Related papers: On the complexity of finding tensor ranks
We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by H{\aa}stad~\cite{H90}. The hardness proof also implies an…
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…
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…
We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system…
We present an $O^*\left(|\mathbb{F}|^{(R-n_*)\left(\sum_d n_d\right)+n_*}\right)$-time algorithm for determining whether a tensor of shape $n_0\times\dots\times n_{D-1}$ over a finite field $\mathbb{F}$ has rank $\le R$, where $n_*:=\max_d…
A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the…
A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of…
We develop a systematic way to solve linear equations involving tensors of arbitrary rank. We start off with the case of a rank $3$ tensor, which appears in many applications, and after finding the condition for a unique solution we derive…
In this paper, we show that the low rank matrix completion problem can be reduced to the problem of finding the rank of a certain tensor.
We study the rank one completion problem for tensors of arbitrary orders. The notion of rank one determinable tensors is introduced. We explore its properties and propose a recursive algorithm for computing rank one tensor completion. This…
Tensor type data are becoming important recently in various application fields. We determine a rank of a tensor T so that A+T is diagonalizable for a given 3-tensor A with 2 slices over the complex and real number field.
In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite extensions of finite fields, enriched with some not published recent results as well as analyzes enhancing the qualitative…
These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will…
Finding the rank of a tensor is a problem that has many applications. Unfortunately it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of…
We consider relative error low rank approximation of $tensors$ with respect to the Frobenius norm: given an order-$q$ tensor $A \in \mathbb{R}^{\prod_{i=1}^q n_i}$, output a rank-$k$ tensor $B$ for which $\|A-B\|_F^2 \leq (1+\epsilon)$OPT,…
A systematic method is presented for the construction and classification of algebras of gauge transformations for arbitrary high rank tensor gauge fields. For every tensor gauge field of a given rank, the gauge transformation will be…
There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…
In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be…
The main contribution of this note is to establish a framework to extend results of tensor functions over specific field to general field. As a consequence of this framework, we extend the existing work to more general settings: \emph{(1)}…
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…