Related papers: Border rank is not multiplicative under the tensor…

In the realm of tensor optimization, the low-rank Tucker decomposition is crucial for reducing the number of parameters and for saving storage. We explore the geometry of Tucker tensor varieties -- the set of tensors with bounded Tucker…

In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special…

We show that the border subrank of a sufficiently general tensor in $(\mathbb{C}^n)^{\otimes d}$ is $\mathcal{O}(n^{1/(d-1)})$ for $n \to \infty$. Since this matches the growth rate $\Theta(n^{1/(d-1)})$ for the generic (non-border) subrank…

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…

We obtain new uniform upper bounds for the (non necessarily symmetric) tensor rank of the multiplication in the extensions of the finite fields $\F_q$ for any prime or prime power $q\geq2$; moreover these uniform bounds lead to new…

Kruskal's theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. In this work, we propose a conjecture in which the k-rank condition of…

The strength of a multivariate homogeneous polynomial is the minimal number of terms in an expression as a sum of products of lower-degree homogeneous polynomials. Partition rank is the analogue for multilinear forms. Both ranks can drop…

A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness…

Let $R$ be a principal ideal local ring of finite length with a finite residue field of odd characteristic. Let $G(R)$ denote either the general linear group or the general unitary group of degree two over $R$. We study the decomposition of…

In this work we give the classification of real ternary cubic forms with respect to rank and border rank up to SL(3)-action and we examine the differences with the complex case.

Low rank tensor representation underpins much of recent progress in tensor completion. In real applications, however, this approach is confronted with two challenging problems, namely (1) tensor rank determination; (2) handling real tensor…

Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of…

Motivated by a problem in computational complexity, we consider the behavior of rank functions for tensors and polynomial maps under random coordinate restrictions. We show that, for a broad class of rank functions called natural rank…

Low-rank tensor approximations have shown great potential for uncertainty quantification in high dimensions, for example, to build surrogate models that can be used to speed up large-scale inference problems (Eigel et al., Inverse Problems…

In this paper, we analyze the existing rules for constructing derivatives of the scalar and tensor functions of the tensor argument with respect to the tensor argument and the theoretical positions underlying the construction of these…

In recent studies, the tensor ring (TR) rank has shown high effectiveness in tensor completion due to its ability of capturing the intrinsic structure within high-order tensors. A recently proposed TR rank minimization method is based on…

We give a self-contained classification of $1_*$-generic minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$. Together with previous results, this gives a classification of all minimal…

Let H be a finite-dimensional Hopf algebra. We give a description of the tensor product of bimodule categories over Rep(H). When the bimodule categories are invertible this description can be given explicitly. We present some consequences…

In [13], Hillar and Lim famously demonstrated that "multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard". Despite many recent advancements, the state-of-the-art methods for…

We present an efficient low-rank approximation algorithm for non-negative tensors. The algorithm is derived from our two findings: First, we show that rank-1 approximation for tensors can be viewed as a mean-field approximation by treating…