Irreducible Nonnegative Tensor Research Articles

A power-like method for finding the spectral radius of a weakly irreducible nonnegative symmetric tensor

A power-like method for finding the spectral radius of a weakly irreducible nonnegative symmetric tensor

Some properties concerning Perron vectors of weakly irreducible nonnegative tensors, and their application to rigorous enclosure

We clarify some properties concerning Perron vectors of weakly irreducible nonnegative (WIN) tensors, and multi-linear systems with non-homogeneous left-hand side whose solutions are sub-vectors of the Perron vectors. Based on the properties, we propose an algorithm for computing interval vectors containing the Perron vectors. This algorithm is applicable to all the WIN tensors, whereas the algorithms previously developed by the author are applicable only to a subclass of the WIN tensors. Numerical results show efficiency of the proposed algorithm.

An algorithm for the spectral radius of weakly essentially irreducible nonnegative tensors

An algorithm for the spectral radius of weakly essentially irreducible nonnegative tensors

A note on Newton–Noda iteration for computing the Perron pair of a weakly irreducible nonnegative tensor

The Newton–Noda iteration (NNI) can be used to compute the Perron pair of a weakly irreducible nonnegative tensor. The method requires the selection of a positive parameter θk in the kth iteration. A practical procedure for determining the parameter θk is the halving procedure, starting with θk=1. The NNI has been shown to be globally and quadratically convergent if the sequence {θk} is bounded below by a positive constant. In this note, we prove the global convergence of NNI (with θk determined by the halving procedure) without assuming that {θk} is bounded below by a positive constant. We can then see that θk=1 for all k sufficiently large and the convergence of NNI is quadratic.

Noda iteration for computing generalized tensor eigenpairs

In this paper, we propose the tensor Noda iteration (NI) and its inexact version for solving the eigenvalue problem of a particular class of tensor pairs called generalized M-tensor pairs. A generalized M-tensor pair consists of a weakly irreducible nonnegative tensor and a nonsingular M-tensor within a linear combination. It is shown that any generalized M-tensor pair admits a unique positive generalized eigenvalue with a positive eigenvector. A modified tensor Noda iteration (MTNI) is developed for extending the Noda iteration for nonnegative matrix eigenproblems. In addition, the inexact generalized tensor Noda iteration method (IGTNI) and the generalized Newton-Noda iteration method (GNNI) are also introduced for more efficient implementations and faster convergence. Under a mild assumption on the initial values, the convergence of these algorithms is guaranteed. The efficiency of these algorithms is illustrated by numerical experiments.

Power Function Method for Finding the Spectral Radius of Weakly Irreducible Nonnegative Tensors

Since the eigenvalue problem of real supersymmetric tensors was proposed, there have been many results regarding the numerical algorithms for computing the spectral radius of nonnegative tensors, among which the NQZ method is the most studied. However, the NQZ method is only suitable for calculating the spectral radius of a special weakly primitive tensor, or a weakly irreducible primitive tensor that satisfies certain conditions. In this paper, by means of diagonal similarrity transformation of tensors, we construct a numerical algorithm for computing the spectral radius of nonnegative tensors with the aid of power functions. This algorithm is suitable for the calculation of the spectral radius of all weakly irreducible nonnegative tensors. Furthermore, it is efficient and can be widely applied.

Perron–Frobenius theory for some classes of nonnegative tensors in the max algebra

An analog of Perron–Frobenius theory is proposed for some classes of nonnegative tensors in the max algebra. In the first part some important characterizations of nonnegative matrices can be extended to nonnegative tensors over max algebra, especially the Perron–Frobenius theorem for weakly irreducible nonnegative tensors and the Collatz–Wielandt minimax theorem for nonnegative tensors. Then, in the second part, an iterative method is proposed for finding the largest max eigenvalue of a nonnegative tensor based on diagonal similar tensors. The iterative method is convergent for weakly irreducible nonnegative tensors. Some numerical results are provided to illustrate the efficiency of the iterative method.

Fast verified computation for positive solutions to [formula omitted]-tensor multi-linear systems and Perron vectors of a kind of weakly irreducible nonnegative tensors

Two fast numerical algorithms are proposed for computing interval vectors containing positive solutions to M-tensor multi-linear systems. The first algorithm involves only two tensor–vector multiplications. The second algorithm is iterative one, and generally gives interval vectors narrower than those by the first algorithm. We also develop two verification algorithms for Perron vectors of a kind of weakly irreducible nonnegative tensors, which we call slightly positive tensors. The first and second algorithms have properties similar to those of the two algorithms for the solutions to the M-tensor systems. We clarify relations between slightly positive tensors and other tensor classes. Numerical results show efficiency of the algorithms.

An iterative method for finding the spectral radius of an irreducible nonnegative tensor

In this paper, an iterative method is proposed for finding the spectral radius of an irreducible nonnegative tensor based on diagonal similar tensors. The iterative method is convergent for irreducible nonnegative tensors and has an explicit linear convergence rate for generalized weakly positive tensors. Some numerical results are provided to illustrate the efficiency of the iterative method.

A Newton-type algorithm for the tensor eigenvalue complementarity problem and some applications

We focus on establishing an algorithm to solve the tensor eigenvalue complementarity problem (TEiCP), and we have two contributions in this paper. First, a smoothing Newton-type algorithm is proposed for the TEiCP based on the CHKS smoothing function. Its global convergence is established under some mild conditions. Numerical experiments are reported to show that the proposed algorithm is efficient and could detect more solutions than some existing methods. Second, we apply the proposed algorithm to solve the eigenvalue problem of nonnegative tensors. We analyze the relationship between the TEiCP and the H-eigenpair and Z-eigenpair problems of an irreducible nonnegative tensor. We show that the TEiCP with an irreducible nonnegative tensor and unit tensor has a unique solution, which is just the unique positive H-eigenpair of the irreducible nonnegative tensor. We also show that the solution set of the TEiCP with an irreducible nonnegative tensor and identity tensor is nonempty and its solutions are positive. Moreover, we can obtain positive Z-eigenpairs of the irreducible nonnegative tensor from these solutions. Finally, we also apply the proposed algorithm to find the unique positive H-eigenpair and a positive Z-eigenpair of an irreducible nonnegative tensor; the numerical results indicate its efficiency and promising performance.

A note on Banach’s results concerning homogeneous polynomials associated with nonnegative tensors

A classic result of Banach states that the supreme of a multivariate homogenous polynomial is equivalent to that of its associated symmetric multilinear form over unit balls. Using the language of higher-order tensors in finite-dimensional spaces, this means that for a symmetric tensor, its largest singular value is in fact equivalent to the largest magnitude of its eigenvalues. This note strengthens Banach’s results on nonnegative higher-order tensors. It is shown that for a symmetric nonnegative irreducible tensor: (1) any singular value admitting a positive singular vector tuple must be an eigenvalue, where the singular vectors in the tuple must be equal to each other; i.e., they amount to an eigenvector; (2) when the order is odd, any nonnegative singular vector tuple corresponding to the largest singular value must also boil down to a positive eigenvector. These results give the flexibility of directly solving tensor eigenvalue problems via solving tensor singular value problems.

Some bounds for the Z-eigenpair of nonnegative tensors

Tensor eigenvalue problem is one of important research topics in tensor theory. In this manuscript, we consider the properties of Z-eigenpair of irreducible nonnegative tensors. By estimating the ratio of the smallest and largest components of a positive Z-eigenvector for a nonnegative tensor, we present some bounds for the eigenvector and Z-spectral radius of an irreducible and weakly symmetric nonnegative tensor. The proposed bounds complement and extend some existing results. Finally, several examples are given to show that such a bound is different from one given in the literature.

A convergent Newton algorithm for computing Z-eigenvalues of an almost nonnegative irreducible tensor

ABSTRACTIn this paper, we compute Z-eigenvalues of a class of tensors by studying the properties of semi-symmetric tensor. We prove that is identical to zero if and only if , where is the associated semi-symmetric tensor of . Based on the semi-symmetric property, an almost nonnegative irreducible tensor is defined. And we use Newton method to compute Z-eigenvalues of this kind of tensor. The convergence of the proposed algorithm can be guaranteed. Numerical results are reported to illustrate the efficiency of our algorithm.

Perron vector analysis for irreducible nonnegative tensors and its applications

<p style='text-indent:20px;'>In this paper, we analyse the Perron vector of an irreducible nonnegative tensor, and present some lower and upper bounds for the ratio of the smallest and largest entries of a Perron vector based on some new techniques, which always improve the existing ones. Applying these new ratio results, we first refine two-sided bounds for the spectral radius of an irreducible nonnegative tensor. In particular, for the matrix case, the new bounds also improve the corresponding ones. Second, we provide a new Ky Fan type theorem, which improves the existing one. Third, we refine the perturbation bound for the spectral radii of nonnegative tensors, from which one may derive a comparison theorem for spectral radii of nonnegative tensors. Numerical examples are given to show the efficiency of the theoretical results.

Symmetry of eigenvalues of Sylvester matrices and tensors

In this article, the index of imprimitivity of an irreducible nonnegative matrix in the famous Perron-Frobenius 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.

A homotopy method for computing the largest eigenvalue of an irreducible nonnegative tensor

In this paper we propose a homotopy method to compute the largest eigenvalue and a corresponding eigenvector of a nonnegative tensor. We prove that it converges to the desired eigenpair when the tensor is irreducible. We also implement the method using an prediction-correction approach for path following. Some numerical results are provided to illustrate the efficiency of the method.

Some Ostrowski-type bound estimations of spectral radius for weakly irreducible nonnegative tensors

ABSTRACT In this paper, by characterizing the ratio of the smallest and largest values of a Perron vector, we propose some Ostrowski-type bound estimations for the spectral radius of weakly irreducible nonnegative tensors which improve the existing results. Based on Brualdi-type eigenvalue inclusion sets and Gershgorin eigenvalue inclusion sets, we establish some new Ky Fan-type theorems for weakly irreducible nonnegative tensors.

The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs

Let A \mathcal {A} be a weakly irreducible nonnegative tensor with spectral radius ρ ( A ) \rho (\mathcal {A}) . Let D \mathfrak {D} (resp., D ( 0 ) \mathfrak {D}^{(0)} ) be the set of normalized diagonal matrices arising from the eigenvectors of A \mathcal {A} corresponding to the eigenvalues with modulus ρ ( A ) \rho (\mathcal {A}) (resp., the eigenvalue ρ ( A ) \rho (\mathcal {A}) ). It is shown that D \mathfrak {D} is an abelian group containing D ( 0 ) \mathfrak {D}^{(0)} as a subgroup, which acts transitively on the set { e i 2 π j ℓ A : j = 0 , 1 , … , ℓ − 1 } \{e^{\mathbf {i}\frac {2 \pi j}{\ell }}\mathcal {A}:j =0,1, \ldots ,\ell -1\} , where | D / D ( 0 ) | = ℓ |\mathfrak {D}/\mathfrak {D}^{(0)}|=\ell and D ( 0 ) \mathfrak {D}^{(0)} is the stabilizer of A \mathcal {A} . The spectral symmetry of A \mathcal {A} is characterized by the group D / D ( 0 ) \mathfrak {D}/\mathfrak {D}^{(0)} , and A \mathcal {A} is called spectral ℓ \ell -symmetric. We obtain structural information about A \mathcal {A} by analyzing the property of D \mathfrak {D} , and especially for connected hypergraphs we get some results on the edge distribution and coloring. If moreover A \mathcal {A} is symmetric, we prove that A \mathcal {A} is spectral ℓ \ell -symmetric if and only if it is ( m , ℓ ) (m,\ell ) -colorable. We characterize the spectral ℓ \ell -symmetry of a tensor by using its generalized traces, and we show that for an arbitrary integer m ≥ 3 m \ge 3 and each positive integer ℓ \ell with ℓ ∣ m \ell \mid m , there always exists an m m -uniform hypergraph G G such that G G is spectral ℓ \ell -symmetric.

Some bounds for H-eigenpairs and Z-eigenpairs of a tensor

In this paper, we consider the H -eigenpairs and Z -eigenpairs of a tensor. By estimating the ratio of the smallest and largest entries in the Perron vector, we present some sharper bounds for the H -spectral radius of a nonnegative irreducible tensor. Similarly, we also obtain some Z -spectral radius bounds of an irreducible weakly symmetric tensor. In addition, by using the technique of matricizing, several bounds for the Z -spectrum are derived. These proposed bounds improve some existing ones, and some numerical examples are given to show the theoretical results.

Continuation methods for computing Z-/H-eigenpairs of nonnegative tensors

In this paper, a homotopy continuation method for the computation of nonnegative Z-/H-eigenpairs of a nonnegative tensor is presented. We show that the homotopy continuation method is guaranteed to compute a nonnegative eigenpair. Additionally, using degree analysis, we show that the number of positive Z-eigenpairs of an irreducible nonnegative tensor is odd. A novel homotopy continuation method is proposed to compute an odd number of positive Z-eigenpairs, and some numerical results are presented.