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丛书名： Data Science

2020年

作者： Nicolas Gillis

ISBN: (数字)9781611976410

ISBN: (纸本)9781611976403

nonnegative matrix factorization (NMF) in its modern form has become a standard tool in the analysis of high-dimensional data sets. This book provides a comprehensive and up-to-date account of the most important aspects of the NMF problem and is the first to detail its theoretical aspects, including geometric interpretation, nonnegative rank, complexity, and uniqueness. It explains why understanding these theoretical insights is key to using this computational tool effectively and meaningfully.

nonnegative matrix factorization

is accessible to a wide audience and is ideal for anyone interested in the workings of NMF,
discusses some new results on the nonnegative rank and the identifiability of NMF, and
makes available MATLAB codes for readers to run the numerical examples presented in the book.

Graduate students starting to work on NMF and researchers interested in better understanding the NMF problem and how they can use it will find this book useful. It can be used in advanced undergraduate and graduate-level courses on numerical linear algebra and on advanced topics in numerical linear algebra and requires only a basic knowledge of linear algebra and optimization.

关键词： nonnegative matrix factorization low-rank matrix approximations nonnegative rank multiplicative updates extended formulations linear dimensionality reduction matrix factorization non-linear optimization blind source separation dictionary learning nested polytope problem identifiability recommender systems topic modeling feature extraction hyperspectral imaging extended formulations audio source separation communication complexity sparsity principal component analysis singular value decomposition computational complexity beta-divergences separability block coordinate descent methods self-modeling curve resolution gene expression analysis matrix factorization

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Graph regularized discriminative nonnegative matrix factorization

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ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE 2025年 139卷

作者： Liu, Zhonghua Zhu, Fa Xiong, Hao Chen, Xingchi Pelusi, Danilo Vasilakos, Athanasios V. Zhejiang Ocean Univ Sch Informat Engn Zhoushan Peoples R China Nanjing Forestry Univ Coll Informat Sci & Technol Nanjing Peoples R China Nanjing Forestry Univ Coll Artificial Intelligence Nanjing Peoples R China Macquarie Univ Ctr Hlth Informat Sydney Australia Peng Cheng Lab Shenzhen Peoples R China IAU Coll Comp Sci & Informat Technol Dept Networks & Commun POB 1982 Dammam 31441 Saudi Arabia Univ Teramo Dept Commun Sci Teramo Italy Univ Agder UiA Ctr AI Res CAIR Grimstad Norway

It is well known that both the label information and the local geometry structure information are very important for image data clustering and classification. However, nonnegative matrix factorization (NMF) and its variants do not fully utilize the information or only use one of them. This paper presents a graph regularized discriminative nonnegative matrix factorization (GDNMF) for image data clustering, in which the local geometrical structure and label information of the observed samples are thoroughly considered. In the objective function of NMF, two constraint terms are added to preserve the above information. One is a sparse graph, which is adaptively constructed to obtain the local geometrical structure information. The other is data label information, which is used to capture discriminative information of the original data. By using local and label information, the proposed regularized discriminative nonnegative matrix factorization indeed improves the discrimination power of matrix decomposition. In addition, the F-norm formulation based cost function of regularized discriminative nonnegative matrix factorization is given, and the update rules for the optimization function of regularized discriminative nonnegative matrix factorization are proved. The experiment results on several public image datasets demonstrate the effectiveness of GDNMF algorithm. The innovation of this paper lies in extending unsupervised NMF to semi-supervised case and adaptively capturing the local structure of data based on sparse graph. However, the proposed method does not take into account the challenges of multiview data processing.

关键词： nonnegative matrix factorization Label information Graph regularized discriminative nonnegative matrix factorization Image 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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An ordered subsets orthogonal nonnegative matrix factorization framework with application to image clustering

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INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS 2025年第3期16卷 1531-1543页

作者： Ma, Limin Tong, Can Qi, Shouliang Yao, Yudong Teng, Yueyang Northeastern Univ Coll Med & Biol Informat Engn Shenyang 110169 Peoples R China Changshu Inst Technol Sch Comp Sci & Engn Suzhou 215500 Peoples R China Stevens Inst Technol Dept Elect & Comp Engn Hoboken NJ 07030 USA Minist Educ Key Lab Intelligent Comp Med Image Shenyang 110169 Peoples R China

nonnegative matrix factorization (NMF) for image clustering attains impressive machine learning performances. However, the current iterative methods for optimizing NMF problems involve numerous matrix calculations and suffer from high computational costs in large-scale images. To address this issue, this paper presents an ordered subsets orthogonal NMF framework (OS-ONMF) that divides the data matrix in an orderly manner into several subsets and performs NMF on each subset. It balances clustering performance and computational efficiency. After decomposition, each ordered subset still contains the core information of the original data. That is, blocking does not reduce image resolutions but can greatly shorten running time. This framework is a general model that can be applied to various existing iterative update algorithms. We also provide a subset selection method and a convergence analysis of the algorithm. Finally, we conducted clustering experiments on seven real-world image datasets. The experimental results showed that the proposed method can greatly shorten the running time without reducing clustering accuracy.

关键词： Acceleration Clustering nonnegative matrix factorization Ordered subsets

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Motif-based mix-order nonnegative matrix factorization for community detection

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PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS 2025年 661卷

作者： Bu, Xiaotong Wang, Gaoxia Hou, Ximei China Three Gorges Univ Coll Math & Phys Yichang 443002 Hubei Peoples R China Anhui Inst Informat Technol Coll Gen Educ & Foreign Languages Wuhu 241000 Anhui Peoples R China Huaibei Normal Univ Expt Middle Sch Huaibei 235000 Anhui Peoples R China

Community structure is one of the important characteristics of complex networks, so it is of great application value to correctly detect community structure in the study of network structure. nonnegative matrix factorization (NMF) has been proved to be an ideal model of the community detection. Traditional NMF only focuses on the first-order structure (such as adjacency matrix), but does not consider the higher-order structure (such as motif adjacency matrix). However, only considering one of them cannot well represent the global structural characteristics of complex networks. In this paper, we propose a new Mixed-Order nonnegative matrix factorization (MONMF) framework, which can model both first-order and higher-order structures. Previous nonnegative matrix factorization is mostly used in undirected networks, but we will study based on a variety of motif types in directed networks, use motifs to capture higher-order structures in networks, and introduce linear and nonlinear methods to combine the adjacency matrix representing the first-order structure with the motif adjacency matrix representing the higher-order structure to construct a new feature matrix of NMF. At the same time, we introduce the missing edge matrix that characterizes the edgeless connection structure of the network, and gives the expression of the motif adjacency matrix of the three-node open simple motif and the three-node open anchor motif. The MONMF operation is mainly performed on different real networks for open simple motifs and open anchor motifs. Compared with the comparison methods, MONMF can significantly improve the performance of community detection in complex networks.

关键词： Community detection nonnegative matrix factorization Motif adjacency matrix Open motif Directed network

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MVNMF: Multiview nonnegative matrix factorization for radio-multigenomic analysis in breast cancer prognosis

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MEDICAL IMAGE ANALYSIS 2025年 103卷

作者： Guan, Jian Fan, Ming Li, Lihua Hangzhou Dianzi Univ Sch Comp Sci & Technol Hangzhou 310018 Peoples R China Hangzhou Dianzi Univ Inst Intelligent Biomed Hangzhou 310018 Peoples R China Minjiang Univ Coll Math & Data Sci Fuzhou 350121 Peoples R China

Radiogenomic research provides a deeper understanding of breast cancer biology by investigating the correlations between imaging phenotypes and genetic data. However, current radiogenomic research primarily focuses on the correlation between imaging phenotypes and single-genomic data (e.g., gene expression data), overlooking the potential of multi-genomics data to unveil more nuances in cancer characterization. To this end, we propose a multiview nonnegative matrix factorization (MVNMF) method for the radio-multigenomic analysis that identifies dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) features associated with multi-genomics data, including DNA copy number alterations, mutations, and mRNAs, each of which is independently predictive of cancer outcomes. MVNMF incorporates subspace learning and multiview regularization into a unified model to simultaneously select features and explore correlations. Subspace learning is utilized to identify representative radiomic features crucial for tumor analysis, while multiview regularization enables the learning of the correlation between the identified radiomic features and multi-genomics data. Experimental results showed that, for overall survival prediction in breast cancer, MVNMF classified patients into two distinct groups characterized by significant differences in survival (p = 0.0012). Furthermore, it achieved better performance with a C-index of 0.698 compared to the method without considering any genomics data (C-index = 0.528). MVNMF is an effective framework for identifying radiomic features linked to multi-genomics data, which improves its predictive power and provides a better understanding of the biological mechanisms underlying observed phenotypes. MVNMF offers a novel framework for prognostic prediction in breast cancer, with the potential to catalyze further radiogenomic/radio-multigenomic studies.

关键词： Multiview learning Multi-omics Radiogenomics nonnegative matrix factorization Breast cancer

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Efficient algorithms of box-constrained nonnegative matrix factorization and its applications in image clustering

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APPLIED NUMERICAL MATHEMATICS 2025年 208卷 176-188页

作者： Guo, Jie Li, Ting Wan, Zhong Li, Jiaoyan Xiao, Yamei Changsha Normal Univ Sch Math Sci Changsha 410111 Hunan Peoples R China Zhejiang Lab Res Ctr Data Hub & Secur Hangzhou 311121 Zhejiang Peoples R China Cent South Univ Sch Math & Stat Changsha 410083 Hunan Peoples R China Hunan Normal Univ Coll Life Sci Changsha 410081 Hunan Peoples R China

nonnegative matrix factorization (NMF) provides an important approach of unsupervising learning, but it faces computational challenges when applied into clustering of high-dimensional datasets. In this paper, a class of novel nonmonotone gradient-descent algorithms are developed for solving box-constrained NMF problems. Unlike existing algorithms in the literature that update each matrix factor individually by fixing another one, our algorithms simultaneously update the paired matrix factors by leveraging adaptive projected Barzilai-Borwein directions and appropriate step sizes generated by the developed nonmonotone line search rules. Theoretically, it is proved that the developed algorithms are well defined and globally convergent. Extensive numerical tests on public image datasets demonstrate that the developed algorithms in this paper outperform the state-of-the-art ones, in terms of clustering performance, computational efficiency, and robustness of mining noisy data.

关键词： Image clustering Optimization modeling Algorithms nonnegative matrix factorization

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Weighted symmetric nonnegative matrix factorization and graph-boosting to improve the attributed graph clustering

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ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE 2025年 142卷

作者： Li, Shunlei Wan, Lili Zhang, Yin Luo, Lixia Chinese Univ Hong Kong Multiscale Med Robot Ctr Ltd Hong Kong Peoples R China Wuhan Donghu Univ Sch Mech & Elect Engn Wuhan 430212 Hubei Peoples R China Wuhan Polytech Univ Sch Elect & Elect Engn Wuhan 430023 Hubei Peoples R China Guilin Univ Elect Technol Sch Optoelect Engn Guilin 541004 Guangxi Peoples R China Hunan Univ Informat Technol Sch Comp Sci & Engn Changsha 410148 Hunan Peoples R China

Graph clustering is a critical task in network analysis, aimed at grouping nodes based on their structural or attributed similarities. In particular, attributed graph clustering, which considers both structural links and node attributes, is essential for complex networks where additional node information is available. nonnegative matrix factorization (NMF) has shown promise in graph clustering;however, it faces limitations when applied to attributed graph clustering, such as an inability to detect outliers, distortion of geometric data point structures, and disregard for attributed information. Moreover, many existing attributed graph clustering methods overlook distant node relationships due to network sparsity, which hinders further performance improvements. To address these challenges, this paper introduces Weighted Symmetric NMF with graph-Boosting for attributed Graph Clustering (WSBGC), an innovative extension of NMF. WSBGC leverages attribute similarity among nodes to create a weighted version of NMF, enabling outlier detection while preserving the geometric structure of data points. Additionally, WSBGC employs graph-boosting, leveraging attribute information to account for distant node relationships and improve clustering accuracy. A graph attention auto-encoder is then used to construct the final clustering model. The effectiveness of WSBGC is validated through extensive experiments on real-world datasets. Notably, our algorithm improves accuracy by 2.5% compared to the best available method, demonstrating its superior clustering performance in attributed graphs.

关键词： Attributed graph Graph clustering Graph-boosting nonnegative matrix factorization Weighted symmetric

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Semi-supervised structured nonnegative matrix factorization for anchor graph embedding

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NEUROCOMPUTING 2025年 639卷

作者： Li, Xiangli Mei, Jianping Mo, Yuanjian Guilin Univ Elect Technol Sch Math & Comp Sci Guilin 541004 Guangxi Peoples R China Guangxi Coll & Univ Key Lab Data Anal & Computat Guilin 541004 Guangxi Peoples R China Ctr Appl Math Guangxi Guilin 541004 Guangxi Peoples R China

Semi-supervised nonnegative matrix factorization (NMF) has been widely used in various clustering tasks due to its reliable performance. The key is how to use effectively a small amount of label information to obtain a more discriminative low-dimensional representation of data. In order to improve the clustering performance of semi-supervised NMF more effectively, this paper proposes a new semi-supervised NMF method, namely semi-supervised structured NMF for anchor graph embedding (AESSNMF). Specifically, AESSNMF uses three kinds of supervision information simultaneously, namely, pointwise constraints, pairwise constraints, and negative label information. Also, in order to handle mixed-sign data, AESSNMF uses a convex NMF form and only imposes nonnegative constraints on the coefficient matrix. AESSNMF constructs an anchor graph to embed the matrix factorization process, rather than performing the matrix factorization directly on the original data. We use the alternating iterative algorithm to optimize the objective function of AESSNMF. We also discuss the relationship between several related NMF based algorithms and AESSNMF. A large number of experimental results show that AESSNMF is superior to other related algorithms.

关键词： Anchor point nonnegative matrix factorization Semi-supervised clustering Label information

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Multi-view data representation via adaptive label propagation nonnegative matrix factorization

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INFORMATION SCIENCES 2025年 700卷

作者： Li, Hui Leng, Chengcai Peng, Jinye Pei, Zhao Basu, Anup Northwest Univ Sch Math Xian 710127 Peoples R China Northwest Univ Sch Informat Sci & Technol Xian 710127 Peoples R China Shaanxi Normal Univ Sch Comp Sci Xian 710119 Peoples R China Univ Alberta Dept Comp Sci Edmonton AB T6G 2E8 Canada

nonnegative matrix factorization (NMF) integrating label propagation (LP) algorithm can enhance the discrimination ability of low-dimensional representations. However, traditional LP-based NMF approaches require the construction of a k-nearest neighbor graph in advance to obtain the weight matrix for LP. This increases the burden of adjusting the neighborhood parameter k and leads to a suboptimal weight matrix. To address these issues, an Adaptive Label Propagation NMF (ALPNMF) framework is designed by jointly minimizing the data reconstruction error and the predicted soft label reconstruction error. First, ALPNMF completes neighbor selection and weight calculation simultaneously, without adjusting the neighborhood parameter k. Second, ALPNMF takes into account the dependence between the graph weight learning and LP, thereby yielding the globally optimal graph weight matrix for LP and the more accurate predicted soft label matrix for guiding the decomposition process. Finally, with the aim of combining the information from diverse views to acquire more efficient low-dimensional representations, we extend ALPNMF to a new multi-view NMF approach named Multi-view data representation via Adaptive Label Propagation NMF (MALPNMF). Experimental results on five benchmark multi-view datasets demonstrate that MALPNMF outperforms other state-of-the-art multi-view NMF methods.

关键词： Multi-view data representation nonnegative matrix factorization Adaptive label propagation Self-expressiveness

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