Abstract:
We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued, pairwise orthogonal vectors. Such decompositions do not generally exist, but we show that some symmetric tensor decomposition problems can be converted to orthogonal problems following the whitening procedure proposed by Anandkumar et al. (2012). If an orthogonal decomposition of an $m$-way $n$-dimensional symmetric tensor exists, we propose a novel method to compute it that reduces to an $n \times n$ symmetric matrix eigenproblem. We provide numerical results demonstrating the effectiveness of the method.

Abstract:
While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention from theoretical computer science and scientific computing. We complement this existing body of literature with an algebro-geometric analysis of the set of orthogonally decomposable tensors. More specifically, we prove that they form a real-algebraic variety defined by polynomials of degree at most four. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary. A key feature of our approach is a surprising connection between orthogonally decomposable tensors and semisimple algebras---associative in the ordinary and symmetric settings and of compact Lie type in the alternating setting.

Abstract:
We consider representations of tensors as sums of decomposable tensors or, equivalently, decomposition of multilinear forms into one--forms. In this short note we show that there exists a particular finite strongly orthogonal decomposition which is essentially unique and yields all critical points of the multilinear form on the torus. In particular, this determines exactly the number of critical points of the multilinear form, giving an affirmative answer to a finiteness conjecture by Friedland.

Abstract:
In cosmological perturbation theory a first major step consists in the decomposition of the various perturbation amplitudes into scalar, vector and tensor perturbations, which mutually decouple. In performing this decomposition one uses -- beside the Hodge decomposition for one-forms -- an analogous decomposition of symmetric tensor fields of second rank on Riemannian manifolds with constant curvature. While the uniqueness of such a decomposition follows from Gauss' theorem, a rigorous existence proof is not obvious. In this note we establish this for smooth tensor fields, by making use of some important results for linear elliptic differential equations.

Abstract:
A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a sum of symmetric outer product of vectors. A rank-1 order-k tensor is the outer product of k non-zero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1.

Abstract:
We first prove two new spectral properties for symmetric nonnegative tensors. We prove a maximum property for the largest H-eigenvalue of a symmetric nonnegative tensor, and establish some bounds for this eigenvalue via row sums of that tensor. We show that if an eigenvalue of a symmetric nonnegative tensor has a positive H-eigenvector, then this eigenvalue is the largest H-eigenvalue of that tensor. We also give a necessary and sufficient condition for this. We then introduce copositive tensors. This concept extends the concept of copositive matrices. Symmetric nonnegative tensors and positive semi-definite tensors are examples of copositive tensors. The diagonal elements of a copositive tensor must be nonnegative. We show that if each sum of a diagonal element and all the negative off-diagonal elements in the same row of a real symmetric tensor is nonnegative, then that tensor is a copositive tensor. Some further properties of copositive tensors are discussed.

Abstract:
We consider the problem of determining the symmetric tensor rank for symmetric tensors with an algebraic geometry approach. We give algorithms for computing the symmetric rank for $2\times ... \times 2$ tensors and for tensors of small border rank. From a geometric point of view, we describe the symmetric rank strata for some secant varieties of Veronese varieties.

Abstract:
Let X be a smooth projective variety over C. We find the natural notion of semistable orthogonal bundle and construct the moduli space, which we compactify by considering also orthogonal sheaves, i.e. pairs (E,\phi), where E is a torsion free sheaf on X and \phi is a symmetric nondegenerate (in the open set where E is locally free) bilinear form on E. We also consider special orthogonal sheaves, by adding a trivialization \psi of the determinant of E such that det(\phi)=\psi^2 ; and symplectic sheaves, by considering a form which is skewsymmetric. More generally, we consider semistable tensors, i.e. multilinear forms on a torsion free sheaf, and construct their projective moduli space using GIT.

Abstract:
In the present paper we show properties of a little-known Laplacian operator acting on symmetric tensors. This operator is an analogue of the well known Hodge-de Rham Laplacian which acts on exterior differential forms. Moreover, this operator admits the Weitzenb\"ock decomposition and we study it using the analytical method, due to Bochner, of proving vanishing theorems for the null space of a Laplace operator admitting a Weitzenb\"ock decomposition and further of estimating its lowest eigenvalue.

Abstract:
Decomposing tensors into orthogonal factors is a well-known task in statistics, machine learning, and signal processing. We study orthogonal outer product decompositions where the factors in the summands in the decomposition are required to be orthogonal across summands, by relating this orthogonal decomposition to the singular value decompositions of the flattenings. We show that it is a non-trivial assumption for a tensor to have such an orthogonal decomposition, and we show that it is unique (up to natural symmetries) in case it exists, in which case we also demonstrate how it can be efficiently and reliably obtained by a sequence of singular value decompositions. We demonstrate how the factoring algorithm can be applied for parameter identification in latent variable and mixture models.