Mohsen Aliabadi, Shmuel Friedland
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 complexity of our algorithm.
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In this paper, we define the minimum (maximum) rank, term rank and the sign nonsingular of tensors. The sufficiency and necessity for the minimum rank of a real tensor to be $1$ is given. And we show that the maximum rank of a tensor is not less than the term rank. We also prove that the minimum rank of a sign nonsingular tensor is not less than the dimension of it. And we get some characterizations of a tensor having sign left or sign right inverses.
We answer to a question posed recently by B. Lovitz and F. Petrov, proving the conjectured sufficient minimality and uniqueness condition of the 3-tensor decomposition.
Cayley's first hyperdeterminant is a straightforward generalization of determinants for tensors. We prove that nonzero hyperdeterminants imply lower bounds on some types of tensor ranks. This result applies to the slice rank introduced by Tao and more generally to partition ranks introduced by Naslund. As an application, we show upper bounds on some generalizations of colored sum-free sets based on constraints related to order polytopes.
We determine the tensor rank of all semifields of order 16 over $\mathbb{F}_2$ and of all semifields of order 81 over $\mathbb{F}_3$. Our results imply that some semifields of order 81 have lower multiplicative complexity than the finite field $\mathbb{F}_{81}$ over $\mathbb{F}_3$. We prove new results on the correspondence between linear codes and tensor rank, including a generalisation of a theorem of Brockett and Dobkin to arbitrary tensors, which makes the problem computationally feasible.
We investigate the computational complexity of tensor rank, a concept that plays fundamental role in different topics of modern applied mathematics. For tensors over any integral domain, we prove that the rank problem is polynomial time equivalent to solving a system of polynomial equations over this integral domain. Our result gives a complete description of the algorithmic complexity of tensor rank and allows one to solve several known open problems. In particular, the tensor rank over $\mathbb{Z}$ turns out to be undecidable, which answers the question posed by Gonzalez and Ja'Ja' in 1980. …
We show that the tensor rank of tensor product of two three-qubit W states is not less than eight. Combining this result with the recent result of M. Christandl, A. K. Jensen, and J. Zuiddam that the tensor rank of tensor product of two three-qubit W states is at most eight, we deduce that the tensor rank of tensor product of two three-qubit W states is eight. We also construct the upper bound of the tensor rank of tensor product of many three-qubit W states.
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 analysis of $5 \times 5$ and $7 \times 7$ determinant and permanent tensors, as well as the symmetric $3 \times 3$ permanent and determinant tensors. We end with some remarks on binary tensors.
The analytic rank of a tensor, first defined by Gowers and Wolf in the context of higher-order Fourier analysis, is defined to be the logarithm of the bias of the tensor. We prove that it is a subadditive measure of rank: that is, the analytic rank of the sum of two tensors is at most the sum of their individual analytic ranks. This analytic property turns out to have surprising applications: (i) common roots of tensors are always positively correlated; and (ii) the slice rank and partition rank, which were defined recently in the resolution of the cap-set problem in Ramsey theory, can be repl…
A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two tensors of order less than $d$, and it has partition rank at most $k$ if it can be written as a sum of $k$ tensors of partition rank 1. In this paper, we prove that if the analytic rank of an order $d$ tensor is at most $r$, then its partition rank is at most $f(r,d,|\mathbb{F}|)$. Previously, this was known with $f$ being an Ackermann-type function in $r$ …
The tensor rank of some Gabidulin codes of small dimension is investigated. In particular, we determine the tensor rank of any rank metric code equivalent to an $8$-dimensional $\mathbb{F}_q$-linear generalized Gabidulin code in $\mathbb{F}_{q}^{4\times4}$. This shows that such a code is never minimum tensor rank. In this way, we detect the first infinite family of Gabidulin codes which are not minimum tensor rank.
Random restrictions of high-rank tensors and polynomial maps
针对计算复杂性中的问题，研究张量与多项式映射的秩函数在随机坐标限制下的行为，证明自然秩函数在稠密集合上的随机限制通常仅使秩减少常数因子。
A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two tensors of order less than $d$, and it has partition rank at most $k$ if it can be written as a sum of $k$ tensors of partition rank 1. In this paper, we prove that if the analytic rank of an order $d$ tensor is at most $r$, then its partition rank is at most $f(r,d,|\mathbb{F}|)$, where, for fixed $d$ and $\mathbb{F}$, $f$ is a polynomial in $r$. This is a…
Constructions of Rank-Metric Codes of Small Tensor Rank
本文研究了秩度量码的张量秩与相关线性码参数之间的关系，提出了张量秩缺陷的概念，并利用代数几何码构造了具有小张量秩缺陷的秩度量码。
Schur complements for tensors and multilinear commutative rank
本文证明了多线性形式矩阵的三种秩概念等价，并修正了相关研究中的缺陷，回答了关于张量秩关系的问题。
Block Tensor Rank of Sum-Rank Metric Codes
本文引入和秩度量码的块张量秩，证明其可加性分解，并基于投影和坐标码导出下界，构造达到Singleton或Griesmer界的码族。
We introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new vertices and edges. We show that tensor surgery is capable of preserving the low rank structure of an initial tensor decomposition and thus allows to prove nontrivial upper bounds on tensor rank, border rank and asymptotic rank of the final tensors. We illustrate our method with a number of examples. Tensor surgery on …
We introduce the $G$-stable rank of a higher order tensors over perfect fields. The $G$-stable rank is related to the Hilbert-Mumford criterion for stability in Geometric Invariant Theory. We will relate the $G$-stable rank to the tensor rank and slice rank. For numerical applications, we express the $G$-stable rank as a solution to an optimization problem. Over the field ${\mathbb F}_3$ we discuss an application to the Cap Set Problem.
Geometric rank of tensors and subrank of matrix multiplication
本文提出几何秩这一新工具，证明其可上界张量子秩与超图独立数，并用于给出矩阵乘法张量（边界）子秩的紧上界。
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