Related papers: Tensors of Nonnegative Rank Two

It has recently been shown that the tensor rank can be strictly submultiplicative under the tensor product, where the tensor product of two tensors is a tensor whose order is the sum of the orders of the two factors. The necessary upper…

In this paper, we extend some classes of structured matrices to higher order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a…

Motivated by (and using tools from) communication complexity, we investigate the relationship between the following two ranks of a $0$-$1$ matrix: its nonnegative rank and its binary rank (the $\log$ of the latter being the unambiguous…

In recent years several classes of structured matrices are extended to classes of tensors in the context of tensor complementarity problem. The tensor complementarity problem is a class of nonlinear complementarity problem where the…

We present an example of a subfield $\mathcal{F}\subset\mathbb{R}$ and a matrix $A$ whose conventional and nonnegative ranks equal five, but the nonnegative rank with respect to $\mathcal{F}$ equals six. In other words, $A$ can be…

Rank-2 and rank-3 tensors are almost all identifiable with only few exceptions. We classify them all together with the dimensions and the structures of all the sets evincing the rank.

We give constructions of n^k x n^k x n tensors of rank at least 2n^k - O(n^(k-1)). As a corollary we obtain an [n]^r shaped tensor with rank at least 2n^(r/2) - O(n^(r/2)-1) when r is odd. The tensors are constructed from a simple recursive…

We study the relationship between the positivity property in a rank 2 cluster algebra, and the property of such an algebra to be tame. More precisely, we show that a rank 2 cluster algebra has a basis of indecomposable positive elements if…

The tensor rank and border rank of the $3 \times 3$ determinant tensor is known to be $5$ if characteristic is not two. In this paper, we show that the tensor rank remains $5$ for fields of characteristic two as well. We also include an…

In this article the notion of the nondecreasing (ND) rank of a matrix or tensor is introduced. A tensor has an ND rank of r if it can be represented as a sum of r outer products of vectors, with each vector satisfying a monotonicity…

This paper presents various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at…

We show that finding rank-$R$ decompositions of a 3D tensor, for $R\le 4$, over a fixed finite field can be done in polynomial time. However, if some cells in the tensor are allowed to have arbitrary values, then rank-2 is NP-hard over the…

A rank-n tensor on a Lorentzian manifold V whose contraction with n arbitrary causal future directed vectors is non-negative is said to have the dominant property. These tensors, up to sign, are called causal tensors, and we determine their…

We prove (without exceptions) the existence of irredundant tensor decompositions with the number of addenda equal to rank $+1$. We also discuss the existence of decompositions with more than the tensor rank terms, which are concise, while…

We develop a framework to analyse invariant decompositions of elements of tensor product spaces. Namely, we define an invariant decomposition with indices arranged on a simplicial complex, and which is explicitly invariant under a group…

Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If $\pi$ is a spacelike 2 plane, let $R(\pi)$ be the associated skew-symmetric curvature operator. We classify the algebraic curvature…

We introduce subspace rank as a tool for studying ranks of tensors and X-rank more generally. We derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in C^2 \otimes C^b \otimes C^b. We…

We prove a lower bound on the rank of tensors constructed from families of linear maps that `expand' the dimension of every subspace. Such families, called {\em dimension expanders} have been studied for many years with several known…

We present a family of flattening methods of tensors which we call Kronecker-Koszul flattenings, generalizing the famous Koszul flattenings and further equations of secant varieties studied among others by Landsberg, Manivel, Ottaviani and…

Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also…