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

Hu, Shenglong; Li, Guoyin; Qi, Liqun; Song, Yisheng

doi:10.1007/s10957-013-0293-9

Cited by 34 publications

(36 citation statements)

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“…Then,

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is an even‐order essentially nonnegative tensor. It follows from the study by Hu et al (Proposition 3.1) that f is SOS (see also a more recent study), and hence, the desired result holds. (2) Initial step.…”

Section: Maximum H‐eigenvalue Of a Symmetric W‐tensormentioning

confidence: 65%

“…In view of the importance of eigenvalues of tensors, many researchers have devoted themselves to the study of numerical methods for eigenvalues of high‐order tensors . Cui et al proposed a sophisticated Jacobian semidefinite relaxation method, which computes all of the real eigenvalues of a small symmetric tensor.…”

Section: Introductionmentioning

confidence: 99%

“…Liu et al and Zhou et al improved the NQZ method so that the refined algorithm is always convergent. Recently, as a more general class than nonnegative tensors, the essentially nonnegative tensors were studied . Hu et al showed that the maximum H ‐eigenvalue of an even‐order essentially nonnegative tensor can be found by solving a polynomial optimization problem, which is equivalently reformulated as a semidefinite programming problem.…”

Section: Introductionmentioning

confidence: 99%

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A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test

Chen

et al. 2017

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SummaryFinding the maximum eigenvalue of a symmetric tensor is an important topic in tensor computation and numerical multilinear algebra. In this paper, we introduce a new class of structured tensors called W-tensors, which not only extends the well-studied nonnegative tensors by allowing negative entries but also covers several important tensors arising naturally from spectral hypergraph theory. We then show that finding the maximum H-eigenvalue of an even-order symmetric W-tensor is equivalent to solving a structured semidefinite program and hence can be validated in polynomial time. This yields a highly efficient semidefinite program algorithm for computing the maximum H-eigenvalue of W-tensors and is based on a new structured sums-of-squares decomposition result for a nonnegative polynomial induced by W-tensors. Numerical experiments illustrate that the proposed algorithm can successfully find the maximum H-eigenvalue of W-tensors with dimension up to 10,000, subject to machine precision. As applications, we provide a polynomial time algorithm for computing the maximum H-eigenvalues of large-size Laplacian tensors of hyperstars and hypertrees, where the algorithm can be up to 13 times faster than the state-of-the-art numerical method introduced by Ng, Qi, and Zhou in 2009. Finally, we also show that the proposed algorithm can be used to test the copositivity of a multivariate form associated with symmetric extended Z-tensors, whose order may be even or odd.

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“…Then,

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Section: Maximum H‐eigenvalue Of a Symmetric W‐tensormentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test

Chen

et al. 2017

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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

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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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“…[8] In the literature, there are research work which one can exactly compute the maximum real eigenvalue for irreducible nonnegative tensors, [9][10][11] nonnegative tensors, [12][13][14] essentially positive tensors, [15] weakly positive tensors [16] and essentially nonnegative tensors. [17] In particular, Hu et al [17] provided upper and lower estimates for the maximum eigenvalue of general symmetric tensors using the polynomial optimization techniques.…”

Section: Introductionmentioning

confidence: 99%

The location of H-eigenvalues of real even order symmetry tensors

Jin

2016

Linear and Multilinear Algebra

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In this article, we derive a new characterization of B-tensors, which allows us to establish an interval containing all the H-eigenvalues of real even order symmetry tensors. This interval, called B-interval, has a nature analogous as the Gerschgorin interval of H-eigenvalues of real even order symmetric tensors and provides some supplement information on the Gerschgorin interval. We further consider a class of tensors whose off-diagonal entries have restricted dispersion, and prove that its B-interval is contained in the Gerschgorin interval. We also obtain a lower bound of the H-eigenvalues determined only by the minimum diagonal element and the minimum off-diagonal element of this class of tensors, which, in its turn, can be used to check the positive definiteness. Finally, we construct a new interval, which is proved that the new interval is better than both Gerschgorin interval and B-interval for an arbitrary real even order symmetry tensor. ARTICLE HISTORY

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

Cited by 34 publications

References 34 publications

A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test

A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test

A survey on the spectral theory of nonnegative tensors

The location of H-eigenvalues of real even order symmetry tensors

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