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Wenyu Sun - One of the best experts on this subject based on the ideXlab platform.
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on semi Definiteness and minimal h eigenvalue of a symmetric space tensor using nonnegative polynomial optimization techniques
Signal Processing-image Communication, 2019Co-Authors: Wenyu SunAbstract:Abstract Verifying the positive semi-Definiteness of a symmetric space tensor is an important and challenging topic in tensor computation. In this paper, we develop two methods to address the problem based on the theory of nonnegative polynomials which enables us to establish semi-definite programs to examine the positive semi-Definiteness of a given symmetric space tensor. Moreover, using the similar idea, we can show that the minimal H -eigenvalue of a symmetric space tensor must be the optimal value of a semi-definite program. Computational results and discussions are provided to illustrate the significance of the results and the effectiveness of the proposed methods.
Haibin Chen - One of the best experts on this subject based on the ideXlab platform.
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Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors
Journal of Industrial & Management Optimization, 2015Co-Authors: Haibin ChenAbstract:Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only if its generating vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its generating vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semi-Definiteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised.
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Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors
arXiv: Spectral Theory, 2014Co-Authors: Haibin ChenAbstract:Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only if its generating vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its generating vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semi-Definiteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. Furthermore, we prove that the Hadamard product of two positive semi-definite (positive definite respectively) symmetric Cauchy tensors is a positive semi-definite (positive definite respectively) tensor, which can be generalized to the Hadamard product of finitely many positive semi-definite (positive definite respectively) symmetric Cauchy tensors. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised.
Bala Rajaratnam - One of the best experts on this subject based on the ideXlab platform.
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retaining positive Definiteness in thresholded matrices
Linear Algebra and its Applications, 2012Co-Authors: Dominique Guillot, Bala RajaratnamAbstract:Abstract Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive Definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive Definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of complete components. This result rigorously demonstrates that, except in special cases, positive Definiteness can be easily lost. We then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive Definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive Definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive Definiteness upon thresholding is governed by complex algebraic conditions.
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retaining positive Definiteness in thresholded matrices
arXiv: Statistics Theory, 2011Co-Authors: Dominique Guillot, Bala RajaratnamAbstract:Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive Definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive Definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of disconnected complete components. This result rigorously demonstrates that, except in special cases, positive Definiteness can be easily lost. We then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive Definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive Definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive Definiteness upon thresholding is governed by complex algebraic conditions.
Dominique Guillot - One of the best experts on this subject based on the ideXlab platform.
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retaining positive Definiteness in thresholded matrices
Linear Algebra and its Applications, 2012Co-Authors: Dominique Guillot, Bala RajaratnamAbstract:Abstract Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive Definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive Definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of complete components. This result rigorously demonstrates that, except in special cases, positive Definiteness can be easily lost. We then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive Definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive Definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive Definiteness upon thresholding is governed by complex algebraic conditions.
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retaining positive Definiteness in thresholded matrices
arXiv: Statistics Theory, 2011Co-Authors: Dominique Guillot, Bala RajaratnamAbstract:Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive Definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive Definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of disconnected complete components. This result rigorously demonstrates that, except in special cases, positive Definiteness can be easily lost. We then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive Definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive Definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive Definiteness upon thresholding is governed by complex algebraic conditions.
Yisheng Song - One of the best experts on this subject based on the ideXlab platform.
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a necessary and sufficient condition of positive Definiteness for 4th order symmetric tensors defined in particle physics
arXiv: Mathematical Physics, 2020Co-Authors: Yisheng SongAbstract:In this paper, we mainly discuss analytical expressions of positive Definiteness for a special 4th order 3-dimensional symmetric tensor defined by the constructed model for a physical phenomenon. Firstly, an analytically necessary and sufficient conditions of 4th order 2-dimensional symmetric tensors are given to test its positive Definiteness. Furthermore, by means of such a result, a necessary and sufficient condition of positive Definiteness is obtained for a special 4th order 3-dimensional symmetric tensor. Such an analytical conditions can be used for verifying the vacuum stability of general scalar potentials of two real singlet scalar fields and the Higgs boson. The positive semi-Definiteness conclusions are presented too.
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Analytical expressions of copositivity for 4th order symmetric tensors and applications.
arXiv: Optimization and Control, 2019Co-Authors: Yisheng SongAbstract:In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking copositivity and positive Definiteness of tensors given by such scalar potentials. Because the tensors given by general scalar potential are 4th order and symmetric, our work mainly focuses on finding precise expressions to test copositivity and positive Definiteness of 4th order tensors in this paper. First of all, an analytically sufficient and necessary condition of positive Definiteness is provided for 4th order 2 dimensional symmetric tensors. For 4th order 3 dimensional symmetric tensors, we give two analytically sufficient conditions of (strictly) cpositivity by using proof technique of reducing orders or dimensions of such a tensor. Furthermore, an analytically sufficient and necessary condition of copositivity is showed for 4th order 2 dimensional symmetric tensors. We also give several distinctly analytically sufficient conditions of (strict) copositivity for 4th order 2 dimensional symmetric tensors. Finally, we apply these results to check lower bound of scalar potentials, and to present analytical vacuum stability conditions for potentials of two real scalar fields and the Higgs boson.