In this paper, a lower bound and an upper bound for the spectral radius of nonnegative tensors are obtained. Our new bounds are tighter than the corresponding bounds obtained by Li et al.(J. Inequal. Appl. 2015). A nu...In this paper, a lower bound and an upper bound for the spectral radius of nonnegative tensors are obtained. Our new bounds are tighter than the corresponding bounds obtained by Li et al.(J. Inequal. Appl. 2015). A numerical example is given to show the effectiveness of theoretical results.展开更多
The nonnegative tensor （matrix） factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization probl...The nonnegative tensor （matrix） factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. To solve the subproblem efficiently, we first exploit a variable regularization term which makes the subproblem far from ill-condition. Second, an augmented Lagrangian alternating direction method is employed to solve this convex and well-conditioned regularized subproblem, and two accelerating skills are also implemented. Some preliminary numerical experiments are performed to show the improvements of the new method.展开更多
Consider the problem of computing the largest eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. ...Consider the problem of computing the largest eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. Moreover, we present a convergent algorithm for calculating the largest eigenvalue for any nonnegative tensors.展开更多
How to extract robust feature is an important research topic in machine learning community. In this paper, we investigate robust feature extraction for speech signal based on tensor structure and develop a new method ...How to extract robust feature is an important research topic in machine learning community. In this paper, we investigate robust feature extraction for speech signal based on tensor structure and develop a new method called constrained Nonnegative Tensor Factorization （cNTF）. A novel feature extraction framework based on the cortical representation in primary auditory cortex （A1） is proposed for robust speaker recognition. Motivated by the neural firing rates model in A1, the speech signal first is represented as a general higher order tensor, cNTF is used to learn the basis functions from multiple interrelated feature subspaces and find a robust sparse representation for speech signal. Computer simulations are given to evaluate the performance of our method and comparisons with existing speaker recognition methods are also provided. The experimental results demonstrate that the proposed method achieves higher recognition accuracy in noisy environment.展开更多
The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor.We analyze the analytic...The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor.We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors.We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors.We finally consider the perturbation of the smallest eigenvalue of nonsingular M-tensors and design a strategy to compute its smallest eigenvalue.We verify our results via random numerical examples.展开更多
In this paper,we present several sharper upper bounds for the M-spectral radius and Z-spectral radius based on the eigenvalues of some unfolding matrices of nonnegative tensors.Meanwhile,we show that these bounds coul...In this paper,we present several sharper upper bounds for the M-spectral radius and Z-spectral radius based on the eigenvalues of some unfolding matrices of nonnegative tensors.Meanwhile,we show that these bounds could be tight for some special tensors.For a general nonnegative tensor which can be transformed into a matrix,we prove the maximal singular value of this matrix is an upper bound of its Z-eigenvalues.Some examples are provided to show these proposed bounds greatly improve some existing ones.展开更多
In this paper,a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors.What is more,we prove this method has an explicit linear convergence rate for indirectly posi...In this paper,a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors.What is more,we prove this method has an explicit linear convergence rate for indirectly positive tensors.Interestingly,the algorithm is exactly the NQZ method(proposed by Ng,Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090–1099,2009)by taking a specific parameter.Furthermore,we give a modified NQZ method,which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron vector.Besides,we promote an inexact power-type algorithm.Finally,some numerical results are reported.展开更多
In this paper,we establish some Brauer-type bounds for the spectral radius of Hadamard product of two nonnegative tensors based on Brauer-type inclusion set,which are shown to be sharper than the existing bounds estab...In this paper,we establish some Brauer-type bounds for the spectral radius of Hadamard product of two nonnegative tensors based on Brauer-type inclusion set,which are shown to be sharper than the existing bounds established in the literature.The validity of the obtained results is theoretically and numerically tested.展开更多
Nonnegative tensor ring(NTR) decomposition is a powerful tool for capturing the significant features of tensor objects while preserving the multi-linear structure of tensor data. The existing algorithms rely on freque...Nonnegative tensor ring(NTR) decomposition is a powerful tool for capturing the significant features of tensor objects while preserving the multi-linear structure of tensor data. The existing algorithms rely on frequent reshaping and permutation operations in the optimization process and use a shrinking step size or projection techniques to ensure core tensor nonnegativity, which leads to a slow convergence rate, especially for large-scale problems. In this paper, we first propose an NTR algorithm based on the modulus method(NTR-MM), which constrains core tensor nonnegativity by modulus transformation. Second, a low-rank approximation(LRA) is introduced to NTR-MM(named LRA-NTR-MM), which not only reduces the computational complexity of NTR-MM significantly but also suppresses the noise. The simulation results demonstrate that the proposed LRA-NTR-MM algorithm achieves higher computational efficiency than the state-of-the-art algorithms while preserving the effectiveness of feature extraction.展开更多
The discrimination of neutrons from gamma rays in a mixed radiation field is crucial in neutron detection tasks.Several approaches have been proposed to enhance the performance and accuracy of neutron-gamma discrimina...The discrimination of neutrons from gamma rays in a mixed radiation field is crucial in neutron detection tasks.Several approaches have been proposed to enhance the performance and accuracy of neutron-gamma discrimination.However,their performances are often associated with certain factors,such as experimental requirements and resulting mixed signals.The main purpose of this study is to achieve fast and accurate neutron-gamma discrimination without a priori information on the signal to be analyzed,as well as the experimental setup.Here,a novel method is proposed based on two concepts.The first method exploits the power of nonnegative tensor factorization(NTF)as a blind source separation method to extract the original components from the mixture signals recorded at the output of the stilbene scintillator detector.The second one is based on the principles of support vector machine(SVM)to identify and discriminate these components.In addition to these two main methods,we adopted the Mexican-hat function as a continuous wavelet transform to characterize the components extracted using the NTF model.The resulting scalograms are processed as colored images,which are segmented into two distinct classes using the Otsu thresholding method to extract the features of interest of the neutrons and gamma-ray components from the background noise.We subsequently used principal component analysis to select the most significant of these features wich are used in the training and testing datasets for SVM.Bias-variance analysis is used to optimize the SVM model by finding the optimal level of model complexity with the highest possible generalization performance.In this framework,the obtained results have verified a suitable bias–variance trade-off value.We achieved an operational SVM prediction model for neutron-gamma classification with a high true-positive rate.The accuracy and performance of the SVM based on the NTF was evaluated and validated by comparing it to the charge comparison method via figure of merit.The results indicate that the proposed approach has a superior discrimination quality(figure of merit of 2.20).展开更多
An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370： 284-294]. In this paper, we establish a linear co...An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370： 284-294]. In this paper, we establish a linear conver- gence rate of the Chang-Qi-Zhou algorithm under a reasonable assumption.展开更多
We study irreducible tensors. the real and complex geometric simplicity of nonnegative First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an even-...We study irreducible tensors. the real and complex geometric simplicity of nonnegative First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an even- order nonnegative irreducible tensor is proved. For an odd-order nonnegative irreducible tensor, sufficient conditions are investigated to ensure the spectral radius to be real geometrically simple. Furthermore, the complex geometric simplicity of nonnegative irreducible tensors is also studied.展开更多
We study iterative methods for solving a set of sparse non-negative tensor equations （multivariate polynomial systems） arising from data mining applications such as information retrieval by query search and communit...We study iterative methods for solving a set of sparse non-negative tensor equations （multivariate polynomial systems） arising from data mining applications such as information retrieval by query search and community discovery in multi-dimensional networks. By making use of sparse and non-negative tensor structure, we develop Jacobi and Gauss-Seidel methods for solving tensor equations. The multiplication of tensors with vectors are required at each iteration of these iterative methods, the cost per iteration depends on the number of non-zeros in the sparse tensors. We show linear convergence of the Jacobi and Gauss-Seidel methods under suitable conditions, and therefore, the set of sparse non-negative tensor equations can be solved very efficiently. Experimental results on information retrieval by query search and community discovery in multi-dimensional networks are presented to illustrate the application of tensor equations and the effectiveness of the proposed methods.展开更多
Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares （SOS） algorithm for computing the minimal H-eigenv...Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares （SOS） algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors （EEN-tensors）, which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a non- negative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.展开更多
We define weakly positive tensors and study the relations among essentially positive tensors, weakly positive tensors, and primitive tensors. In particular, an explicit linear convergence rate of the Liu-Zhou-Ibrahim...We define weakly positive tensors and study the relations among essentially positive tensors, weakly positive tensors, and primitive tensors. In particular, an explicit linear convergence rate of the Liu-Zhou-Ibrahim（LZI） algorithm for finding the largest eigenvalue of an irreducible nonnegative tensor, is established for weakly positive tensors. Numerical results are given to demonstrate linear convergence of the LZI algorithm for weakly positive tensors.展开更多
In this article, the index of imprimitivity of an irreducible nonnegative matrix in the famous PerronFrobenius theorem is studied within a more general framework, both in a more general tensor setting and in a more na...In this article, the index of imprimitivity of an irreducible nonnegative matrix in the famous PerronFrobenius theorem is studied within a more general framework, both in a more general tensor setting and in a more natural spectral symmetry perspective. A k-th order tensor has symmetric spectrum if the set of eigenvalues is symmetric under a group action with the group being a subgroup of the multiplicative group of k-th roots of unity. A sufficient condition, in terms of linear equations over the quotient ring, for a tensor possessing symmetric spectrum is given, which becomes also necessary when the tensor is nonnegative, symmetric and weakly irreducible, or an irreducible nonnegative matrix. Moreover, it is shown that for a weakly irreducible nonnegative tensor, the spectral symmetries are the same when either counting or ignoring multiplicities of the eigenvalues. In particular, the spectral symmetry(index of imprimitivity) of an irreducible nonnegative Sylvester matrix is completely resolved via characterizations with the indices of its positive entries. It is shown that the spectrum of an irreducible nonnegative Sylvester matrix can only be 1-symmetric or 2-symmetric, and the exact situations are fully described. With this at hand, the spectral symmetry of a nonnegative two-dimensional symmetric tensor with arbitrary order is also completely characterized.展开更多
The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real ...The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of/k,S-singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of lk,S-singular values /vectors, some properties of the related /k＇S-spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.展开更多
We show that an （eventually） strongly increasing and positively homogeneous mapping T defined on a Banach space can be turned into an Edelstein contraction with respect to Hilbert＇s projective metric. By applying t...We show that an （eventually） strongly increasing and positively homogeneous mapping T defined on a Banach space can be turned into an Edelstein contraction with respect to Hilbert＇s projective metric. By applying the Edelstein contraction theorem, a nonlinear version of the famous Krein- Rutman theorem is presented, and a simple iteration process {T^kx/｜｜T^kx｜｜} （ x ∈ P^＋） is given for finding a positive eigenvector with positive eigenvalue of T. In particular, the eigenvalue problem of a nonnegative tensor A can be viewed as the fixed point problem of the Edelstein contraction with respect to Hilbert＇s projective metric. As a result, the nonlinear Perron-Frobenius property of a nonnegative tensor A is reached easily.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11601473)the Doctoral Innovation Found of Hebei University of Engineering(Grant No.112/SJ010002142)。
文摘In this paper, a lower bound and an upper bound for the spectral radius of nonnegative tensors are obtained. Our new bounds are tighter than the corresponding bounds obtained by Li et al.(J. Inequal. Appl. 2015). A numerical example is given to show the effectiveness of theoretical results.
文摘The nonnegative tensor （matrix） factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. To solve the subproblem efficiently, we first exploit a variable regularization term which makes the subproblem far from ill-condition. Second, an augmented Lagrangian alternating direction method is employed to solve this convex and well-conditioned regularized subproblem, and two accelerating skills are also implemented. Some preliminary numerical experiments are performed to show the improvements of the new method.
文摘Consider the problem of computing the largest eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. Moreover, we present a convergent algorithm for calculating the largest eigenvalue for any nonnegative tensors.
基金supported by the National Natural Science Foundation of China under Grant No.60775007the National Basic Research 973 Program of China under Grant No.2005CB724301the Science and Technology Commission of Shanghai Municipality under Grant No.08511501701
文摘How to extract robust feature is an important research topic in machine learning community. In this paper, we investigate robust feature extraction for speech signal based on tensor structure and develop a new method called constrained Nonnegative Tensor Factorization （cNTF）. A novel feature extraction framework based on the cortical representation in primary auditory cortex （A1） is proposed for robust speaker recognition. Motivated by the neural firing rates model in A1, the speech signal first is represented as a general higher order tensor, cNTF is used to learn the basis functions from multiple interrelated feature subspaces and find a robust sparse representation for speech signal. Computer simulations are given to evaluate the performance of our method and comparisons with existing speaker recognition methods are also provided. The experimental results demonstrate that the proposed method achieves higher recognition accuracy in noisy environment.
基金the National Natural Science Foundation of China(No.11271084)International Cooperation Project of Shanghai Municipal Science and Technology Commission(No.16510711200).
文摘The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor.We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors.We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors.We finally consider the perturbation of the smallest eigenvalue of nonsingular M-tensors and design a strategy to compute its smallest eigenvalue.We verify our results via random numerical examples.
基金the National Natural Science Foundation of China(No.11271206)the Natural Science Foundation of Tianjin(No.12JCYBJC31200).
文摘In this paper,we present several sharper upper bounds for the M-spectral radius and Z-spectral radius based on the eigenvalues of some unfolding matrices of nonnegative tensors.Meanwhile,we show that these bounds could be tight for some special tensors.For a general nonnegative tensor which can be transformed into a matrix,we prove the maximal singular value of this matrix is an upper bound of its Z-eigenvalues.Some examples are provided to show these proposed bounds greatly improve some existing ones.
基金the Ph.D.Candidate Research Innovation Fund of Nankai University.Qing-Zhi Yang’s work was supported by the National Natural Science Foundation of China(No.11271206)Doctoral Fund of Chinese Ministry of Education(No.20120031110024).
文摘In this paper,a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors.What is more,we prove this method has an explicit linear convergence rate for indirectly positive tensors.Interestingly,the algorithm is exactly the NQZ method(proposed by Ng,Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090–1099,2009)by taking a specific parameter.Furthermore,we give a modified NQZ method,which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron vector.Besides,we promote an inexact power-type algorithm.Finally,some numerical results are reported.
基金This work was supported by the National Natural Science Foundation of China(Grant No.11671228).
文摘In this paper,we establish some Brauer-type bounds for the spectral radius of Hadamard product of two nonnegative tensors based on Brauer-type inclusion set,which are shown to be sharper than the existing bounds established in the literature.The validity of the obtained results is theoretically and numerically tested.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.62073087,61973087 and 61973090)the Key-Area Research and Development Program of Guangdong Province(Grant No.2019B010154002)。
文摘Nonnegative tensor ring(NTR) decomposition is a powerful tool for capturing the significant features of tensor objects while preserving the multi-linear structure of tensor data. The existing algorithms rely on frequent reshaping and permutation operations in the optimization process and use a shrinking step size or projection techniques to ensure core tensor nonnegativity, which leads to a slow convergence rate, especially for large-scale problems. In this paper, we first propose an NTR algorithm based on the modulus method(NTR-MM), which constrains core tensor nonnegativity by modulus transformation. Second, a low-rank approximation(LRA) is introduced to NTR-MM(named LRA-NTR-MM), which not only reduces the computational complexity of NTR-MM significantly but also suppresses the noise. The simulation results demonstrate that the proposed LRA-NTR-MM algorithm achieves higher computational efficiency than the state-of-the-art algorithms while preserving the effectiveness of feature extraction.
基金L’Ore´al-UNESCO for the Women in Science Maghreb Program Grant Agreement No.4500410340.
文摘The discrimination of neutrons from gamma rays in a mixed radiation field is crucial in neutron detection tasks.Several approaches have been proposed to enhance the performance and accuracy of neutron-gamma discrimination.However,their performances are often associated with certain factors,such as experimental requirements and resulting mixed signals.The main purpose of this study is to achieve fast and accurate neutron-gamma discrimination without a priori information on the signal to be analyzed,as well as the experimental setup.Here,a novel method is proposed based on two concepts.The first method exploits the power of nonnegative tensor factorization(NTF)as a blind source separation method to extract the original components from the mixture signals recorded at the output of the stilbene scintillator detector.The second one is based on the principles of support vector machine(SVM)to identify and discriminate these components.In addition to these two main methods,we adopted the Mexican-hat function as a continuous wavelet transform to characterize the components extracted using the NTF model.The resulting scalograms are processed as colored images,which are segmented into two distinct classes using the Otsu thresholding method to extract the features of interest of the neutrons and gamma-ray components from the background noise.We subsequently used principal component analysis to select the most significant of these features wich are used in the training and testing datasets for SVM.Bias-variance analysis is used to optimize the SVM model by finding the optimal level of model complexity with the highest possible generalization performance.In this framework,the obtained results have verified a suitable bias–variance trade-off value.We achieved an operational SVM prediction model for neutron-gamma classification with a high true-positive rate.The accuracy and performance of the SVM based on the NTF was evaluated and validated by comparing it to the charge comparison method via figure of merit.The results indicate that the proposed approach has a superior discrimination quality(figure of merit of 2.20).
文摘An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370： 284-294]. In this paper, we establish a linear conver- gence rate of the Chang-Qi-Zhou algorithm under a reasonable assumption.
文摘We study irreducible tensors. the real and complex geometric simplicity of nonnegative First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an even- order nonnegative irreducible tensor is proved. For an odd-order nonnegative irreducible tensor, sufficient conditions are investigated to ensure the spectral radius to be real geometrically simple. Furthermore, the complex geometric simplicity of nonnegative irreducible tensors is also studied.
文摘We study iterative methods for solving a set of sparse non-negative tensor equations （multivariate polynomial systems） arising from data mining applications such as information retrieval by query search and community discovery in multi-dimensional networks. By making use of sparse and non-negative tensor structure, we develop Jacobi and Gauss-Seidel methods for solving tensor equations. The multiplication of tensors with vectors are required at each iteration of these iterative methods, the cost per iteration depends on the number of non-zeros in the sparse tensors. We show linear convergence of the Jacobi and Gauss-Seidel methods under suitable conditions, and therefore, the set of sparse non-negative tensor equations can be solved very efficiently. Experimental results on information retrieval by query search and community discovery in multi-dimensional networks are presented to illustrate the application of tensor equations and the effectiveness of the proposed methods.
基金This work was done during the first authors＇ postdoctoral period in Qufu Normal University. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11601261, 11671228） and the Natural Science Foundation of Shandong Province （No. ZR2016AQ12）.
文摘Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares （SOS） algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors （EEN-tensors）, which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a non- negative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.
基金Acknowledgments. This first author＇s work was supported by the National Natural Science Foundation of China （Grant No. 10871113）. This second author＇s work was supported by the Hong Kong Research Grant Council.
文摘We define weakly positive tensors and study the relations among essentially positive tensors, weakly positive tensors, and primitive tensors. In particular, an explicit linear convergence rate of the Liu-Zhou-Ibrahim（LZI） algorithm for finding the largest eigenvalue of an irreducible nonnegative tensor, is established for weakly positive tensors. Numerical results are given to demonstrate linear convergence of the LZI algorithm for weakly positive tensors.
基金supported by National Natural Science Foundation of China(Grant No.11771328)Young Elite Scientists Sponsorship Program by Tianjin,and Innovation Research Foundation of Tianjin University(Grant Nos.2017XZC-0084 and 2017XRG-0015)。
文摘In this article, the index of imprimitivity of an irreducible nonnegative matrix in the famous PerronFrobenius theorem is studied within a more general framework, both in a more general tensor setting and in a more natural spectral symmetry perspective. A k-th order tensor has symmetric spectrum if the set of eigenvalues is symmetric under a group action with the group being a subgroup of the multiplicative group of k-th roots of unity. A sufficient condition, in terms of linear equations over the quotient ring, for a tensor possessing symmetric spectrum is given, which becomes also necessary when the tensor is nonnegative, symmetric and weakly irreducible, or an irreducible nonnegative matrix. Moreover, it is shown that for a weakly irreducible nonnegative tensor, the spectral symmetries are the same when either counting or ignoring multiplicities of the eigenvalues. In particular, the spectral symmetry(index of imprimitivity) of an irreducible nonnegative Sylvester matrix is completely resolved via characterizations with the indices of its positive entries. It is shown that the spectrum of an irreducible nonnegative Sylvester matrix can only be 1-symmetric or 2-symmetric, and the exact situations are fully described. With this at hand, the spectral symmetry of a nonnegative two-dimensional symmetric tensor with arbitrary order is also completely characterized.
文摘The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of/k,S-singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of lk,S-singular values /vectors, some properties of the related /k＇S-spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.
基金Acknowledgements The authors would like to thank the anonymous referees for their useful comments and valuable suggestions. This work was supported by the Hong Kong Research Grant Council （Grant Nos. PolyU 501808, 501909, 502510, 502111） and the first author was supported partly by the National Natural Science Foundation of China （Grant Nos. 11071279, 11171094, 11271112）.
文摘We show that an （eventually） strongly increasing and positively homogeneous mapping T defined on a Banach space can be turned into an Edelstein contraction with respect to Hilbert＇s projective metric. By applying the Edelstein contraction theorem, a nonlinear version of the famous Krein- Rutman theorem is presented, and a simple iteration process {T^kx/｜｜T^kx｜｜} （ x ∈ P^＋） is given for finding a positive eigenvector with positive eigenvalue of T. In particular, the eigenvalue problem of a nonnegative tensor A can be viewed as the fixed point problem of the Edelstein contraction with respect to Hilbert＇s projective metric. As a result, the nonlinear Perron-Frobenius property of a nonnegative tensor A is reached easily.
