Related papers: Tensors of Nonnegative Rank Two

The M-matrix is an important concept in matrix theory, and has many applications. Recently, this concept has been extended to higher order tensors [18]. In this paper, we establish some important properties of M-tensors and nonsingular…

We study odd-dimensional modular tensor categories and maximally non-self dual (MNSD) modular tensor categories of low rank. We give lower bounds for the ranks of modular tensor categories in terms of the rank of the adjoint subcategory and…

Determining the matrix multiplication exponent $\omega$ is one of the greatest open problems in theoretical computer science. We show that it is impossible to prove $\omega = 2$ by starting with structure tensors of modules of fixed degree…

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with…

Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(k + 1)-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this…

Let K be a field of characteristic different from 2 and let V be a vector space of dimension n over K. Let M be a non-zero subspace of symmetric bilinear forms defined on V x V and let r=rank(M) denote the set of different positive integers…

In this paper we parameterize non-negative matrices of sum one and rank at most two. More precisely, we give a family of parameterizations using the least possible number of parameters. We also show how these parameterizations relate to a…

Using the relationship between totally nonnegative matrices and directed acyclic weighted planar networks, we show that $2\times 2$ minors of minor matrices of totally nonnegative matrices are also nonnegative. We give a combinatorial…

We prove that for every positive integer $d \ge 2$ there exist polynomial functions $F_d, G_d: \mathbb{N} \to \mathbb{N}$ such that for each positive integer $r$, every order-$d$ tensor $T$ over an arbitrary field and with partition rank at…

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…

The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field $\F$ is at most $k$ over the algebraic closure of $\F$, where $k$ is a given positive integer. We estimate the arithmetic…

We develop a notion of {\em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $n\times n$…

In this paper, we propose two new classes of tensors: double B-tensors and quasi-double B-tensors, give some properties of double B-tensors and quasi-double B-tensors, discuss their relationships with B-tensors and positive definite tensors…

This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over…

We develop a theory of levels for irreducible representations of symmetric groups of degree $n$ analogous to the theory of levels for finite classical groups. A key property of level is that the level of a character, provided it is not too…

The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is…

The orthogonal decomposition factorizes a tensor into a sum of an orthogonal list of rankone tensors. We present several properties of orthogonal rank. We find that a subtensor may have a larger orthogonal rank than the whole tensor and…

Nonparametric extension of tensor regression is proposed. Nonlinearity in a high-dimensional tensor space is broken into simple local functions by incorporating low-rank tensor decomposition. Compared to naive nonparametric approaches, our…

In this work we study different notions of ranks and approximation of tensors. We consider the tensor rank, the nuclear rank and we introduce the notion of symmetric decomposable rank, a notion of rank defined only on symmetric tensors. We…

We classify closed, simply-connected non-negatively curved 5-manifolds admitting an (almost) effective, isometric $T^3$ or $T^2$ action. As a direct consequence, we show that for any manifold, of dimensions up to and including 9 under the…