Dynamic mode decomposition (DMD) has been applied for analyzing various nonlinear dynamic systems. In this study, we clarify the relationship among the variants of DMD and nonnegative matrix factorization (NMF), which...
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ISBN:
(纸本)9798350344868;9798350344851
Dynamic mode decomposition (DMD) has been applied for analyzing various nonlinear dynamic systems. In this study, we clarify the relationship among the variants of DMD and nonnegative matrix factorization (NMF), which is also widely used for audio source separation and document clustering. We show that the decomposition form of DMD is a special case of complex NMF (CNMF), and both are equivalent when certain constraint conditions are added to CNMF. The equivalence of the objective functions of nonnegative DMD and CNMF under the aforementioned constraints is also proven. This theoretical analysis provides guidelines for an effective initial value setting of CNMF and for a new construction method that is positioned between CNMF and DMD.
We introduce a new method based on nonnegative matrix factorization, Neural NMF, for detecting latent hierarchical structure in data. Datasets with hierarchical structure arise in a wide variety of fields, such as doc...
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We introduce a new method based on nonnegative matrix factorization, Neural NMF, for detecting latent hierarchical structure in data. Datasets with hierarchical structure arise in a wide variety of fields, such as document classification, image processing, and bioinformatics. Neural NMF recursively applies NMF in layers to discover overarching topics encompassing the lower-level features. We derive a backpropagation optimization scheme that allows us to frame hierarchical NMF as a neural network. We test Neural NMF on a synthetic hierarchical dataset, the 20 Newsgroups dataset, and the MyLymeData symptoms dataset. Numerical results demonstrate that Neural NMF outperforms other hierarchical NMF methods on these data sets and offers better learned hierarchical structure and interpretability of topics.
It's a huge workload to label the edema area in the diagnosis of knee edema at present, based on the excellent ability of nonnegative matrix factorization to process high-dimensional data and the characteristics o...
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ISBN:
(纸本)9783031671913;9783031671920
It's a huge workload to label the edema area in the diagnosis of knee edema at present, based on the excellent ability of nonnegative matrix factorization to process high-dimensional data and the characteristics of less requirement for prior information, we combine nonnegative matrix factorization with the diagnosis of knee edema to determine whether there is edema and find the corresponding edema area. In this paper, we propose a new weakly-supervised method named FSNMF algorithm, by adjusting the corresponding eigenvalues based on different targets, so as to ensure the effectiveness of disease image classification and edema region feature extraction. In our paper, experiments have been carried out on the proposed algorithm to show the performance of our FSNMF algorithm. Through experiments on public data sets, we verify the classification accuracy of the proposed method is improved compared with the previous NMF classification algorithm and the accuracy of edema identification using FSNMF is also excellent.
nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for analyzing nonnegative data. A key aspect of NMF is the choice of the objective function that depends on the noise model (or sta...
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nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for analyzing nonnegative data. A key aspect of NMF is the choice of the objective function that depends on the noise model (or statistics of the noise) assumed on the data. In many applications, the noise model is unknown and difficult to estimate. In this paper, we define a multi-objective NMF (MO-NMF) problem, where several objectives are combined within the same NMF model. We propose to use Lagrange duality to judiciously optimize for a set of weights to be used within the framework of the weighted-sum approach, that is, we minimize a single objective function which is a weighted sum of the all objective functions. We design a simple algorithm based on multiplicative updates to minimize this weighted sum. We show how this can be used to find distributionally robust NMF (DR-NMF) solutions, that is, solutions that minimize the largest error among all objectives, using a dual approach solved via a heuristic inspired from the Frank-Wolfe algorithm. We illustrate the effectiveness of this approach on synthetic, document and audio data sets. The results show that DR-NMF is robust to our incognizance of the noise model of the NMF problem.
nonnegative matrix factorization (NMF)-based models have been proven to be highly effective and scalable in addressing collaborative filtering (CF) problems in the recommender system (RS). Since RS requires tremendous...
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nonnegative matrix factorization (NMF)-based models have been proven to be highly effective and scalable in addressing collaborative filtering (CF) problems in the recommender system (RS). Since RS requires tremendous user data to provide personalized information services, the issue of data privacy has gained prominence. Although the differential privacy (DP) technique has been widely applied to RS, the requirement of nonnegativity makes it difficult to successfully incorporate DP into NMF. In this paper, a differentially private NMF (DPNMF) method is proposed by perturbing the coefficients of the polynomial expression of the objective function, which achieves a good trade-off between privacy protection and recommendation quality. Moreover, to alleviate the influence of the noises added by DP on the items with sparse ratings, an imputation-based DPNMF (IDPNMF) method is proposed. Theoretic analyses and experimental results on several benchmark datasets show that the proposed schemes have good performance and can achieve better recommendation quality on large-scale datasets. Therefore, our schemes have high potential to implement privacy-preserving RS based on big data. (C) 2022 Elsevier Inc. All rights reserved.
nonnegative matrix factorization (NMF) has become a popular technique for dimensionality reduction, and been widely used in machine learning, computer vision, and data mining. Existing unsupervised NMF methods impose ...
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nonnegative matrix factorization (NMF) has become a popular technique for dimensionality reduction, and been widely used in machine learning, computer vision, and data mining. Existing unsupervised NMF methods impose the intrinsic geometric constraint on the encoding matrix, which only indirectly affects the base matrix. Moreover, they ignore the global structure of the data space. To address these issues, in this paper we propose a novel unsupervised NMF learning framework, called Robust Graph regularized nonnegative matrix factorization (RGNMF). RGNMF constructs a sparse graph imposed on the basis matrix to catch the global structure and preserve the discriminative information. And it models the local structure by building a k-NN graph constrained on the encoding matrix, which gains the compact representation. Consequently, RGNMF not only respects the global structure, but also depicts the local structure. In addition, it employs such a L-2,L-1-norm cost function to decompose the basis matrix and encoding matrix that its robustness can be improved. Further, it imposes the L-2,L-1-norm constraint on the basis matrix to choose the discriminative feature. Hence, RGNMF can gain the robust discriminative representation by combining structure learning and L-2,L-1-norm constraints imposed on the basis matrix and encoding matrix. Extensive experiments on real-world problems demonstrate that RGNMF achieves better clustering results than the state-of-the-art approaches.
Kernel nonnegative matrix factorization (KNMF) has emerged as a promising nonlinear data representation method, especially for applications with small sample sizes. Existing methods are usually based on a single kerne...
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Kernel nonnegative matrix factorization (KNMF) has emerged as a promising nonlinear data representation method, especially for applications with small sample sizes. Existing methods are usually based on a single kernel function, representing samples by the global or local features learned by the matrixfactorization algorithm. In this paper, a combined kernel method is proposed and applied to data representation for small-sample face recognition in particular. Based on the combined kernel, which is a linear combination of the fractional power inner-product kernel and a newly defined Gaussian-type kernel, the new KNMF method is able to extract both global and local nonlinear features from the inputs. An efficient gradient decent algorithm is derived to solve the combined kernel nonnegative matrix factorization (CKNMF) problem, and a rigorous convergence proof is presented. The proposed method is experimentally evaluated on small image datasets, and the results demonstrate its superior performance than the state-of-the-art KNMF methods.
nonnegative matrix factorization (NMF) has been proved to be a powerful method in data processing and has also shown success in applications such as feature extraction and image representation. In this paper, we propo...
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ISBN:
(纸本)9781665458412
nonnegative matrix factorization (NMF) has been proved to be a powerful method in data processing and has also shown success in applications such as feature extraction and image representation. In this paper, we propose two symmetric matrix-based methods, Symcom and Symize, to achieve square strategy (SQR) in SQR-NMF. This integration process allows the matrix to preserve symmetry property associated with images to enhance image reconstruction. Simulation results show that Symcom performs better on super wide or super long data matrices and Symize achieves better results on symmetrical data matrices.
nonnegative matrix factorization (NMF) was a classic model for dimensional reduction. Manhattan NMF is a variant version of NMF that uses a L-1-norm cost function as the objective function instead of the L-2-norm cost...
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ISBN:
(纸本)9798350321456
nonnegative matrix factorization (NMF) was a classic model for dimensional reduction. Manhattan NMF is a variant version of NMF that uses a L-1-norm cost function as the objective function instead of the L-2-norm cost function. Manhattan NMF can be formulated as a nonconvex nonsmooth optimization problem. An algorithm framework for solving the Manhattan NMF problem based on the alternating direction method of multiplication is presented to us. Compared with the existed algorithm, our proposed algorithm is more effective by experiments on synthetic and real data sets.
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