arXiv subjects

Shenglong Hu

Publications and source records attributed to Shenglong Hu.

30 records · Page 2Linked to original sources

arXiv2014

Relations of the Nuclear Norms of a Tensor and its Matrix Flattenings

For a $3$-tensor of dimensions $I_1\times I_2\times I_3$, we show that the nuclear norm of its every matrix flattening is a lower bound of the tensor nuclear norm, and which in turn is upper bounded by $\sqrt{\min\{I_i : i\neq j\}}$ times the nuclear norm of the matrix flattening in mode $j$ for all $j=1,2,3$. The results can be generalized to $N$-tensors with any $N\geq 3$. Both the lower and upper bounds for the tensor nuclear norm are sharp in the case $N=3$. A computable criterion for the lower bound being tight is given as well.

math.NA↗

arXiv2014

A Tensor Analogy of Yuan's Theorem of the Alternative and Polynomial Optimization with Sign structure

Yuan's theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan's theorem of the alternative to the symmetric tensor setting. As an application, we establish that the optimal value of a class of nonconvex polynomial optimization problems with suitable sign structure (or more explicitly, with essentially non-positive coefficients) can be computed by a related convex conic programming problem, and the optimal solution of these nonconvex polynomial optimization problems can be recovered from the corresponding solution of the convex conic programming problem. Moreover, we obtain that this class of nonconvex polynomial optimization problems enjoy exact sum-of-squares relaxation, and so, can be solved via a single semidefinite programming problem.

math.OC↗

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

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

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.

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↗

arXiv2011

Finding the Spectral Radius of a Nonnegative Tensor

In this paper, we introduce a new class of nonnegative tensors --- strictly nonnegative tensors. A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa. We show that the spectral radius of a strictly nonnegative tensor is always positive. We give some sufficient and necessary conditions for the six well-conditional classes of nonnegative tensors, introduced in the literature, and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors. We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility. We show that for a nonnegative tensor T, there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible; and the spectral radius of T can be obtained from those spectral radii of the induced tensors. In this way, we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption. The preliminary numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.

math.NA↗

arXiv2011

E-Determinants of Tensors

We generalize the concept of the symmetric hyperdeterminants for symmetric tensors to the E-determinants for general tensors. We show that the E-determinant inherits many properties of the determinant of a matrix. These properties include: solvability of polynomial systems, the E-determinat of the composition of tensors, product formula for the E-determinant of a block tensor, Hadamard's inequality, Gersgrin's inequality and Minikowski's inequality. As a simple application, we show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution. We investigate the characteristic polynomial of a tensor through the E-determinant. Explicit formulae for the coefficients of the characteristic polynomial are given when the dimension is two.

math.NA↗