion/maximum number of iterations is achieved. We mathematically formulate (SNMF)-N-3 as a constrained optimization problem, and provide an alternative optimization algorithm to solve it with the theoretical convergence guaranteed. Extensive experimental results on 10 commonly used benchmark datasets demonstrate the significant advantage of our (SNMF)-N-3 over 14 state-of-the-art methods in terms of 5 quantitative metrics. The source code is publicly available at https://***/jyh-learning/SSSNMF.

关键词： matrix decomposition symmetric matrices Optimization Clustering methods Faces Sensitivity Dimensionality reduction symmetric nonnegative matrix factorization dimensionality reduction clustering

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