Real Symmetric Tensor Research Articles

Extreme Ratio Between Spectral and Frobenius Norms of Nonnegative Tensors

One of the fundamental problems in multilinear algebra, the minimum ratio between the spectral and Frobenius norms of tensors, has received considerable attention in recent years. While most values are unknown for real and complex tensors, the asymptotic order of magnitude and tight lower bounds have been established. However, little is known about nonnegative tensors. In this paper, we present an almost complete picture of the ratio for nonnegative tensors. In particular, we provide a tight lower bound that can be achieved by a wide class of nonnegative tensors under a simple necessary and sufficient condition, which helps to characterize the extreme tensors and obtain results such as the asymptotic order of magnitude. We show that the ratio for symmetric tensors is no more than that for general tensors multiplied by a constant depending only on the order of tensors, hence determining the asymptotic order of magnitude for real, complex, and nonnegative symmetric tensors. We also find that the ratio is in general different from the minimum ratio between the Frobenius and nuclear norms for nonnegative tensors, a sharp contrast to the case for real tensors and complex tensors.

Exact analytic expressions of real tensor eigenvalue distributions of Gaussian tensor model for small N

We obtain exact analytic expressions of real tensor eigenvalue/vector distributions of real symmetric order-three tensors with Gaussian distributions for N ≤ 8. This is achieved by explicitly computing the partition function of a zero-dimensional boson–fermion system with four interactions. The distributions are expressed by combinations of polynomial, exponential, and error functions as results of feasible complicated bosonic integrals that appear after fermionic integrations. By extrapolating the expressions and also using a previous result, we guess a large-N expression. The expressions are compared with Monte Carlo simulations, and precise agreement and good agreement are obtained with the exact and the large-N expressions, respectively. Understanding the feasibility of the integration is left for future study, which would provide a general-N analytic formula.

Dehomogenization for completely positive tensors

A real symmetric tensor is completely positive (CP) if it is a sum of symmetric tensor powers of nonnegative vectors. We propose a dehomogenization approach for studying CP tensors. This gives new Moment-SOS relaxations for CP tensors. Detection for CP tensors and the linear conic optimization with CP tensor cones can be solved more efficiently by the dehomogenization approach.

Some new criteria for judging $ \mathcal{H} $-tensors and their applications

<abstract><p>$ \mathcal{H} $-tensors play a key role in identifying the positive definiteness of even-order real symmetric tensors. Some criteria have been given since it is difficult to judge whether a given tensor is an $ \mathcal{H} $-tensor, and their range of judgment has been limited. In this paper, some new criteria, from an increasing constant $ k $ to scale the elements of a given tensor can expand the range of judgment, are obtained. Moreover, as an application of those new criteria, some sufficient conditions for judging positive definiteness of even-order real symmetric tensors are proposed. In addition, some numerical examples are presented to illustrate those new results.</p></abstract>

Robust Eigenvectors of Symmetric Tensors

The {\em tensor power method} generalizes the matrix power method to higher order arrays, or tensors. Like in the matrix case, the fixed points of the tensor power method are the eigenvectors of the tensor. While every real symmetric matrix has an eigendecomposition, the vectors generating a symmetric decomposition of a real symmetric tensor are not always eigenvectors of the tensor. In this paper we show that whenever an eigenvector {\em is} a generator of the symmetric decomposition of a symmetric tensor, then (if the order of the tensor is sufficiently high) this eigenvector is {\em robust} , i.e., it is an attracting fixed point of the tensor power method. We exhibit new classes of symmetric tensors whose symmetric decomposition consists of eigenvectors. Generalizing orthogonally decomposable tensors, we consider {\em equiangular tight frame decomposable} and {\em equiangular set decomposable} tensors. Our main result implies that such tensors can be decomposed using the tensor power method.

Real tensor eigenvalue/vector distributions of the Gaussian tensor model via a four-fermi theory

Abstract Eigenvalue distributions are important dynamic quantities in matrix models, and it is an interesting challenge to study corresponding quantities in tensor models. We study real tensor eigenvalue/vector distributions for real symmetric order-three random tensors with a Gaussian distribution as the simplest case. We first rewrite this problem as the computation of a partition function of a four-fermi theory with R replicated fermions. The partition function is exactly computed for some small-N,R cases, and is shown to precisely agree with Monte Carlo simulations. For large-N, it seems difficult to compute it exactly, and we apply an approximation using a self-consistency equation for two-point functions and obtain an analytic expression. It turns out that the real tensor eigenvalue distribution obtained by taking R = 1/2 is simply the Gaussian within this approximation. We compare the approximate expression with Monte Carlo simulations, and find that, if an extra overall factor depending on N is multiplied to the the expression, it agrees well with the Monte Carlo results. It is left for future study to improve the approximation for large-N to correctly derive the overall factor.

Signed distributions of real tensor eigenvectors of Gaussian tensor model via a four-fermi theory

Eigenvalue distributions are important dynamical quantities in matrix models, and it is a challenging problem to derive them in tensor models. In this paper, we consider real symmetric order-three tensors with Gaussian distributions as the simplest case, and derive an explicit formula for signed distributions of real tensor eigenvectors: Each real tensor eigenvector contributes to the distribution by ±1, depending on the sign of the determinant of an associated Hessian matrix. The formula is expressed by the confluent hypergeometric function of the second kind, which is obtained by computing a partition function of a four-fermi theory. The formula can also serve as lower bounds of real eigenvector distributions (with no signs), and their tightness/looseness are discussed by comparing with Monte Carlo simulations. Large-N limits are taken with the characteristic oscillatory behavior of the formula being preserved.

Maximum relative distance between real rank-two and rank-one tensors

It is shown that the relative distance in Frobenius norm of a real symmetric order-d tensor of rank-two to its best rank-one approximation is upper bounded by sqrt{1-(1-1/d)^{d-1}}. This is achieved by determining the minimal possible ratio between spectral and Frobenius norm for symmetric tensors of border rank two, which equals left( 1-{1}/{d}right) ^{(d-1)/{2}}. These bounds are also verified for arbitrary real rank-two tensors by reducing to the symmetric case.

Practical Criteria for H-Tensors and Their Application

Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. H-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying H-tensors are given in the literature. As an application, several sufficient conditions of the positive definiteness for an even-order real symmetric tensor were obtained. Numerical examples are given to illustrate the effectiveness of the proposed method.

An iterative scheme for identifying the positive semi-definiteness of even-order real symmetric H-tensor

H-tensor plays an important role in identifying the positive (semi-) definiteness of even-order real symmetric tensor. In this paper, we propose an algorithm for identifying general H-tensor and prove the algorithm will terminate within finite iterative steps. As application, we give an iterative method for identifying positive semi-definiteness of the even order symmetric tensor. At last, some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed.

An optimal $ Z $-eigenvalue inclusion interval for a sixth-order tensor and its an application

<abstract><p>An optimal $ Z $-eigenvalue inclusion interval for a sixth-order tensor is presented. As an application, a sufficient condition for the positive definiteness of a sixth-order real symmetric tensor (also a homogeneous polynomial form) is obtained, which is used to judge the asymptotically stability of time-invariant polynomial systems.</p></abstract>

Spectral norm of a symmetric tensor and its computation

We show that the spectral norm of a $d$-mode real or complex symmetric tensor in $n$ variables can be computed by finding the fixed points of the corresponding polynomial map. For a generic complex symmetric tensor the number of fixed points is finite, and we give upper and lower bounds for the number of fixed points. For $n=2$ we show that these fixed points are the roots of a corresponding univariate polynomial of degree at most $(d-1)^2+1$, except certain cases, which are completely analyzed. In particular, for $n=2$ the spectral norm of $d$-symmetric tensor is polynomially computable in $d$ with a given relative precision. For a fixed $n>2$ we show that the spectral norm of a $d$-mode symmetric tensor is polynomially computable in $d$ with a given relative precision with respect to the Hilbert-Schmidt norm of the tensor. These results show that the geometric measure of entanglement of $d$-mode symmetric qunits on $\mathbb{C}^n$ are polynomially computable for a fixed $n$.

$Z$-Eigenvalue Localization Sets for Even Order Tensors and Their Applications

Firstly, a new Gersgorin-type $Z$ -eigenvalue localization set with parameters for even order tensors is presented. As an application, some sufficient conditions for the positive (semi-)definiteness of even order real symmetric tensors are obtained. Secondly, by selecting appropriate parameters an optimal set is obtained and proved to be tighter than some existing results. Thirdly, as another application, new upper bounds for the $Z$ -spectral radius of even order weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results.

How many eigenvalues of a random symmetric tensor are real?

This article answers a question posed by Draisma and Horobet, who asked for a closed formula for the expected number of real eigenvalues of a random real symmetric tensor drawn from the Gaussian distribution relative to the Bombieri norm. This expected value is equal to the expected number of real critical points on the unit sphere of a Kostlan polynomial. We also derive an exact formula for the expected absolute value of the determinant of a matrix from the Gaussian Orthogonal Ensemble.

Signed k-uniform hypergraphs and tensors

A signed k-graph is a k-graph with an incidence function on each vertex and its incident edge. In this paper we extend some concepts of the adjacency and Laplacian tensor from k-graphs to signed k-graphs. Based on the spectral properties of the real symmetric tensor, we investigate the H-spectrum of adjacency and Laplacian tensors for signed k-graphs. Finally the inverse Perron value of any vertex is present. Some bounds for the inverse Perron value are established in terms of some invariants including the number of vertices (edges), the degree sequence.

<inline-formula><tex-math id="M1">\begin{document}$ E $\end{document}</tex-math></inline-formula>-eigenvalue localization sets for tensors

Several existing $Z$-eigenvalue localization sets for tensors are first generalized to $E$-eigenvalue localization sets. And then two tighter $ E$-eigenvalue localization sets for tensors are presented. As applications, a sufficient condition for the positive definiteness of fourth-order real symmetric tensors, a sufficient condition for the positive semi-definiteness of fourth-order real symmetric tensors, and a new upper bound for the $ Z$-spectral radius of weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results.

Some criteria for identifying $${{\mathcal {H}}}$$-tensors and its applications

We present some new criteria for \({{\mathcal {H}}}\)-tensors by constructing different positive diagonal matrices and applying some techniques of inequalities. As an application, some sufficient conditions for the positive definiteness of an even-order real symmetric tensor are derived. Some numerical examples are presented to illustrate the theoretical results.

E-eigenvalue Localization Sets for Fourth-Order Tensors

A Gersgorin-type E-eigenvalue localization set and two Brauer-type E-eigenvalue localization sets for fourth-order tensors are presented. As applications, some sufficient conditions for the positive (semi-)definiteness of fourth-order real symmetric tensors and some upper bounds for the Z-spectral radius of fourth-order weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results.

Some results on Brauer-type and Brualdi-type eigenvalue inclusion sets for tensors

In this paper, we establish the comparison between Brauer-type eigenvalue inclusion set given by Bu et al. (Linear Algebra Appl 512:234–248, 2017) and a Brualdi-type eigenvalue inclusion set given by Bu et al. (Linear Algebra Appl 480:168–175, 2015), and investigate the relationship of some Brauer-type eigenvalue inclusion sets provided by Li et al. (Linear Multilinear Algebra 64:587–601, 2016a) and Bu et al. (2017). In particular, we provide a sufficient condition such that the Brualdi-type eigenvalue inclusion set is tighter than some Brauer-type eigenvalue inclusion sets. Moreover, some sufficient conditions depending on entries of the given tensor for identifying strong $${\mathcal {M}}$$ -tensor and the positive definiteness of an even-order real symmetric tensor are presented. To verify our theoretical results and show their effectiveness, some numerical examples are given.

Formula omitted]-equivariant and translation covariant continuous tensor valuations

The space of continuous, SL(m,C)-equivariant, m≥2, and translation covariant valuations taking values in the space of real symmetric tensors on Cm≅R2m of rank r≥0 is completely described. The classification involves the moment tensor valuation for r≥1 and is analogous to the known classification of the corresponding tensor valuations that are SL(2m,R)-equivariant, although the method of proof cannot be adapted.