Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors

Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only if its generating vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its generating vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semi-definiteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. Furthermore, we prove that the Hadamard product of two positive semi-definite (positive definite respectively) symmetric Cauchy tensors is a positive semi-definite (positive definite respectively) tensor, which can be generalized to the Hadamard product of finitely many positive semi-definite (positive definite respectively) symmetric Cauchy tensors. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised

Centrosymmetric, Skew Centrosymmetric and Centrosymmetric Cauchy Tensors

Recently, Zhao and Yang introduced centrosymmetric tensors. In this paper, we further introduce skew centrosymmetric tensors and centrosymmetric Cauchy tensors, and discuss properties of these three classes of structured tensors. Some sufficient and necessary conditions for a tensor to be centrosymmetric or skew centrosymmetric are given. We show that, a general tensor can always be expressed as the sum of a centrosymmetric tensor and a skew centrosymmetric tensor. Some sufficient and necessary conditions for a Cauchy tensor to be centrosymmetric or skew centrosymmetric are also given. Spectral properties on H-eigenvalues and H-eigenvectors of centrosymmetric, skew centrosymmetric and centrosymmetric Cauchy tensors are discussed. Some further questions on these tensors are raised

Three Dimensional Strongly Symmetric Circulant Tensors

In this paper, we give a necessary and sufficient condition for an even order three dimensional strongly symmetric circulant tensor to be positive semi-definite. In some cases, we show that this condition is also sufficient for this tensor to be sum-of-squares. Numerical tests indicate that this is also true in the other cases

SOS-Hankel Tensors: Theory and Application

Hankel tensors arise from signal processing and some other applications. SOS (sum-of-squares) tensors are positive semi-definite symmetric tensors, but not vice versa. The problem for determining an even order symmetric tensor is an SOS tensor or not is equivalent to solving a semi-infinite linear programming problem, which can be done in polynomial time. On the other hand, the problem for determining an even order symmetric tensor is positive semi-definite or not is NP-hard. In this paper, we study SOS-Hankel tensors. Currently, there are two known positive semi-definite Hankel tensor classes: even order complete Hankel tensors and even order strong Hankel tensors. We show complete Hankel tensors are strong Hankel tensors, and even order strong Hankel tensors are SOS-Hankel tensors. We give several examples of positive semi-definite Hankel tensors, which are not strong Hankel tensors. However, all of them are still SOS-Hankel tensors. Does there exist a positive semi-definite non-SOS-Hankel tensor? The answer to this question remains open. If the answer to this question is no, then the problem for determining an even order Hankel tensor is positive semi-definite or not is solvable in polynomial-time. An application of SOS-Hankel tensors to the positive semi-definite tensor completion problem is discussed. We present an ADMM algorithm for solving this problem. Some preliminary numerical results on this algorithm are reported

Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere

For an $n$-variate order-$d$ tensor $A$, define $A_{\max} := \sup_{\| x \|_2 = 1} \langle A , x^{\otimes d} \rangle$ to be the maximum value taken by the tensor on the unit sphere. It is known that for a random tensor with i.i.d $\pm 1$ entries, $A_{\max} \lesssim \sqrt{n\cdot d\cdot\log d}$ w.h.p. We study the problem of efficiently certifying upper bounds on $A_{\max}$ via the natural relaxation from the Sum of Squares (SoS) hierarchy. Our results include: - When $A$ is a random order-$q$ tensor, we prove that $q$ levels of SoS certifies an upper bound $B$ on $A_{\max}$ that satisfies $$B ~~~~\leq~~ A_{\max} \cdot \biggl(\frac{n}{q^{\,1-o(1)}}\biggr)^{q/4-1/2} \quad \text{w.h.p.}$$ Our upper bound improves a result of Montanari and Richard (NIPS 2014) when $q$ is large. - We show the above bound is the best possible up to lower order terms, namely the optimum of the level-$q$ SoS relaxation is at least $$A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ .$$ - When $A$ is a random order-$d$ tensor, we prove that $q$ levels of SoS certifies an upper bound $B$ on $A_{\max}$ that satisfies $$B ~~\leq ~~ A_{\max} \cdot \biggl(\frac{\widetilde{O}(n)}{q}\biggr)^{d/4 - 1/2} \quad \text{w.h.p.}$$ For growing $q$, this improves upon the bound certified by constant levels of SoS. This answers in part, a question posed by Hopkins, Shi, and Steurer (COLT 2015), who established the tight characterization for constant levels of SoS

A constructive arbitrary-degree Kronecker product decomposition of tensors

We propose the tensor Kronecker product singular value decomposition~(TKPSVD) that decomposes a real $k$-way tensor $\mathcal{A}$ into a linear combination of tensor Kronecker products with an arbitrary number of $d$ factors $\mathcal{A} = \sum_{j=1}^R \sigma_j\, \mathcal{A}^{(d)}_j \otimes \cdots \otimes \mathcal{A}^{(1)}_j$. We generalize the matrix Kronecker product to tensors such that each factor $\mathcal{A}^{(i)}_j$ in the TKPSVD is a $k$-way tensor. The algorithm relies on reshaping and permuting the original tensor into a $d$-way tensor, after which a polyadic decomposition with orthogonal rank-1 terms is computed. We prove that for many different structured tensors, the Kronecker product factors $\mathcal{A}^{(1)}_j,\ldots,\mathcal{A}^{(d)}_j$ are guaranteed to inherit this structure. In addition, we introduce the new notion of general symmetric tensors, which includes many different structures such as symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors.Comment: Rewrote the paper completely and generalized everything to tensor