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A Unified Convergence Analysis of the multiplicative update algorithm for Regularized Nonnegative Matrix Factorization

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2018年第1期66卷 129-138页

作者： Zhao, Renbo Tan, Vincent Y. F. NUS Dept Elect & Comp Engn Singapore 117583 Singapore NUS Dept Ind Syst Engn & Management Singapore 117576 Singapore NUS Dept Math Singapore 119076 Singapore

The multiplicative update (MU) algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizers. However, theoretical convergence guarantees have only been derived for a few special divergences without regularization. In this work, we provide a conceptually simple, self-contained, and unified proof for the convergence of the MU algorithm applied on NMF with a wide range of divergences and regularizers. Our main result shows the sequence of iterates (i.e., pairs of basis and coefficient matrices) produced by the MU algorithm converges to the set of stationary points of the nonconvex NMF optimization problem. Our proof strategy has the potential to open up new avenues for analyzing similar problems in machine learning and signal processing.

关键词： Nonnegative matrix factorization multiplicative update algorithm convergence analysis nonconvex optimization stationary points

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A UNIFIED CONVERGENCE ANALYSIS OF THE multiplicative update algorithm FOR NONNEGATIVE MATRIX FACTORIZATION

A UNIFIED CONVERGENCE ANALYSIS OF THE MULTIPLICATIVE UPDATE ...

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

作者： Zhao, Renbo Tan, Vincent Y. F. Natl Univ Singapore Dept Elect & Comp Engn Singapore Singapore Natl Univ Singapore Dept Math Singapore Singapore

ISBN: (纸本)9781509041176

The multiplicative update (MU) algorithm has been used extensively to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizations. However, theoretical convergence guarantees have only been derived for a few special divergences. In this work, we provide a conceptually simple, self-contained, and unified proof for the convergence of the MU algorithm applied on NMF with a wide range of divergences and regularizations. Our result shows the sequence of iterates (i.e., pairs of basis and coefficient matrices) produced by the MU algorithm converges to the set of stationary points of the NMF (optimization) problem. Our proof strategy has the potential to open up new avenues for analyzing similar problems.

关键词： Nonnegative Matrix Factorization multiplicative update algorithm Convergence Analysis Nonconvex Optimization Stationary Points

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A unified convergence analysis of the multiplicative update algorithm for nonnegative matrix factorization

A unified convergence analysis of the multiplicative update ...

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IEEE International Conference on Acoustics, Speech and Signal Processing

作者： Renbo Zhao Vincent Y. F. Tan Department of Electrical and Computer Engineering & Department of Mathematics National University of Singapore Singapore

ISBN: (纸本)9781509041183

关键词： Nonnegative Matrix Factorization multiplicative update algorithm Convergence Analysis Nonconvex Optimization Stationary Points matrix decomposition NMF protocol updating algorithm Converge Nonnegative Multiplication proof strategy

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A convergence algorithm for graph co-regularized transfer learning

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Science China(Information Sciences) 2023年第3期66卷 150-159页

作者： Zuyuan YANG Naiyao LIANG Zhenni LI Shengli XIE Guangdong Key Laboratory of Io T Information Technology School of Automation Guangdong University of Technology Key Laboratory of iDetection and Manufacturing-IoT Ministry of Education Guangdong-Hong Kong-Macao Joint Laboratory for Smart Discrete Manufacturing

Transfer learning is an important technology in addressing the problem that labeled data in a target domain are difficult to collect using extensive labeled data from the source domain. Recently,an algorithm named graph co-regularized transfer learning(GTL) has shown a competitive performance in transfer learning. However, its convergence is affected by the used approximate scheme, degenerating learned results. In this paper, after analyzing convergence conditions, we propose a novel update rule using the multiplicative update rule and develop a new algorithm named improved GTL(IGTL) with a strict convergence guarantee. Moreover, to prove the convergence of our method, we design a special auxiliary function whose value is intimately related to that of the objective function. Finally, the experimental results on the synthetic dataset and two real-world datasets confirm that the proposed IGTL is convergent and performs better than the compared methods.

关键词： transfer learning convergence analysis non-negative matrix factorization multiplicative update algorithm optimization

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A dynamic hypergraph regularized non-negative tucker decomposition framework for multiway data analysis

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INTERNATIONAL JOURNAL OF MACHINE LEARNING AND CYBERNETICS 2022年第12期13卷 3691-3710页

作者： Huang, Zhenhao Zhou, Guoxu Qiu, Yuning Yu, Yuyuan Dai, Haolei Guangdong Univ Technol Sch Automat Guangzhou Peoples R China Guangdong Hong Kong MacaoJoint Lab Smart Discrete Guangzhou Peoples R China

Non-negative tensor decomposition has achieved significant success in machine learning due to its superiority in extracting the non-negative parts-based features and physically meaningful latent components from high-order data. To improve its representation ability, hypergraph has been incorporated into the tensor decomposition model to capture the nonlinear manifold structure of data. However, previous hypergraph regularized tensor decomposition methods rely on the original data space. This may result in inaccurate manifold structure and representation performance degeneration when original data suffer from noise corruption. To solve these problems, in this paper, we propose a dynamic hypergraph regularized non-negative Tucker decomposition (DHNTD) method for multiway data analysis. Specifically, to take full advantage of the multilinear structure and nonlinear manifold of tensor data, we learn the dynamic hypergraph and non-negative low-dimensional representation in a unified framework. Moreover, we develop a multiplicative update (MU) algorithm to solve our optimization problem and theoretically prove its convergence. Experimental results in clustering tasks using six image datasets demonstrate the superiority of our proposed method compared with the state-of-the-art methods.

关键词： Non-negative tensor decomposition Manifold structure Dynamic hypergraph multiplicative update algorithm

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Structure preserving non-negative matrix factorization for dimensionality reduction

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COMPUTER VISION AND IMAGE UNDERSTANDING 2013年第9期117卷 1175-1189页

作者： Li, Zechao Liu, Jing Lu, Hanqing Chinese Acad Sci Natl Lab Pattern Recognit Inst Automat Beijing 100190 Peoples R China

The problem of dimensionality reduction is to map data from high dimensional spaces to low dimensional spaces. In the process of dimensionality reduction, the data structure, which is helpful to discover the latent semantics and simultaneously respect the intrinsic geometric structure, should be preserved. In this paper, to discover a low-dimensional embedding space with the nature of structure preservation and basis compactness, we propose a novel dimensionality reduction algorithm, called Structure Preserving Non-negative Matrix Factorization (SPNMF). In SPNMF, three kinds of constraints, namely local affinity, distant repulsion, and embedding basis redundancy elimination, are incorporated into the NMF framework. SPNMF is formulated as an optimization problem and solved by an effective iterative multiplicative update algorithm. The convergence of the proposed update solutions is proved. Extensive experiments on both synthetic data and six real world data sets demonstrate the encouraging performance of the proposed algorithm in comparison to the state-of-the-art algorithms, especially some related works based on NMF. Moreover, the convergence of the proposed updating rules is experimentally validated. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Dimensionality reduction Non-negative matrix factorization Structure preserving Basis compactness multiplicative update algorithm

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Spectral unmixing applied to fast identification of γ-emitting radionuclides using NaI(Tl) detectors

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APPLIED RADIATION AND ISOTOPES 2020年 158卷 109068-109068页

作者： Paradis, H. Bobin, C. Bobin, J. Bouchard, J. Lourenco, V. Thiam, C. Andre, R. Ferreux, L. Ott, A. de Vismes Thevenin, M. CEA LIST LNE LNHB F-91191 Gif Sur Yvette France CEA IRFU Lab Cosmol & Stat F-91191 Gif Sur Yvette France IRSN Lab Mesures Nucl F-78116 Le Vesinet France IRSN Lab Metrol Radioact Environm F-91400 Orsay France CEA IRAMIS Serv Phys Etat Condense F-91191 Gif Sur Yvette France

Spectral unmixing was investigated for fast spectroscopic identification in y-emitter mixtures at low-statistics in the case of measurements performed to prevent illegal nuclear material trafficking or for in situ environmental analysis following a radiological or nuclear accident. For that purpose, a multiplicative update algorithm based on full-spectrum analysis was tested in the case of a 3 '' x3 '' NaI(Tl) detector. Automatic decision-making was addressed using Monte Carlo calculations of decision thresholds and detection limits. The first results obtained with a portable instrument equipped with a 3 '' x3 '' NaI(Tl) detector designed for the control of food samples by non-expert users following a radiological or nuclear accident, are also presented.

关键词： Spectroscopic measurements NaI(Tl) detector Environmental analysis Automatic decision-making Spectral unmixing Inverse problem multiplicative update algorithm

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Sublinear Optimization for Machine Learning

Sublinear Optimization for Machine Learning

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IEEE 51st Annual Symposium on Foundations of Computer Science (FOCS)

作者： Clarkson, Kenneth L. Hazan, Elad Woodruff, David P. IBM Almaden Res Ctr San Jose CA 95120 USA Technion Israel Inst Technol Dept Ind Engn IL-32000 Haifa Israel

ISBN: (纸本)9780769542447

We give sublinear-time approximation algorithms for some optimization problems arising in machine learning, such as training linear classifiers and finding minimum enclosing balls. Our algorithms can be extended to some kernelized versions of these problems, such as SVDD, hard margin SVM, and L-2-SVM, for which sublinear-time algorithms were not known before. These new algorithms use a combination of a novel sampling techniques and a new multiplicative update algorithm. We give lower bounds which show the running times of many of our algorithms to be nearly best possible in the unit-cost RAM model. We also give implementations of our algorithms in the semi-streaming setting, obtaining the first low pass polylogarithmic space and sublinear time algorithms achieving arbitrary approximation factor.

关键词： SVM arbitrary approximation factor classification computational complexity learning (artificial intelligence) linear classifier machine learning machine learning multiplicative update algorithm optimisation optimization pattern classification polylogarithmic space polynomial approximation sampling technique sublinear algorithms sublinear optimization sublinear time approximation support vector machine support vector machines

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An Efficient Parallelization Model for Sparse Non-negative Matrix Factorization Using cuSPARSE Library on Multi-GPU Platform 21st

An Efficient Parallelization Model for Sparse Non-negative M...

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21st International Conference on algorithms and Architectures for Parallel Processing (ICA3PP)

作者： Moumni, Hatem Hamdi-Larbi, Olfa Univ Tunis El Manar Fac Sci Tunis LIPS FST Lab Tunis 2092 Tunisia Taibah Univ Coll Business Madinah Medina Saudi Arabia

ISBN: (纸本)9783030953881;9783030953874

Positive or Non-negative Matrix Factorization (NMF) is an effective technique and has been widely used for Big Data representation. It aims to find two non-negative matrices W and H whose product provides an optimal approximation to the original input data matrix A, such that A approximate to W * H. Although, NMF plays an important role in several applications, such as machine learning, data analysis and biomedical applications. Due to the sparsity that is caused by missing information in many high-dimension scenes (e.g., social networks, recommender systems and DNA gene expressions), the NMF method cannot mine a more accurate representation from the explicit information. Therefore, the Sparse Non-negative Matrix Factorization (SNMF) can incorporate the intrinsic geometry of the data, which is combined with implicit information. Thus, SNMF can realize a more compact representation for the sparse data. In this paper, we study the Sparse Non-negative Matrix Factorization (SNMF). We use multiplicative update algorithm (MUA) that computes the factorization by applying update on both matrices W and H. Accordingly, to address these issue, we propose a two models to implement a parallel version of SNMF on GPUs using NVIDIA CUDA framework. To optimize SNMF, we use cuSPARSE optimized library to compute the algebraic operations in MUA where sparse matrices A, W and H are stored in Compressed Sparse Row (CSR) format. At last, our contribution is validated through a series of experiments achieved on two input sets i.e. a set of randomly generated matrices and a set of benchmark matrices from real applications with different sizes and densities. We show that our algorithms allow performance improvements compared to baseline implementations. The speedup on multi-GPU platform can exceed 11x as well as the Ratio can exceed 91%.

关键词： SNMF Unsupervised learning algorithm multiplicative update algorithm Sparse matrix Compressed formats Optimization

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Metrological approach of γ-emitting radionuclides identification at low statistics: application of sparse spectral unmixing to scintillation detectors

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METROLOGIA 2021年第1期58卷

作者： Andre, Remi Bobin, Christophe Bobin, Jerome Xu, Jiaxin de Vismes Ott, Anne CEA LIST LNE LNHB F-91191 Gif Sur Yvette France CEA IRFU Lab Cosmol & Stat F-91191 Gif Sur Yvette France IRSN Lab Metrol Radioact Environm F-91400 Orsay France

This paper presents a metrological approach of spectral unmixing for automatic identification and quantitative analysis of gamma-emitting radionuclides in natural background radiation at low statistics. Based on full-spectrum analysis, the proposed method relies on the maximum likelihood estimation based on Poisson statistics that accounts for the spectral signatures of the gamma-emitters to be identified and natural background. In order to obtain robust decision-making at low statistics, a sparsity constraint is implemented along with counting estimation given by spectral unmixing. In contrast with the standard approach, this technique relies on a single decision threshold applied for a likelihood ratio test. Standard deviations on estimated counting are determined using the Fisher information matrix. The robustness of decision-making and counting estimation was investigated by means of Monte Carlo calculations based on experimental spectral signatures of two types of scintillation detectors [NaI(Tl), plastic]. This study demonstrates that sparse spectral unmixing is a reliable method for gamma-spectra analysis based on low-level measurements. The sparsity constraint acts as an efficient technique for decision-making in the case of complex mixtures of gamma-emitters with significant contribution of natural background. This method also yields unbiased counting estimation related to the identified radionuclides. Reliable assessment of standard deviations are obtained and the Gaussian approximation of the coverage intervals is validated. The proposed method can be applied either by non-expert users for automatic analysis of gamma-spectra or to help experts in decision-making in the case of complex mixtures of gamma-emitters at low statistics.

关键词： gamma-spectra measurements radionuclide identification automatic decision-making sparse spectral unmixing maximum likelihood estimation multiplicative update algorithm

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