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orthogonal nonnegative matrix factorization problems for clustering: A new formulation and a competitive algorithm

ANNALS OF OPERATIONS RESEARCH 2024年第3期339卷 1481-1497页

作者： Dehghanpour, Ja'far Mahdavi-Amiri, Nezam Sharif Univ Technol Fac Math Sci POB 11155-9415 Tehran Iran

orthogonal nonnegative matrix factorization (ONMF) with orthogonality constraints on a matrix has been found to provide better clustering results over existing clustering problems. Because of the orthogonality constraint, this optimization problem is difficult to solve. Many of the existing constraint-preserving methods deal directly with the constraints using different techniques such as matrix decomposition or computing exponential matrices. Here, we propose an alternative formulation of the ONMF problem which converts the orthogonality constraints into non-convex constraints. To handle the non-convex constraints, a penalty function is applied. The penalized problem is a smooth nonlinear programming problem with quadratic (convex) constraints that can be solved by a proper optimization method. We first make use of an optimization method with two gradient projection steps and then apply a post-processing technique to construct a partition of the clustering problem. Comparative performance analysis of our proposed approach with other available clustering methods on randomly generated test problems and hard synthetic data-sets shows the outperformance of our approach, in terms of the obtained misclassification error rate and the Rand index.

关键词： orthogonal nonnegative matrix factorization Isoperimetry problem Clustering Optimization problem with orthogonality constraints

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Randomized Algorithms for orthogonal nonnegative matrix factorization

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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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Variational Bayesian orthogonal nonnegative matrix factorization Over the Stiefel Manifold

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IEEE TRANSACTIONS ON IMAGE PROCESSING 2022年 31卷 5543-5558页

作者： Rahiche, Abderrahmane Cheriet, Mohamed Univ Quebec Synchromedia Lab Ecole Technol Super ETS Montreal PQ H3C 3P8 Canada

nonnegative matrix factorization (NMF) is one of the best-known multivariate data analysis techniques. The NMF uniqueness and its rank selection are two major open problems in this field. The solutions uniqueness issue can be addressed by imposing the orthogonality condition on NMF. This constraint yields sparser part-based representations and improved performance in clustering and source separation tasks. However, existing orthogonal NMF algorithms rely mainly on non-probabilistic frameworks that ignore the noise inherent in real-life data and lack variable uncertainties. Thus, in this work, we investigate a new probabilistic formulation of orthogonal NMF (ONMF). In the proposed model, we impose the orthogonality through a directional prior distribution defined on the Stiefel manifold called von Mises-Fisher distribution. This manifold consists of a set of directions that comply with the orthogonality condition that arises in many applications. Moreover, our model involves an automatic relevance determination (ARD) prior to address the model order selection issue. We devised an efficient variational Bayesian inference algorithm to solve the proposed ONMF model, which allows fast processing of large datasets. We evaluated the proposed model, called VBONMF, on the task of blind decomposition of real-world multispectral images of ancient documents. The numerical experiments demonstrate its efficiency and competitiveness compared to the state-of-the-art approaches.

关键词： Bayes methods Probabilistic logic Manifolds matrix decomposition Task analysis Source separation Data models Blind source separation multispectral images orthogonal nonnegative matrix factorization Stiefel manifold variational Bayes von Mises-Fisher distribution

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Blind Decomposition of Multispectral Document Images Using orthogonal nonnegative matrix factorization

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IEEE TRANSACTIONS ON IMAGE PROCESSING 2021年 30卷 5997-6012页

作者： Rahiche, Abderrahmane Cheriet, Mohamed Univ Quebec Synchromedia Lab Ecole Technol Super ETS Montreal PQ H3C 1K3 Canada

This paper addresses the challenge of Multispectral (MS) document image segmentation, which is an essential step for subsequent document image analysis. Most previous studies have focused only on binary (text/non-text) separation. They also rely on handcrafted features and techniques dedicated to conventional images that do not take advantage of MS images' spectral richness. In this work, we reformulate this task as a source separation problem, whereby we target the blind decomposition of entire MS document images via a new orthogonal nonnegative matrix factorization (ONMF). On the one hand, we incorporate orthogonality constraint as a Riemannian optimization on the Stiefel manifold. On the other hand, based on which factor we impose the orthogonality constraint, i.e., either on the endmember matrix, abundance matrix, or both, we propose three ONMF models to investigate this issue and determine which model is more suitable for this study. Minimizing the three models subject to nonnegativity and orthogonality constraints simultaneously is very challenging. Therefore, we extend the alternating direction method of multipliers scheme to solve them. We evaluated our models on synthetic Hyperspectral (HS) images and real-world MS document images. The experimental results confirm the effectiveness of the proposed models and demonstrate their generalization power compared to state-of-the-art techniques.

关键词： Manifolds Optimization Data models matrix decomposition Task analysis Image segmentation Image analysis Multispectral images orthogonal nonnegative matrix factorization Stiefel manifold document image segmentation Blind source separation

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Multi-block Bregman proximal alternating linearized minimization and its application to orthogonal nonnegative matrix factorization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2021年第3期79卷 681-715页

作者： Ahookhosh, Masoud Hien, Le Thi Khanh Gillis, Nicolas Patrinos, Panagiotis Univ Antwerp Dept Math Middelheimlaan 1 B-2020 Antwerp Belgium Univ Mons Fac Polytechn Dept Math & Operat Res Rue Houdain 9 B-7000 Mons Belgium Katholieke Univ Leuven Dept Elect Engn ESAT STADIUS Kasteelpark Arenberg 10 B-3001 Leuven Belgium

We introduce and analyze BPALM and A-BPALM, two multi-block proximal alternating linearized minimization algorithms using Bregman distances for solving structured nonconvex problems. The objective function is the sum of a multi-block relatively smooth function (i.e., relatively smooth by fixing all the blocks except one) and block separable (nonsmooth) nonconvex functions. The sequences generated by our algorithms are subsequentially convergent to critical points of the objective function, while they are globally convergent under the KL inequality assumption. Moreover, the rate of convergence is further analyzed for functions satisfying the Lojasiewicz's gradient inequality. We apply this framework to orthogonal nonnegative matrix factorization (ONMF) that satisfies all of our assumptions and the related subproblems are solved in closed forms, where some preliminary numerical results are reported.

关键词： Nonsmooth nonconvex optimization Proximal alternating linearized minimization Bregman distance Multi-block relative smoothness KL inequality orthogonal nonnegative matrix factorization

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KERNEL orthogonal nonnegative matrix factorization: APPLICATION TO MULTISPECTRAL DOCUMENT IMAGE DECOMPOSITION

KERNEL ORTHOGONAL NONNEGATIVE MATRIX FACTORIZATION: APPLICAT...

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IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

作者： Rahiche, Abderrahmane Cheriet, Mohamed Ecole Technol Super ETS Synchromedia Lab Montreal PQ Canada

ISBN: (纸本)9781728176055

As a nonlinear extension of the standard nonnegative matrix factorization (NMF), kernel-based variants have demonstrated to be more effective for discovering meaningful latent features from raw data. However, many existing kernel methods allow only obtaining the basis matrix in the projected feature space, which prevents its inverse mapping back to the original space as requested in many applications. In this work, we propose a new kernel orthogonal NMF method that does not suffer from the pre-image issue. We incorporate the orthogonality constraint as an optimization problem over the Stiefel manifold to improve the sparsity and the model's clustering properties. We solve the proposed model with an efficient optimization approach based on the alternating direction method of multipliers (ADMM) scheme and the projected gradients method. We validate our model on the task of blind decomposition of real-world Multispectral (MS) document images. Our experiments demonstrate the competitiveness of our proposed model in comparison to the state-of-the-art techniques.

关键词： orthogonal nonnegative matrix factorization Kernel Multispectral document image blind decomposition Stiefel manifold

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Gyral Growth Patterns of Macaque Brains Revealed by Scattered orthogonal nonnegative matrix factorization 11th

Gyral Growth Patterns of Macaque Brains Revealed by Scattere...

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11th International Workshop on Machine Learning in Medical Imaging (MLMI)

作者： Zhang, Songyao Du, Lei Lv, Jinglei He, Zhibin Jiang, Xi Guo, Lei Wang, Li Liu, Tianming Shen, Dinggang Li, Gang Zhang, Tuo Northwestern Polytech Univ Sch Automat Xian Peoples R China Univ Sydney Sydney Imaging Brain & Mind Ctr Camperdown NSW Australia Univ Sydney Sch Biomed Engn Camperdown NSW Australia Univ Elect Sci & Technol China Sch Life Sci & Technol MOE Key Lab Neuroinformat Chengdu Peoples R China Univ N Carolina Dept Radiol Chapel Hill NC 27599 USA Univ Georgia Dept Comp Sci Cort Architecture Imaging & Discovery Lab Athens GA 30602 USA Univ Georgia Bioimaging Res Ctr Athens GA 30602 USA

ISBN: (纸本)9783030598600;9783030598617

Cerebral cortex development undergoes a variety of alternate processes, providing valuable information to study the developmental mechanism of the cortical folding and the structural and functional architectures. Many longitudinal studies are performed on the development of sulci using features like sulcal depth, but the gyral system is less studied. To fill the gap, we propose a novel feature, termed gyral height, to quantify the longitudinal developmental patterns of gyri. Another practical problem is the difficulty of obtaining data for all time points for all subjects, even in animal datasets, such as the macaque neurodevelopment dataset in this work. Therefore, we develop a novelmethod by introducing a scattered factor to the orthogonal nonnegative matrix factorization to align data both longitudinally and cross-sectionally. By this method, the gyral height feature maps are decomposed into orthogonal cortical clusters which encode spatiotemporal patterns. Close relations are found between these clusters and anatomical, structural connective and functional metrics, suggesting the potential of the novel cortical feature and the method in investigating the brain development.

关键词： Gyral height orthogonal nonnegative matrix factorization

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orthogonal nonnegative matrix factorization BY SPARSITY AND NUCLEAR NORM OPTIMIZATION

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SIAM JOURNAL ON matrix ANALYSIS AND APPLICATIONS 2018年第2期39卷 856-875页

作者： Pan, Junjun Ng, Michael K. Hong Kong Baptist Univ Dept Math Kowloon Tong Hong Kong Peoples R China

In this paper, we study orthogonal nonnegative matrix factorization. We demonstrate the coefficient matrix can be sparse and low-rank in the orthogonal nonnegative matrix factorization. By using these properties, we propose to use a sparsity and nuclear norm minimization for the factorization and develop a convex optimization model for finding the coefficient matrix in the factorization. Numerical examples including synthetic and real-world data sets are presented to illustrate the effectiveness of the proposed algorithm and demonstrate that its performance is better than other testing methods.

关键词： orthogonal nonnegative matrix factorization sparsity nuclear norm convex optimization document clustering hyperspectral image unmixing

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Spatially Coherent Clustering Based on orthogonal nonnegative matrix factorization

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JOURNAL OF IMAGING 2021年第10期7卷 194-194页

作者： Fernsel, Pascal Univ Bremen Ctr Ind Math D-28359 Bremen Germany

Classical approaches in cluster analysis are typically based on a feature space analysis. However, many applications lead to datasets with additional spatial information and a ground truth with spatially coherent classes, which will not necessarily be reconstructed well by standard clustering methods. Motivated by applications in hyperspectral imaging, we introduce in this work clustering models based on orthogonal nonnegative matrix factorization (ONMF), which include an additional Total Variation (TV) regularization procedure on the cluster membership matrix to enforce the needed spatial coherence in the clusters. We propose several approaches with different optimization techniques, where the TV regularization is either performed as a subsequent post-processing step or included into the clustering algorithm. Finally, we provide a numerical evaluation of 12 different TV regularized ONMF methods on a hyperspectral dataset obtained from a matrix-assisted laser desorption/ionization imaging measurement, which leads to significantly better clustering results compared to classical clustering models.

关键词： orthogonal nonnegative matrix factorization clustering spatial coherence hyperspectral data MALDI imaging

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CELL SUBCLASS IDENTIFICATION IN SINGLE-CELL RNA-SEQUENCING DATA USING orthogonal nonnegative matrix factorization

CELL SUBCLASS IDENTIFICATION IN SINGLE-CELL RNA-SEQUENCING D...

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IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

作者： Wang, Shuai Wu, Peng Zhou, Manqi Chang, Tsung-Hui Wu, Song Chinese Univ Hong Kong Sch Sci & Engn Shenzhen 518172 Peoples R China Shenzhen Res Inst Big Data Shenzhen 518172 Peoples R China Shenzhen Univ Inst Urol Surg Shenzhen 518000 Peoples R China

ISBN: (纸本)9781538646588

Identification of cell subclasses using single-cell RNA-Sequencing (scRNA-Seq) data is of paramount importance since it uncovers the hidden biological processes within the cell population. While the nonnegative matrix factorization (NMF) model has been reported to be effective in various unsupervised clustering tasks, it may still produce inappropriate results for some scRNA-Seq datasets with heterogeneous structures. In this paper, we propose the use of an orthogonally constrained NMF (ONMF) model for the subclass identification problem of scRNA-Seq datasets. The ONMF model in general can provide improved clustering performance, but is challenging to solve. We present a computationally efficient algorithm based on optimization techniques of variable splitting and alternating direction method of multipliers (ADMM). Through two scRNA-Seq datasets, we show that the proposed method can yield promising performance in identifying cell subclasses and detecting key genes over the existing methods. Moreover, the key genes identified by the proposed method are shown biologically significant via the gene set enrichment analysis.

关键词： Cell subclass identification single-cell RNA-Seq orthogonal nonnegative matrix factorization unsupervised clustering gene extraction

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