相关论文: Identifiability for a class of symmetric tensors

We study the problem of finding orthogonal low-rank approximations of symmetric tensors. In the case of matrices, the approximation is a truncated singular value decomposition which is then symmetric. Moreover, for rank-one approximations…

Asymptotic tensor rank is notoriously difficult to determine. Indeed, determining its value for the $2\times 2$ matrix multiplication tensor would determine the matrix multiplication exponent, a long-standing open problem. On the other…

We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor…

The separability detecting problem of mixed states is one of the fundamental problems in quantum information theory. In the last 20 years, almost all methods are based on the sufficient or necessary conditions for entanglement. However, in…

We study cubic surfaces as symmetric tensors of format $4 \times 4 \times 4$. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The…

We study symmetric tensor decompositions, i.e. decompositions of the input symmetric tensor T of order 3 as sum of r 3rd-order tensor powers of u_i where u_i are vectors in \C^n. In order to obtain efficient decomposition algorithms, it is…

High-dimensional data in the form of tensors are challenging for kernel classification methods. To both reduce the computational complexity and extract informative features, kernels based on low-rank tensor decompositions have been…

This paper addresses the detection of a low rank high-dimensional tensor corrupted by an additive complex Gaussian noise. In the asymptotic regime where all the dimensions of the tensor converge towards $+\infty$ at the same rate, existing…

Finding the maximum eigenvalue of a symmetric tensor is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a semi-definite program algorithm for computing the maximum $H$-eigenvalue of a…

We study highest weight vectors for symmetric and alternating spaces of tensors, whose dimensions are given by generalized Kronecker coefficients. We describe the algebraic relations for classical constructions of corresponding spanning…

Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric…

We give sufficient conditions on a symmetric tensor S in S^dF^n to satisfy the equality: the symmetric rank of S, denoted as srank(S), is equal to the rank of S, denoted as rank(S). This is done by considering the rank of the unfolded S…

The Euclidean Kerr metric is conformal, in two distinct ways, to a Kahler metric, with conformal factors determined by the repeated eigenvalue of the two chiral halves of the Weyl curvature. A Lorentzian analogue holds, where the…

In this paper, the problem of the identification of the symmetry class of a given tensor is asked. Contrary to classical approaches which are based on the spectral properties of the linear operator describing the elasticity, our setting is…

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…

Nonnegative tensor factorization has applications in statistics, computer vision, exploratory multiway data analysis and blind source separation. A symmetric nonnegative tensor, which has a symmetric nonnegative factorization, is called a…

We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank…

It is easily checkable if a given tensor is a B tensor, or a B$_0$ tensor or not. In this paper, we show that a symmetric B tensor can always be decomposed to the sum of a strictly diagonally dominated symmetric M tensor and several…

The first author with B. Sturmfels studied the variety of matrices with eigenvectors in a given linear subspace, called Kalman variety. We extend that study from matrices to symmetric tensors, proving in the tensor setting the…

Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. In a recent paper [arXiv:0907.2994v1] we discussed how to…