In this paper, we present an eigenvalue method for testing positive definiteness of a multivariate form. This problem plays an important role in the stability study of nonlinear autonomous systems ...via Lyapunov's direct method in automatic control. At first we apply the D'Andrea-Dickenstein version of the classical Macaulay formulas of the resultant to compute the symmetric hyperdeterminant of an even order supersymmetric tensor. By using the supersymmetry property, we give detailed computation procedures for the Bezoutians and specified ordering of monomials in this approach. We then use these formulas to calculate the characteristic polynomial of a fourth order three dimensional supersymmetric tensor and give an eigenvalue method for testing positive definiteness of a quartic form of three variables. Some numerical results of this method are reported.
Eigenvalues and invariants of tensors Qi, Liqun
Journal of mathematical analysis and applications,
01/2007, Letnik:
325, Številka:
2
Journal Article
Recenzirano
Odprti dostop
A tensor is represented by a supermatrix under a co-ordinate system. In this paper, we define E-eigenvalues and E-eigenvectors for tensors and supermatrices. By the resultant theory, we define the ...E-characteristic polynomial of a tensor. An E-eigenvalue of a tensor is a root of the E-characteristic polynomial. In the regular case, a complex number is an E-eigenvalue if and only if it is a root of the E-characteristic polynomial. We convert the E-characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor, i.e., they are invariant under co-ordinate system changes. We call them principal invariants of that tensor. The maximum number of principal invariants of
mth order
n-dimensional tensors is a function of
m and
n. We denote it by
d
(
m
,
n
)
and show that
d
(
1
,
n
)
=
1
,
d
(
2
,
n
)
=
n
,
d
(
m
,
2
)
=
m
for
m
⩾
3
and
d
(
m
,
n
)
⩽
m
n
−
1
+
⋯
+
m
for
m
,
n
⩾
3
. We also define the rank of a tensor. All real eigenvectors associated with nonzero E-eigenvalues are in a subspace with dimension equal to its rank.
Finding the sparsest solutions to a tensor complementarity problem is generally NP-hard due to the nonconvexity and noncontinuity of the involved
ℓ
0
norm. In this paper, a special type of tensor ...complementarity problems with
Z
-tensors has been considered. Under some mild conditions, we show that to pursuit the sparsest solutions is equivalent to solving polynomial programming with a linear objective function. The involved conditions guarantee the desired exact relaxation and also allow to achieve a global optimal solution to the relaxed nonconvex polynomial programming problem. Particularly, in comparison to existing exact relaxation conditions, such as RIP-type ones, our proposed conditions are easy to verify.
The class of quaternion matrix optimization (QMO) problems, with quaternion matrices as decision variables, has been widely used in color image processing and other engineering areas in recent years. ...However, optimization theory for QMO is far from adequate. The main objective of this paper is to provide necessary theoretical foundations on optimality analysis, in order to enrich the contents of optimization theory and to pave way for the design of efficient numerical algorithms as well. We achieve this goal by conducting a thorough study on the first-order and second-order (sub)differentiation of real-valued functions in quaternion matrices, with a newly introduced operation called R-product as the key tool for our calculus. Combining with the classical optimization theory, we establish the first-order and the second-order optimality analysis for QMO. Particular treatments on convex functions, the
ℓ
0
-norm and the rank function in quaternion matrices are tailored for a sparse low rank QMO model, arising from color image denoising, to establish its optimality conditions via stationarity.
In this paper we define the best rank-one approximation ratio of a tensor space. It turns out that in the finite dimensional case this provides an upper bound for the quotient of the residual of the ...best rank-one approximation of any tensor in that tensor space and the norm of that tensor. This upper bound is strictly less than one, and it gives a convergence rate for the greedy rank-one update algorithm. For finite dimensional general tensor spaces, third order finite dimensional symmetric tensor spaces, and finite biquadratic tensor spaces, we give positive lower bounds for the best rank-one approximation ratio. For finite symmetric tensor spaces and finite dimensional biquadratic tensor spaces, we give upper bounds for this ratio. PUBLICATION ABSTRACT
In this paper, we first introduce a new class of structured tensors, named strictly positive semidefinite tensors, and show that a strictly positive semidefinite tensor is not an
R
0
tensor. We focus ...on the stochastic tensor complementarity problem (STCP), where the expectation of the involved tensor is a strictly positive semidefinite tensor. We denote such an STCP as a monotone STCP. Based on three popular NCP functions, the
min
function, the
Fischer–Burmeister (FB)
function and the
penalized FB
function, as well as the special structure of the monotone STCP, we introduce three new NCP functions and establish the expected residual minimization (ERM) formulation of the monotone STCP. We show that the solution set of the ERM problem is nonempty and bounded if the solution set of the expected value (EV) formulation for such an STCP is nonempty and bounded. Moreover, an approximate regularized model is proposed to weaken the conditions for nonemptiness and boundedness of the solution set of the ERM problem in practice. Numerical results indicate that the performance of the ERM method is better than that of the EV method.
In this paper we propose an iterative method for calculating the largest eigenvalue of an irreducible nonnegative tensor. This method is an extension of a method of Collatz (1942) for calculating the ...spectral radius of an irreducible nonnegative matrix. Numerical results show that our proposed method is promising. We also apply the method to studying higher-order Markov chains. PUBLICATION ABSTRACT
We first prove two new spectral properties for symmetric nonnegative tensors. We prove a maximum property for the largest H-eigenvalue of a symmetric nonnegative tensor, and establish some bounds for ...this eigenvalue via row sums of that tensor. We show that if an eigenvalue of a symmetric nonnegative tensor has a positive H-eigenvector, then this eigenvalue is the largest H-eigenvalue of that tensor. We also give a necessary and sufficient condition for this. We then introduce copositive tensors. This concept extends the concept of copositive matrices. Symmetric nonnegative tensors and positive semi-definite tensors are examples of copositive tensors. The diagonal elements of a copositive tensor must be nonnegative. We show that if each sum of a diagonal element and all the negative off-diagonal elements in the same row of a real symmetric tensor is nonnegative, then that tensor is a copositive tensor. Some further properties of copositive tensors are discussed.
Recovery of network traffic data from incomplete observed data is an important issue in internet engineering and management. In this paper, by fully combining the temporal stability and periodicity ...features in internet traffic data, a new separable optimization model for internet data recovery is proposed, which is based upon the T-product factorization and the rapid discrete Fourier transform of tensors. The separable structural features presented in the model provide the possibility to design more efficient parallel algorithms. Moreover, by using generalized inverse matrices, an easy-to-operate and effective algorithm is proposed. In theory, we prove that under suitable conditions, every accumulation point of the sequence generated by the proposed algorithm is a stationary point of the established model. Numerical simulation results carried on the widely used real-world internet network datasets, show that the proposed method outperforms state-of-the-art competitions.
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