Positive Maps and Separable Matrices

Abstract

Abstract. A linear map between real symmetric matrix spaces is positive if all positive semidefinite matrices are mapped to positive semidefinite ones. A real symmetric matrix is separable if it can be written as a summation of Kronecker products of positive semidefinite matrices. This paper studies how to check if a linear map is positive or not and how to check if a matrix is separable or not. We propose numerical algorithms, based on Lasserre's type of semidefinite relaxations, for solving such questions. To check the positivity of a linear map, we construct a hierarchy of semidefinite relaxations for minimizing the associated bi-quadratic form over the unit spheres. We show that the positivity can be detected by solving a finite number of such semidefinite relaxations. To check the separability of a matrix, we construct a hierarchy of semidefinite relaxations. If it is not separable, we can get a mathematical certificate for that; if it is, we can get a decomposition for the separability.

Hankel Tensor Decompositions and Ranks

“…The task of checking existence of µ in Theorem 6.4 is a truncated moment problem. We refer to [25,26,31,33,34] for related work. Interestingly, separable Hermitian tensors can also be characterized by the Hermitian flattening map m. As in (4.2), the decomposition (6.5) is equivalent to that (6.7) m(A) = r i=1 λ i u 1 i (u 1 i ) * ⊠ · · · ⊠ u m i (u m i ) * .…”

Best Nonnegative Rank-One Approximations of Tensors

“…Problem (7) covers all instances of minimizing a multi-form over the intersection of the multi-sphere and the nonnegative orthants, since the cases with odd d i 's can be equivalently formulated into (7) as in Section 3.1. Polynomial optimization over the multi-sphere is one research direction in recent years, see [32,35,39,41] and references therein. Moreover, in [32] a biquadratic optimization over the joint sphere (multi-sphere with p = 2) with one group variables being nonnegative is discussed as well.…”

“…The basic idea of the SOS relaxation in [26] is by relaxing the rank characterization of a moment vector y ∈ R ν(d,n 1 ,...,np) . Without the nonnegativity constraint, it is classically relaxed as M (y) 0, i.e., the positive semidefiniteness of the moment matrix, see [26,35,41]. It can be shown that the dual problem under this method is an SDP problem obtained by representing a polynomial as a sum of squares (SOS).…”

Positive definiteness of paired symmetric tensors and elasticity tensors

“…In order to judge positive definiteness of a fourth order three dimensional paired symmetric (elasticity) tensor, from Corollary 3.12 we may compute the smallest M -eigenvalue of the concerned tensor. Some methods can be applied to do it, such as the one given in [17]. In this section, by using the special structure of the paired symmetric (elasticity) tensor, we propose a sequential sedefinite programming method for computing the smallest M -eigenvalue of a fourth order three dimensional paired symmetric (elasticity) tensor, by which we can judge whether a fourth order three dimensional paired symmetric (elasticity) tensor is positive definite or not.…”

“…Positive definiteness of polynomials has been being an important issue in many areas, which has been discussed extensively. In [4,17,20], the authors studied positive definiteness conditions of fourth order paired symmetric tensors, which plays an important role in elasticity theory. In Section 4, following the ideas given in [4,20], we further discuss positive definiteness of fourth order three dimensional (strongly) paired symmetric tensors and propose several necessary and sufficient conditions for which the concerned tensor is positive definite.…”