Haibin Chen

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

Summary nFinding 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.

Further results on Cauchy tensors and Hankel tensors

In this article, we present various new results on Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries. Furthermore, we prove that all even order generalized Cauchy tensors with positive entries are completely positive tensors, which means every such that generalized Cauchy tensor can be decomposed as the sum of nonnegative rank-1 tensors. We also establish that all the H-eigenvalues of nonnegative Cauchy tensors are nonnegative. Secondly, we present new mathematical properties of Hankel tensors. We prove that an even order Hankel tensor is Vandermonde positive semi-definite if and only if its associated plane tensor is positive semi-definite. We also show that, if the Vandermonde rank of a Hankel tensor A is less than the dimension of the underlying space, then positive semi-definiteness of A is equivalent to the fact that A is a complete Hankel tensor, and so, is further equivalent to the SOS property of A . Thirdly, we introduce a new class of structured tensors called Cauchy-Hankel tensors, which is a special case of Cauchy tensors and Hankel tensors simultaneously. Sufficient and necessary conditions are established for an even order Cauchy-Hankel tensor to be positive definite.

Tensor Eigenvalue Complementarity Problems

This chapter is a companion chapter of Chap. 4. In this chapter, we mainly discuss tensor eigenvalue complementarity problems (TEiCP). It is a generalization of the matrix eigenvalue complementarity problem (EiCP), which possess a broad range of interesting applications.

Third Order Tensors in Physics and Mechanics

Third order tensors have wide applications in physics and mechanics. Examples include piezoelectric tensors in crystal study, third order symmetric traceless tensors in liquid crystal study and third order susceptibility tensors in nonlinear optics study. On the other hand, the Levi-Civita tensor is famous in tensor calculus.

Higher Order Tensors in Quantum Physics

In this chapter, we will apply tensor analysis to the quantum entanglement problem and the classicality problem of spin states in quantum physics.

Hankel Tensor Computation and Exponential Data Fitting

Hankel structures are widely used in real-world problems arising from signal processing, automatic control, and geophysics. For example, a Hankel matrix was formulated to analyze the time-domain signals in nuclear magnetic resonance spectroscopy, which is crucial for brain tumour detection.

Fourth Order Tensors in Physics and Mechanics

Fourth order tensors have also wide applications in physics and mechanics. Examples include the piezo-optical tensor, the elasto-optical tensor and the flexoelectric tensor. The most well-known fourth order tensor is the elasticity tensor (Huang et al., Tensor Analysis (in Chinese). Tsinghua University Press, Beijing, 2003, [134], Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices. Clarendon Press, Oxford, 1985, [212], Zou et al., J. Mech. Phys. Solids. 58:346–372, 2010, [318]). It is closely related to the strong ellipticity condition in nonlinear mechanics.

Higher Order Diffusion Tensor Imaging

Diffusion tensor imaging (DTI) is one of the most promising medical imaging models, and the most applicable technique in modern clinical medicine. While, there are limitations to DTI, which becomes useless in non-isotropic materials. As a resolution, diffusion kurtosis imaging (DKI) is proposed as a new model in medical engineering, which can characterize the non-Gaussian diffusion behavior in tissues, and in which a diffusion kurtosis (DK) tensor is involved. A DK tensor is a fourth order three dimensional symmetric tensor. In this chapter, we will apply the spectral theory of tensors to this particular type of tensors arising from medical imaging and derive some applications.

Tensor Complementarity Problems

Complementarity problems encompass several important classes of mathematical optimization problems, e.g., linear programming, quadratic programming, linear conic optimization problems, etc. Actually, we always solve an optimization problem via its optimality condition, which usually turns out to be a complementarity problem, e.g., KKT system.