Qun Wang

Large scale tensors, including large scale Hankel tensors, have many applications in science and engineering. In this paper, we propose an inexact curvilinear search optimization method to compute Z- and H-eigenvalues of $m$th order $n$ dimensional Hankel tensors, where $n$ is large. Owing to the fast Fourier transform, the computational cost of each iteration of the new method is about $\mathcal{O}(mn\log(mn))$. Using the Cayley transform, we obtain an effective curvilinear search scheme. Then, we show that every limiting point of iterates generated by the new algorithm is an eigen-pair of Hankel tensors. Without the assumption of a second-order sufficient condition, we analyze the linear convergence rate of iterate sequence by the Kurdyka-{\L}ojasiewicz property. Finally, numerical experiments for Hankel tensors, whose dimension may up to one million, are reported to show the efficiency of the proposed curvilinear search method.

A positive semi-definite (PSD) tensor which is not a sum-of-squares (SOS) tensor is called a PSD non-SOS (PNS) tensor. Is there a fourth order four dimensional PNS Hankel tensor? Until now, this question is still an open problem. Its answer has both theoretical and practical meanings. We assume that the generating vector $v$ of the Hankel tensor $A$ is symmetric. Under this assumption, we may fix the fifth element $v_4$ of $v$ at $1$. We show that there are two surfaces $M_0$ and $N_0$ with the elements $v_2, v_6, v_1, v_3, v_5$ of $v$ as variables, such that $M_0 \ge N_0$, $A$ is SOS if and only if $v_0 \ge M_0$, and $A$ is PSD if and only if $v_0 \ge N_0$, where $v_0$ is the first element of $v$. If $M_0 = N_0$ for a point $P = (v_2, v_6, v_1, v_3, v_5)^\top$, then there are no fourth order four dimensional PNS Hankel tensors with symmetric generating vectors for such $v_2, v_6, v_1, v_3, v_5$. Then, we call such a point $P$ PNS-free. We show that a $45$-degree planar closed convex cone, a segment, a ray and an additional point are PNS-free. Numerical tests check various grid points, and find that they are also PNS-free.

Anti-circulant tensors have applications in exponential data fitting. They are special Hankel tensors. In this paper, we extend the definition of anti-circulant tensors to generalized anti-circulant tensors by introducing a circulant index $r$ such that the entries of the generating vector of a Hankel tensor are circulant with module $r$. In the special case when $r =n$, where $n$ is the dimension of the Hankel tensor, the generalized anticirculant tensor reduces to the anti-circulant tensor. Hence, generalized anti-circulant tensors are still special Hankel tensors. For the cases that $GCD(m, r) =1$, $GCD(m, r) = 2$ and some other cases, including the matrix case that $m=2$, we give necessary and sufficient conditions for positive semi-definiteness of even order generalized anti-circulant tensors, and show that in these cases, they are SOS tensors. This shows that, in these cases, there are no PNS (positive semidefinite tensors which are not sum of squares) Hankel tensors.

In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected $k$-uniform hypergraph $G$, where $k \ge 3$, reaches its upper bound $2\Delta(G)$, where $\Delta(G)$ is the largest degree of $G$, if and only if $G$ is regular. Thus the largest Laplacian H-eigenvalue of $G$, reaches the same upper bound, if and only if $G$ is regular and odd-bipartite. We show that an $s$-cycle $G$, as a $k$-uniform hypergraph, where $1 \le s \le k-1$, is regular if and only if there is a positive integer $q$ such that $k=q(k-s)$. We show that an even-uniform $s$-path and an even-uniform non-regular $s$-cycle are always odd-bipartite. We prove that a regular $s$-cycle $G$ with $k=q(k-s)$ is odd-bipartite if and only if $m$ is a multiple of $2^{t_0}$, where $m$ is the number of edges in $G$, and $q = 2^{t_0}(2l_0+1)$ for some integers $t_0$ and $l_0$. We identify the value of the largest signless Laplacian H-eigenvalue of an $s$-cycle $G$ in all possible cases. When $G$ is odd-bipartite, this is also its largest Laplacian H-eigenvalue. We introduce supervertices for hypergraphs, and show the components of a Laplacian H-eigenvector of an odd-uniform hypergraph are equal if such components corresponds vertices in the same supervertex, and the corresponding Laplacian H-eigenvalue is not equal to the degree of the supervertex. Using this property, we show that the largest Laplacian H-eigenvalue of an odd-uniform generalized loose $s$-cycle $G$ is equal to $\Delta(G)=2$. We also show that the largest Laplacian H-eigenvalue of a $k$-uniform tight $s$-cycle $G$ is not less than $\Delta(G)+1$, if the number of vertices is even and $k=4l+3$ for some nonnegative integer $l$.