arXiv subjects

Liqun Qi

Publications and source records attributed to Liqun Qi.

At least 181 records · Page 10Linked to original sources

arXiv2013

M-Tensors and Nonsingular M-Tensors

The M-matrix is an important concept in matrix theory, and has many applications. Recently, this concept has been extended to higher order tensors [18]. In this paper, we establish some important properties of M-tensors and nonsingular M-tensors. An M-tensor is a Z-tensor. We show that a Z-tensor is a nonsingular M-tensor if and only if it is semi-positive. Thus, a nonsingular M-tensor has all positive diagonal entries; and an M-tensor, regarding as the limitation of a series of nonsingular M-tensors, has all nonnegative diagonal entries. We introduce even-order monotone tensors and present their spectral properties. In matrix theory, a Z-matrix is a nonsingular M-matrix if and only if it is monotone. This is no longer true in the case of higher order tensors. We show that an even-order monotone Z-tensor is an even-order nonsingular M-tensor but not vice versa. An example of an even-order nontrivial monotone Z-tensor is also given.

math.NA↗

arXiv2013

Convergence of a Second Order Markov Chain

In this paper, we consider convergence properties of a second order Markov chain. Similar to a column stochastic matrix is associated to a Markov chain, a so called {\em transition probability tensor} $P$ of order 3 and dimension $n$ is associated to a second order Markov chain with $n$ states. For this $P$, define $F_P$ as $F_P(x):=Px^{2}$ on the $n-1$ dimensional standard simplex $Δ_n$. If 1 is not an eigenvalue of $\nabla F_P$ on $Δ_n$ and $P$ is irreducible, then there exists a unique fixed point of $F_P$ on $Δ_n$. In particular, if every entry of $P$ is greater than $\frac{1}{2n}$, then 1 is not an eigenvalue of $\nabla F_P$ on $Δ_n$. Under the latter condition, we further show that the second order power method for finding the unique fixed point of $F_P$ on $Δ_n$ is globally linearly convergent and the corresponding second order Markov process is globally $R$-linearly convergent.

math.NA↗

arXiv2013

Some new trace formulas of tensors with applications in spectral hypergraph theory

We give some graph theoretical formulas for the trace $Tr_k(\mathbb {T})$ of a tensor $\mathbb {T}$ which do not involve the differential operators and auxiliary matrix. As applications of these trace formulas in the study of the spectra of uniform hypergraphs, we give a characterization (in terms of the traces of the adjacency tensors) of the $k$-uniform hypergraphs whose spectra are $k$-symmetric, thus give an answer to a question raised in [3]. We generalize the results in [3, Theorem 4.2] and [5, Proposition 3.1] about the $k$-symmetry of the spectrum of a $k$-uniform hypergraph, and answer a question in [5] about the relation between the Laplacian and signless Laplacian spectra of a $k$-uniform hypergraph when $k$ is odd. We also give a simplified proof of an expression for $Tr_2(\mathbb {T})$ and discuss the expression for $Tr_3(\mathbb {T})$.

math.SP↗

arXiv2013

H$^+$-Eigenvalues of Laplacian and Signless Laplacian Tensors

We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H$^+$-eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, and H$^{++}$-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor and the adjacency tensor has at most one H$^{++}$-eigenvalue, but has several other H$^+$-eigenvalues. We identify their largest and smallest H$^+$-eigenvalues, and establish some maximum and minimum properties of these H$^+$-eigenvalues. We then define analytic connectivity of a uniform hypergraph and discuss its application in edge connectivity.

math.SP↗

arXiv2013

Nonnegative Tensor Factorization, Completely Positive Tensors and an Hierarchical Elimination Algorithm

Nonnegative tensor factorization has applications in statistics, computer vision, exploratory multiway data analysis and blind source separation. A symmetric nonnegative tensor, which has a symmetric nonnegative factorization, is called a completely positive (CP) tensor. The H-eigenvalues of a CP tensor are always nonnegative. When the order is even, the Z-eigenvalue of a CP tensor are all nonnegative. When the order is odd, a Z-eigenvector associated with a positive (negative) Z-eigenvalue of a CP tensor is always nonnegative (nonpositive). The entries of a CP tensor obey some dominance properties. The CP tensor cone and the copositive tensor cone of the same order are dual to each other. We introduce strongly symmetric tensors and show that a symmetric tensor has a symmetric binary decomposition if and only if it is strongly symmetric. Then we show that a strongly symmetric, hierarchically dominated nonnegative tensor is a CP tensor, and present a hierarchical elimination algorithm for checking this. Numerical examples are also given.

math.NA↗

arXiv2013

The Largest Laplacian and Signless Laplacian H-Eigenvalues of a Uniform Hypergraph

In this paper, we show that the largest Laplacian H-eigenvalue of a $k$-uniform nontrivial hypergraph is strictly larger than the maximum degree when $k$ is even. A tight lower bound for this eigenvalue is given. For a connected even-uniform hypergraph, this lower bound is achieved if and only if it is a hyperstar. However, when $k$ is odd, it happens that the largest Laplacian H-eigenvalue is equal to the maximum degree, which is a tight lower bound. On the other hand, tight upper and lower bounds for the largest signless Laplacian H-eigenvalue of a $k$-uniform connected hypergraph are given. For a connected $k$-uniform hypergraph, the upper (respectively lower) bound of the largest signless Laplacian H-eigenvalue is achieved if and only if it is a complete hypergraph (respectively a hyperstar). The largest Laplacian H-eigenvalue is always less than or equal to the largest signless Laplacian H-eigenvalue. When the hypergraph is connected, the equality holds here if and only if $k$ is even and the hypergraph is odd-bipartite.

math.SP↗

arXiv2013

Cored Hypergraphs, Power Hypergraphs and Their Laplacian H-Eigenvalues

In this paper, we introduce the class of cored hypergraphs and power hypergraphs, and investigate the properties of their Laplacian H-eigenvalues. From an ordinary graph, one may generate a $k$-uniform hypergraph, called the $k$th power hypergraph of that graph. Power hypergraphs are cored hypergraphs, but not vice versa. Hyperstars, hypercycles, hyperpaths are special cases of power hypergraphs, while sunflowers are a subclass of cored hypergraphs, but not power graphs in general. We show that the largest Laplacian H-eigenvalue of an even-uniform cored hypergraph is equal to its largest signless Laplacian H-eigenvalue. Especially, we find out these largest H-eigenvalues for even-uniform sunflowers. Moreover, we show that the largest Laplacian H-eigenvalue of an odd-uniform sunflower, hypercycle and hyperpath is equal to the maximum degree, i.e., 2. We also compute out the H-spectra of the class of hyperstars. When $k$ is odd, the H-spectra of the hypercycle of size 3 and the hyperpath of length 3 are characterized as well.

math.SP↗

arXiv2013

The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph

In this paper, we show that the eigenvectors of the zero Laplacian and signless Lapacian eigenvalues of a $k$-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector of the zero Laplacian or signless Lapacian eigenvalue have the same modulus. Moreover, under a {\em canonical} regularization, the phases of the components of these eigenvectors only can take some uniformly distributed values in $\{\{exp}(\frac{2jπ}{k})\;|\;j\in [k]\}$. These eigenvectors are divided into H-eigenvectors and N-eigenvectors. Eigenvectors with minimal support is called {\em minimal}. The minimal canonical H-eigenvectors characterize the even (odd)-bipartite connected components of the hypergraph and vice versa, and the minimal canonical N-eigenvectors characterize some multi-partite connected components of the hypergraph and vice versa.

math.SP↗

arXiv2013

Eigenvalue analysis of constrained minimization problem for homogeneous polynomial

In this paper, the concepts of Pareto $H$-eigenvalue and Pareto $Z$-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a symmetric tensor has at least one Pareto $H$-eigenvalue (Pareto $Z$-eigenvalue). Furthermore, the minimum Pareto $H$-eigenvalue (or Pareto $Z$-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor $\mathcal{A}$ is copositive if and only if every Pareto $H$-eigenvalue ($Z-$eigenvalue) of $\mathcal{A}$ is non-negative.

math.OC↗

arXiv2013

The E-Eigenvectors of Tensors

We first show that the eigenvector of a tensor is well-defined. The differences between the eigenvectors of a tensor and its E-eigenvectors are the eigenvectors on the nonsingular projective variety $\mathbb S=\{\mathbf x\in\mathbb P^n\;|\;\sum\limits_{i=0}^nx_i^2=0\}$. We show that a generic tensor has no eigenvectors on $\mathbb S$. Actually, we show that a generic tensor has no eigenvectors on a proper nonsingular projective variety in $\mathbb P^n$. By these facts, we show that the coefficients of the E-characteristic polynomial are algebraically dependent. Actually, a certain power of the determinant of the tensor can be expressed through the coefficients besides the constant term. Hence, a nonsingular tensor always has an E-eigenvector. When a tensor $\mathcal T$ is nonsingular and symmetric, its E-eigenvectors are exactly the singular points of a class of hypersurfaces defined by $\mathcal T$ and a parameter. We give explicit factorization of the discriminant of this class of hypersurfaces, which completes Cartwright and Strumfels' formula. We show that the factorization contains the determinant and the E-characteristic polynomial of the tensor $\mathcal T$ as irreducible factors.

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arXiv2012

Symmetric Nonnegative Tensors and Copositive Tensors

We first prove two new spectral properties for symmetric nonnegative tensors. We prove a maximum property for the largest H-eigenvalue of a symmetric nonnegative tensor, and establish some bounds for this eigenvalue via row sums of that tensor. We show that if an eigenvalue of a symmetric nonnegative tensor has a positive H-eigenvector, then this eigenvalue is the largest H-eigenvalue of that tensor. We also give a necessary and sufficient condition for this. We then introduce copositive tensors. This concept extends the concept of copositive matrices. Symmetric nonnegative tensors and positive semi-definite tensors are examples of copositive tensors. The diagonal elements of a copositive tensor must be nonnegative. We show that if each sum of a diagonal element and all the negative off-diagonal elements in the same row of a real symmetric tensor is nonnegative, then that tensor is a copositive tensor. Some further properties of copositive tensors are discussed.

math.SP↗

arXiv2012

E-Characteristic Polynomials of Tensors

In this paper, we show that the coefficients of the E-characteristic polynomial of a tensor are orthonormal invariants of that tensor. When the dimension is 2, some simplified formulas of the E-characteristic polynomial are presented. A re- sultant formula for the constant term of the E-characteristic polynomial is given. We then study the set of tensors with infinitely many eigenpairs and the set of irregular tensors, and prove both the sets have codimension 2 as subvarieties in the projective space of tensors. This makes our perturbation method workable. By using the perturbation method and exploring the difference between E-eigenvalues and eigenpair equivalence classes, we present a simple formula for the coefficient of the leading term of the E-characteristic polynomial, when the dimension is 2.

math.SP↗

arXiv2012

Nonmonotone Barzilai-Borwein Gradient Algorithm for $\ell_1$-Regularized Nonsmooth Minimization in Compressive Sensing

This paper is devoted to minimizing the sum of a smooth function and a nonsmooth $\ell_1$-regularized term. This problem as a special cases includes the $\ell_1$-regularized convex minimization problem in signal processing, compressive sensing, machine learning, data mining, etc. However, the non-differentiability of the $\ell_1$-norm causes more challenging especially in large problems encountered in many practical applications. This paper proposes, analyzes, and tests a Barzilai-Borwein gradient algorithm. At each iteration, the generated search direction enjoys descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of the $\ell_1$-norm. Moreover, a nonmonotone line search technique is incorporated to find a suitable stepsize along this direction. The algorithm is easily performed, where the values of the objective function and the gradient of the smooth term are required at per-iteration. Under some conditions, the proposed algorithm is shown to be globally convergent. The limited experiments by using some nonconvex unconstrained problems from CUTEr library with additive $\ell_1$-regularization illustrate that the proposed algorithm performs quite well. Extensive experiments for $\ell_1$-regularized least squares problems in compressive sensing verify that our algorithm compares favorably with several state-of-the-art algorithms which are specifically designed in recent years.

math.OC↗

arXiv2012

The Quantum Eigenvalue Problem and Z-Eigenvalues of Tensors

The quantum eigenvalue problem arises in the study of the geometric measure of the quantum entanglement. In this paper, we convert the quantum eigenvalue problem to the Z-eigenvalue problem of a real symmetric tensor. In this way, the theory and algorithms for Z-eigenvalues can be applied to the quantum eigenvalue problem. In particular, this gives an upper bound for the number of quantum eigenvalues. We show that the quantum eigenvalues appear in pairs, i.e., if a real number $λ$ is a quantum eigenvalue of a square symmetric tensor $Ψ$, then $-λ$ is also a quantum eigenvalue of $Ψ$. When $Ψ$ is real, we show that the entanglement eigenvalue of $Ψ$ is always greater than or equal to the Z-spectral radius of $Ψ$, and that in several cases the equality holds. We also show that the ratio between the entanglement eigenvalue and the Z-spectral radius of a real symmetric tensor is bounded above in a real symmetric tensor space of fixed order and dimension.

math.SP↗

arXiv2012

Geometric measure of entanglement of multipartite mixed states

The geometric measure of entanglement of a pure state, defined by its distance to the set of pure separable states, is extended to multipartite mixed states. We characterize the nearest disentangled mixed state to a given mixed state with respect to this measure by a system of equations. The entanglement eigenvalue for a mixed state is introduced. For a given mixed state, we show that its nearest disentangled mixed state is associated with its entanglement eigenvalue.

quant-ph↗

arXiv2012

The geometric measure of entanglement of pure states with nonnegative amplitudes and the spectral theory of nonnegative tensors

The geometric measure of entanglement for a symmetric pure state with nonnegative amplitudes has attracted much attention. On the other hand, the spectral theory of nonnegative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of nonnegative tensors can be applied to the study of the geometric measure of entanglement for a pure state with nonnegative amplitudes. Especially, an elimination method for computing the geometric measure of entanglement for symmetric pure multipartite qubit or qutrit states with nonnegative amplitudes is given. For symmetric pure multipartite qudit states with nonnegative amplitudes, a numerical algorithm with randomization is presented and proven to be convergent. We show that for the geometric measure of entanglement for pure states with nonnegative amplitudes, the nonsymmetric ones can be converted to the symmetric ones.

quant-ph↗

arXiv2012

M-tensors and The Positive Definiteness of a Multivariate Form

We study M-tensors and various properties of M-tensors are given. Specially, we show that the smallest real eigenvalue of M-tensor is positive corresponding to a nonnegative eigenvector. We propose an algorithm to find the smallest positive eigenvalue and then apply the property to study the positive definiteness of a multivariate form.

math.NA↗

arXiv2012

The Minimum Hartree Value for the Quantum Entanglement Problem

A general $n$-partite state $| Ψ>$ of a composite quantum system can be regarded as an element in a Hilbert tensor product space $\HH = \otimes_{k=1}^n \HH_k$, where the dimension of $\HH_k$ is $d_k$ for $k = 1,..., n$. Without loss of generality we may assume that $d_1 \le...\le d_n$. A separable (Hartree) $n$-partite state $| ϕ>$ can be described by $| ϕ> = \otimes_{k=1}^n | ϕ^{(k)}>$ with $| ϕ^{(k)}> \in \HH_k$. We show that $σ:= \min \{< Ψ| ϕ_Ψ> : | Ψ> \in \HH,.$ $. < Ψ| Ψ> = 1\}$ is a positive number, where $| ϕ_Ψ>$ is the nearest separable state to $| Ψ>$. We call $σ$ the minimum Hartree value of $\HH$. We further show that $σ\ge 1/{\sqrt{d_1... d_{n-1}}}$. Thus, the geometric measure of the entanglement content of $Ψ$, $\| | Ψ> - | ϕ_Ψ> \| \le \sqrt{2-2σ} \le \sqrt{2-2(1/{\sqrt{d_1...d_{n-1}}})}$.

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