Contents
Chapter 1 Introduction
Eigenvalues problems of higher order tensors
Related polynomial optimization problems
Applications
Spectral properties and algorithms: a literature review
The organization of this book
Chapter 2 Spectral Properties of H-eigenvalue Problems of a Nonnegative Square Tensor
Introduction
From nonnegative matrices to nonnegative tensors
Nonnegative irreducible tensors and primitive tensors
Perron-Frobenius theorem for nonnegative tensors and related results
Geometric simplicity
The Collatz-Wielandt formula
Other related results
Some properties for nonnegative weakly irreducible tensors
Weak irreducibility
Generalization from nonnegative irreducible tensors to nonnegative
weakly irreducible tensors
Chapter 3 Algorithms for Finding the Largest H-eigenvalue of a
Nonnegative Square Tensor
Introduction
A polynomial-time approach for computing the spectral radius
Two algorithms and convergence analysis
An inexact power-type algorithm
A one-step inner iteration power-type algorithm
Numerical experiments
Experiments on the polynomial-time approach
Experiments on the inexact algorithms
Chapter 4 Spectral Properties and Algorithms of H-singular Value Problems of a Nonnegative Rectangular Tensor
Introduction
Preliminaries
Some conclusions concerning the singular value of a nonnegative
rectangular tensor
Primitivity and the convergence of the CQZ method for ˉnding the
largest singular value of a nonnegative rectangular tensor
Algorithms for computing the largest singular value of a nonnegative
rectangular tensor
A polynomial-time algorithm
An inexact algorithm
A solving method of the largest singular value based on the symmetric
embedding
Singular values of a rectangular tensor
Singular values of a general tensor
Chapter 5 Properties and Algorithms of Z-eigenvalue Problems of a Symmetric Tensor
Introduction
Some spectral properties
The Collatz-Wielandt formula
Bounds on the Z-spectral radius
The reformulation problem and the no duality gap result
The reformulation problem
Dual problem of RP
No duality gap result
Relaxations and algorithms
Nuclear norm regularized convex relaxation of RP and the proximal
augmented Lagrangian method
The truncated nuclear norm regularization and the approximation
Alternating least eigenvalue method for ˉnding a global minima
Numerical results
Chapter 6 Solving Biquadratic Optimization Problems via
Semideˉnite Relaxation
Introduction
Semideˉnite relaxations and approximate bounds
The nonnegative case
The square-free case and the positive semideˉnite case
Approximation al
內容試閱：
Chapter 1
Introduction
Tensor is a hot topic in the past decade Nowadays， many real world problems can be modeled as tensor problems， just to name a few: signal processing[36， 101]， data analysis[17， 32]， chemometrics[12， 13， 111]， hypergraph theory[34， 66]， diusion magnetic resonance imaging MRI [3， 6， 44]， quantum entanglement in quantum physics [35]， higher order Markov chains[107] and elastic materials analysis[50， 77] Speciˉcally， a tensor can be viewed as a multiarray: if a vector a = [a1; ; an]T 2 Rn is a one- way-array， a matrix A = aij 2 Rn×n is a two-way-array， then a tensor of order m dimension n: A = ai1 in is a multiarray Here \order" refers to the number of indices of each entry of A， eg， a matrix is a tensor of order 2 Just imagine we have l matrices of size m × n By stacking them up together， we get a cube， which is a l × m × n tensor If we merge k such tensors together， then we get a hyper-cube， which is a k × l × m × n tensor， and so on Particularly， if k = l = m = n = ， ie， all the dimensions of a tensor are the same， then the tensor is called a \square" tensor， which generalizes the \square" matrix and is an important type of tensors studied in this book
We are particularly interested in the spectral properties of eigenvalue problems of higher order tensor and related polynomial optimization problems In this book， the main concerns are the following ˉve topics:
Spectral properties and algorithms of H-eigenvalue problems;
Spectral properties and algorithms of H-singular value problems;
Properties and algorithms of Z-eigenvalue problems;
Approximation methods of biquadratic optimization problems;
Approximation methods of trilinear optimization problems
We will introduce the related concepts and problems in the rest of this chapter
1.1 Eigenvalues problems of higher order tensors
It is well known that eigenvalues play an important role in matrix theory For a matrix A 2 Cn×n， if there exists a pair ; x 2 C × Cn with x 6= 0 such that
Ax = x;
then is called an eigenvalue of A， and x is an eigenvector corresponding to To study tensor problems， one may naturally ask a question: can we deˉne eigenvalues and eigenvectors on tensors? The answer is positive In 2005， the concept of eigen- values and eigenvectors of a symmetric tensor with order even was introduced by Qi[119]
To be more speciˉc， let A = ai1 im be an order m n-th dimensional real square tensor If there is a complex number and a nonzero complex vector x that are solutions of the following homogeneous polynomial equations:
Axm1 = x[m1]; 11
then is called an eigenvalue of A and x the eigenvector of A associated with In problem 11， Axm1 and x[m1] are vectors， whose i-th entries are given by
respectively If and x are restricted in the real ˉeld， then ; x is called an H- eigenpaire If an eigenvalue is not an H-eigenvalue， we call it an N-eigenvalue of A Besides the H-eigenvalues， Qi deˉned the Z-eigenvalues[119]: a real number and a real vector x are called Z-eigenvalue of A and a Z-eigenvector of A associated with the Z-eigenvalue respectively， if they are solutions of the following system:
1.2
Here k . k is the Euclidean norm If and x are complex， then they are called E-eiganvalue and E-eigenvector
In the same year， Lim[94] independently deˉned eigenvalues for general real ten- sors in the real ˉeld In his work， the lk eigenvalues are H-eigenvalues， while the l2 eigenvalues of tensors are the Z-eigenvalues Note that in the case of m = 2， both the H-eigenvalue and Z-eigenvalue collapse to the eigenvalues of a matrix
The concept of singular values and singular vectors can be generalized to higher order tensors as well Let us recall in the real case that for a matrix A 2 Rn1×n2 ， if there exists a triple ; x1; x2 2 R × Rn1 × Rn2 such that
Ax2 = x1 and ATx1 = x2;
then is called a singular value of A， and x1 and x2 are the left and right singular vector corresponding to ， respectively
For higher order tensors， for example， consider a third order tensor A2Rn1×n2×n3 The corresponding singular value problem can be deˉned as[94]
where
For other higher order tensors， the deˉnition is similar
There are other kinds of eigenvaluessingular values for other types of tensors Assume that p; q; n1 and n2 are positive integers， and n1; n2 2 We call A = ai1 ipj1 jq ， where ai1 ipj1 jq 2 R， for ik = 1; ; n1， k = 1; ; p， and jk = 1; ; n2， k = 1; ; q， a real p; q-th order n1×n2 dimensional rectangular tensor， or simply a real rectangular tensor When p = q = 1， A is simply a real n1 × n2 rectangular matrix Denote M = p + q If there is a complex number and two nonzero complex vectors x and y such that:
Axp1yq = x[M1];
Axpyq1 = y[M1];
where Axp1yq is a vector in Rn1 whose i-th entry is given by
and Axpyq1 is a vector in Rn2 whose j-th entry is given by
then is called the singular value of A and x; y are the left and right eigenvectors of A
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