Computing hypermatrix spectra with the Poisson product formula

doi:10.1080/03081087.2014.910207

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Computing hypermatrix spectra with the Poisson product formula

Journal Article•DOI•

Computing hypermatrix spectra with the Poisson product formula

Joshua Cooper¹, Aaron Dutle¹•Institutions (1)

University of South Carolina¹

04 May 2015-Linear & Multilinear Algebra (Taylor & Francis)-Vol. 63, Iss: 5, pp 956-970

TL;DR: In this article, the spectrum of the all-one hypermatrix was computed using the Poisson product formula, which includes a complete description of the eigenvalues' multiplicities, a seemingly elusive aspect of the spectral theory of tensors.

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Abstract: We compute the spectrum of the ‘all ones’ hypermatrix using the Poisson product formula. This computation includes a complete description of the eigenvalues’ multiplicities, a seemingly elusive aspect of the spectral theory of tensors. We also give a distributional picture of the spectrum as a point-set in the complex plane. Finally, we use the technique to analyse the spectrum of ‘sunflower hypergraphs’, a class that has played a prominent role in extremal hypergraph theory.

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Journal Article•DOI•

A survey on the spectral theory of nonnegative tensors

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Kung-Ching Chang¹, Liqun Qi², Tan Zhang³•Institutions (3)

Peking University¹, Hong Kong Polytechnic University², Murray State University³

01 Dec 2013-Numerical Linear Algebra With Applications

TL;DR: This is a survey paper on the recent development of the spectral theory of nonnegative tensors and its applications: higher order Markov chains, spectral Theory of hypergraphs, and the quantum entanglement.

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Abstract: SUMMARY 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. Copyright © 2013 John Wiley & Sons, Ltd.

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122 citations

Journal Article•DOI•

Brualdi-type eigenvalue inclusion sets of tensors

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Changjiang Bu¹, Yuanpeng Wei¹, Lizhu Sun², Jiang Zhou¹•Institutions (2)

Harbin Engineering University¹, Harbin Institute of Technology²

01 Sep 2015-Linear Algebra and its Applications

TL;DR: By using digraphs of tensors, the authors gave Brualdi-type eigenvalue inclusion sets for tensors and also gave some applications of their result to nonsingularity and positive definiteness.

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40 citations

Journal Article•DOI•

Some characterizations of M-tensors via digraphs

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Jiang Zhou¹, Lizhu Sun², Yuanpeng Wei¹, Changjiang Bu¹•Institutions (2)

Harbin Engineering University¹, Harbin Institute of Technology²

15 Apr 2016-Linear Algebra and its Applications

TL;DR: For a tensor A = (a i 1 ⋯ i m ) ∈ C n × ⋩ × n, the associated digraph of A has the vertex set V ( A ) = { 1, …, n }, and the arc set of Γ A is E ( A) = { ( i, j ) | a i i 2 ⋌ i m ≠ 0, j ∈ { i 2, …, i m } ≠ { i, …-, i m ∈ [ i,, j ] ≠

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11 citations

Journal Article•DOI•

Adjacency spectra of random and complete hypergraphs

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Joshua Cooper¹•Institutions (1)

University of South Carolina¹

01 Jul 2020-Linear Algebra and its Applications

TL;DR: In this paper, it was shown that the spectral radius of the complete k-uniform hypergraph for k = 2, 3 is close to that of an appropriately scaled all-ones hypermatrix.

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10 citations

Cites background from "Computing hypermatrix spectra with ..."

...completely described in [7], differing only in the vanishing fraction of entries corresponding to repeated indices (i....
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...We employ the following result from [7]:...
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Journal Article•DOI•

Laplacian and signless Laplacian Z-eigenvalues of uniform hypergraphs

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Changjiang Bu¹, Yamin Fan¹, Jiang Zhou¹•Institutions (1)

Harbin Engineering University¹

17 May 2016-Frontiers of Mathematics in China

TL;DR: In this article, it was shown that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless LaplACian Z-eigenvalues.

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Abstract: We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest (signless) Laplacian Z-eigenvalue of a hypergraph. For a k-uniform hyperstar with d edges (2d ⩾ k ⩾ 3), we show that its largest (signless) Laplacian Z-eigenvalue is d.

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9 citations

Cites background from "Computing hypermatrix spectra with ..."

...Recently, the research on spectral theory of hypergraphs has attracted extensive attention [1–3,5–7,9,13,16,17]....
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References

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Book•

Discriminants, Resultants, and Multidimensional Determinants

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Izrailʹ Moiseevich Gelʹfand, Mikhail Kapranov, Andrei Zelevinsky

10 May 2013

TL;DR: The Cayley method of studying discriminants was used by Cayley as discussed by the authors to study the Cayley Method of Discriminants and Resultants for Polynomials in One Variable and for forms in Several Variables.

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Abstract: Preface.- Introduction.- General Discriminants and Resultants.- Projective Dual Varieties and General Discriminants.- The Cayley Method of Studying Discriminants.- Associated Varieties and General Resultants.- Chow Varieties.- Toric Varieties.- Newton Polytopes and Chow Polytopes.- Triangulations and Secondary Polytopes.- A-Resultants and Chow Polytopes of Toric Varieties.- A-Discriminants.- Principal A-Discriminants.- Regular A-Determinants and A-Discriminants.- Classical Discriminants and Resultants.- Discriminants and Resultants for Polynomials in One Variable.- Discriminants and Resultants for Forms in Several Variables.- Hyperdeterminants.- Appendix A. Determinants.- Appendix B. A. Cayley: On the Theory of Elimination.- Bibliography.- Notes and References.- List of Notation.- Index

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2,306 citations

"Computing hypermatrix spectra with ..." refers background in this paper

...It is referred to as the “Poisson product formula,” and can be found in any of [3, 6, 10]....
[...]
...Using the multipolynomial resultant [3, 6], one can determine when a collection of n homogeneous equations in n variables has a non-trivial solution over an algebraically closed field....
[...]

Book•

Using Algebraic Geometry

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David A. Cox, John Little, Donal O'Shea

01 Jan 1998

TL;DR: The Berlekamp-Massey-Sakata Decoding Algorithm is used for solving Polynomial Equations and for computations in Local Rings.

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Abstract: Introduction.- Solving Polynomial Equations.- Resultants.- Computation in Local Rings.- Modules.- Free Resolutions.- Polytopes, Resultants, and Equations.- Integer Programming, Combinatorics, and Splines.- Algebraic Coding Theory.- The Berlekamp-Massey-Sakata Decoding Algorithm.

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1,726 citations

Book•

Random walks and electric networks

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Peter G. Doyle¹, J. Laurie Snell¹•Institutions (1)

Dartmouth College¹

01 Dec 1984

TL;DR: The goal will be to interpret Polya’s beautiful theorem that a random walker on an infinite street network in d-dimensional space is bound to return to the starting point when d = 2, but has a positive probability of escaping to infinity without returning to the Starting Point when d ≥ 3, and to prove the theorem using techniques from classical electrical theory.

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Abstract: Probability theory, like much of mathematics, is indebted to physics as a source of problems and intuition for solving these problems. Unfortunately, the level of abstraction of current mathematics often makes it difficult for anyone but an expert to appreciate this fact. Random Walks and Electric Networks looks at the interplay of physics and mathematics in terms of an example the relation between elementary electric network theory and random walks where the mathematics involved is at the college level.

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1,632 citations

Additional excerpts

...[14]), that such a walk will end at a + bi , a, b ∈ Z, with probability 4−n ( n (n + a + b)/2 )( n (n − a + b)/2 )...
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Journal Article•DOI•

Eigenvalues of a real supersymmetric tensor

[...]

Liqun Qi¹•Institutions (1)

Hong Kong Polytechnic University¹

01 Dec 2005-Journal of Symbolic Computation

TL;DR: It is shown that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvaluesare roots of another one- dimensional polynomials associated with the symmetric hyperdeterminant.

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1,378 citations

"Computing hypermatrix spectra with ..." refers background in this paper

...The study of the spectrum of hypermatrices (also termed ‘tensors’ in the literature) has seen rapid development in the few years since its origin1 in the works of Qi and Lim [2,3] in 2005....
[...]
...Linear and Multilinear Algebra, 2015 Vol. 63, No. 5, 956–970, http://dx.doi.org/10.1080/03081087.2014.910207 Computing hypermatrix spectra with the Poisson product formula Joshua Cooper and Aaron Dutle∗ Department of Mathematics, University of South Carolina, Columbia, SC, USA Communicated by L.-H. Lim (Received 22 July 2013; accepted 25 March 2014) We compute the spectrum of the ‘all ones’ hypermatrix using the Poisson product formula....
[...]
...Therefore, cov(Yr ) = 12 I , and, by the Multidimensional Central Limit Theorem (see, e.g. [15]), Xr√ n D−→ N (0, 1)....
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...Another approach that was initiated in [3], though remaining relatively unexplored, is an algebraic one....
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...Introduction The study of the spectrum of hypermatrices (also termed ‘tensors’ in the literature) has seen rapid development in the few years since its origin1 in the works of Qi and Lim [2,3] in 2005....
[...]

Proceedings Article•DOI•

Singular values and eigenvalues of tensors: a variational approach

[...]

Lek-Heng Lim¹•Institutions (1)

Stanford University¹

13 Dec 2005

TL;DR: A theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigen values is proposed.

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Abstract: We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly useful in generalizing certain areas where the spectral theory of matrices has traditionally played an important role. For illustration, we will discuss a multilinear generalization of the Perron-Frobenius theorem

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850 citations

"Computing hypermatrix spectra with ..." refers background in this paper

...The study of the spectrum of hypermatrices (also termed ‘tensors’ in the literature) has seen rapid development in the few years since its origin1 in the works of Qi and Lim [2,3] in 2005....
[...]
...Linear and Multilinear Algebra, 2015 Vol. 63, No. 5, 956–970, http://dx.doi.org/10.1080/03081087.2014.910207 Computing hypermatrix spectra with the Poisson product formula Joshua Cooper and Aaron Dutle∗ Department of Mathematics, University of South Carolina, Columbia, SC, USA Communicated by L.-H. Lim (Received 22 July 2013; accepted 25 March 2014) We compute the spectrum of the ‘all ones’ hypermatrix using the Poisson product formula....
[...]
...Therefore, cov(Yr ) = 12 I , and, by the Multidimensional Central Limit Theorem (see, e.g. [15]), Xr√ n D−→ N (0, 1)....
[...]
...Note that Fi, j = λx2 i, j for every i ∈ [n] and all j ∈ [2], which is the set of equations used to define the characteristic polynomial of the hypermatrix with all zero entries....
[...]
...Introduction The study of the spectrum of hypermatrices (also termed ‘tensors’ in the literature) has seen rapid development in the few years since its origin1 in the works of Qi and Lim [2,3] in 2005....
[...]