The dominant eigenvalue of an essentially nonnegative tensor

Zhang, Liping; Qi, Liqun; Luo, Ziyan; Xu, Yi

doi:10.1002/nla.1880

Cited by 20 publications

(28 citation statements)

References 26 publications

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“…The class of essentially non-negative tensor was introduced in [26]. From the definition, it is clear that any non-negative tensor is essentially non-negative while the converse may not be true in general.…”

Section: Essentially Non-negative Tensormentioning

confidence: 99%

“…As pointed out by one of the referees, one of the important characterizations of the essentially non-negative matrix is the following invariant property: e tA (R n + ) ⊆ R n + for all t ≥ 0 and for all essentially non-negative (n × n) matrix A. Although some interesting log-convexity results were discussed in [26], it is not clear whether the above interesting invariant property can be extended to essentially non-negative tensors or not. One of the key difficulty is that it is not clear how to define an appropriate analog of matrix exponential for the tensor case.…”

Section: Essentially Non-negative Tensormentioning

confidence: 99%

“…Indeed, from the definition, if A is an essentially non-negative tensor, then A + λI is a nonnegative tensor where λ = max 1≤i≤n {|A i,i,...,i |} and I is the identity tensor. Since λ 1 (A + λI) = λ 1 (A) + λ, this suggests that one could develop power method as in [17,18,26] to find the maximum eigenvalue of an essentially non-negative tensor, and establish the convergence of the power method whenever the tensor is further assumed to be irreducible. On the other hand, our current approach shows that finding the maximum eigenvalue of an essentially non-negative tensor is equivalent to solving a semi-definite linear programming problem, and so, can be solved efficiently (for example by interior point method).…”

Section: Remark 33 (Other Approaches For Finding the Maximum Eigenvmentioning

confidence: 99%

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Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

et al. 2013

J Optim Theory Appl

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Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, when the tensor is non-negative in the sense that all of its entries are non-negative, efficient numerical schemes have been proposed to calculate the maximum eigenvalue based on a Perron-Frobenius type theorem for non-negative tensors. In this paper, we consider a new class of tensors called essentially non-negative tensors, which extends the non-negative tensors, and examine the maximum eigenvalue of an essentially non-negative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially non-negative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper as well as lower estimate for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor.

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Section: Essentially Non-negative Tensormentioning

confidence: 99%

Section: Essentially Non-negative Tensormentioning

confidence: 99%

Section: Remark 33 (Other Approaches For Finding the Maximum Eigenvmentioning

confidence: 99%

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Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

et al. 2013

J Optim Theory Appl

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“…Further developments can be found in . Extensions to rectangular nonnegative tensors, essentially nonnegative tensors, copositive tensors, completely positive tensors, and M ‐tensors can be found in .…”

Section: The H‐spectral Theory For Nonnegative Tensorsmentioning

confidence: 99%

A survey on the spectral theory of nonnegative tensors

Chang

Zhang

2013

Numerical Linear Algebra App

107

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This is a survey paper on the recent development of the spectral theory of nonnegative tensors and its applications. After a brief review of the basic definitions on tensors, the H -eigenvalue problem and the Z-eigenvalue problem for tensors are studied separately. To the H -eigenvalue problem for nonnegative tensors, the whole Perron-Frobenius theory for nonnegative matrices is completely extended, while to the Z-eigenvalue problem, there are many distinctions and are studied carefully in details. Numerical methods are also discussed. Three kinds of applications are studied: higher order Markov chains, spectral theory of hypergraphs, and the quantum entanglement.The spectral theory for nonnegative matrices has had a profound impact on both theoretical and applicable mathematics. The centerpiece of this theory resides on the classical Perron-Frobenius theorems as well as some of their important consequences. In this expository article, we intend to give a brief survey to incorporate some of the most recent developments of the various spectral theories for nonnegative tensors along these lines. There is a large volume of published and unpublished work in this field; we apologize in advance if we fail to cite the work of some of our peers as any such oversight is unintentional. SPECTRAL THEORY OF NONNEGATIVE MATRICESA second-order n-dimensional real (or complex) tensor A is the n n real (or complex) matrix A D .a ij /. It can also be viewed as a linear endomorphism on R n (or C n ); hence, the eigenvalue problem for A is a linear problem. In particular, the spectral radius r.A/ of A is defined to be r.A/ D maxfj j j is a real or complex eigenvalue of Ag.According to Gelfand's formula,where jj jj denotes the operator norm. Thus, r.A/ is an intrinsic property of A as it is entirely determined by A itself.An m-order n-dimensional real tensor A consists of n m entries in R:1. r.A/ > 0 is an eigenvalue. 2. There exists a positive vector x 0 > 0, that is, all components of x 0 are positive, such that Ax 0 D r.A/x 0 . 3. (Uniqueness) If is an eigenvalue with a nonnegative eigenvector, then D r.A/. Remarks 1. The notion of symmetric tensor is referred to Definition 4.10 later. In fact, the symmetric assumption on A in [2] in this statement is superfluous. Corresponding to the n homogenous polynomials .Ax m 1 / 1 , , .Ax m 1 / n in n variables x D .x 1 , , x n /, one defines the determinant to be the resultant of these polynomials: det.A/ D Res .Ax m 1 / 1 , .Ax m 1 / n , then the characteristic polynomial becomes . / D det.A I/. Now, we return to eigenvalues for tensors. The proof of Lemma 4.6 in [41] also leads to the following. Lemma 3.23. A nonnegative tensor A is irreducible if and only if the operator T A is semistrongly positive. WD Ax m .kC1/ .Because˛is large enough, the function f A .x/ C˛kxk m becomes convex on R n . For even m, the convergence of the sequence k is guaranteed.

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“…Optimization of the dominant eigenvalue of an essentiallynonnegative, or Metzler, matrix over trace-preserving or fixed-trace diagonal perturbations has been studied [1], [2], as part of a broader effort on the fast eigen-decomposition of these matrices [3]- [5]. In particular, [1] exploits the convexity of the dominant eigenvalue with respect to the diagonal entries [6] to minimize the dominant eigenvalue with respect to trace-preserving diagonal perturbations.…”

Section: Introductionmentioning

confidence: 99%

Dominant eigenvalue minimization with trace preserving diagonal perturbation: Subset design problem

Torres

Roy

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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We study the problem of minimizing the dominant eigenvalue of an essentially-nonnegative matrix with respect to a trace-preserving or fixed-trace diagonal perturbation, in the case where only a subset of the diagonal entries can be perturbed. The spectrum of the perturbed matrix at the optimum is characterized. A constructive algorithm for computing the optimal diagonal trace-preserving perturbation is developed, using the spectral result together with line-sum-symmetrization arguments. A number of graph-theoretic results are developed on the optimal perturbation and how it changes if further entries are contrained, in part using properties of the Perron complement of nonnegative matrices.

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The dominant eigenvalue of an essentially nonnegative tensor

Cited by 20 publications

References 26 publications

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

A survey on the spectral theory of nonnegative tensors

Dominant eigenvalue minimization with trace preserving diagonal perturbation: Subset design problem

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