Orthogonal invariant sets of the diffusion tensor and the development of a curvilinear set suitable for low-anisotropy tissues.

We develop a curvilinear invariant set of the diffusion tensor which may be applied to Diffusion Tensor Imaging measurements on tissues and porous media. This new set is an alternative to the more common invariants such as fractional anisotropy and the diffusion mode. The alternative invariant set possesses a different structure to the other known invariant sets; the second and third members of the curvilinear set measure the degree of orthotropy and oblateness/prolateness, respectively. The proposed advantage of these invariants is that they may work well in situations of low diffusion anisotropy and isotropy, as is often observed in tissues such as cartilage. We also explore the other orthogonal invariant sets in terms of their geometry in relation to eigenvalue space; a cylindrical set, a spherical set (including fractional anisotropy and the mode), and a log-Euclidean set. These three sets have a common structure. The first invariant measures the magnitude of the diffusion, the second and third invariants capture aspects of the anisotropy; the magnitude of the anisotropy and the shape of the diffusion ellipsoid (the manner in which the anisotropy is realised). We also show a simple method to prove the orthogonality of the invariants within a set

The E-Eigenvectors of Tensors

We first show that the eigenvector of a tensor is well-defined. The differences between the eigenvectors of a tensor and its E-eigenvectors are the eigenvectors on the nonsingular projective variety $\mathbb S=\{\mathbf x\in\mathbb P^n\;|\;\sum\limits_{i=0}^nx_i^2=0\}$. We show that a generic tensor has no eigenvectors on $\mathbb S$. Actually, we show that a generic tensor has no eigenvectors on a proper nonsingular projective variety in $\mathbb P^n$. By these facts, we show that the coefficients of the E-characteristic polynomial are algebraically dependent. Actually, a certain power of the determinant of the tensor can be expressed through the coefficients besides the constant term. Hence, a nonsingular tensor always has an E-eigenvector. When a tensor $\mathcal T$ is nonsingular and symmetric, its E-eigenvectors are exactly the singular points of a class of hypersurfaces defined by $\mathcal T$ and a parameter. We give explicit factorization of the discriminant of this class of hypersurfaces, which completes Cartwright and Strumfels' formula. We show that the factorization contains the determinant and the E-characteristic polynomial of the tensor $\mathcal T$ as irreducible factors.Comment: 17 page

A Tensor Analogy of Yuan's Theorem of the Alternative and Polynomial Optimization with Sign structure

Yuan's theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan's theorem of the alternative to the symmetric tensor setting. As an application, we establish that the optimal value of a class of nonconvex polynomial optimization problems with suitable sign structure (or more explicitly, with essentially non-positive coefficients) can be computed by a related convex conic programming problem, and the optimal solution of these nonconvex polynomial optimization problems can be recovered from the corresponding solution of the convex conic programming problem. Moreover, we obtain that this class of nonconvex polynomial optimization problems enjoy exact sum-of-squares relaxation, and so, can be solved via a single semidefinite programming problem.Comment: acceted by Journal of Optimization Theory and its application, UNSW preprint, 22 page