Related papers: A base change framework for tensor functions
In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite fields and we establish new asymptotical and not asymptotical upper bounds about it.
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…
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…
In this paper, we obtain new bounds for the tensor rank of multiplication in any extension of $\F_2$. In particular, it also enables us to obtain the best known asymptotic bound. In this aim, we use the generalized algorithm of type…
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…
Tensor parameters that are amortized or regularized over large tensor powers, often called "asymptotic" tensor parameters, play a central role in several areas including algebraic complexity theory (constructing fast matrix multiplication…
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.
A tensor is a multi-way array that can represent, in addition to a data set, the expression of a joint law or a multivariate function. As such it contains the description of the interactions between the variables corresponding to each of…
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…
The concept of tensor rank, introduced in the twenties, has been popularized at the beginning of the seventies. This has allowed to carry out Factor Analysis on arrays with more than two indices. The generic rank may be seen as an upper…
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 note, we present an elementary proof of the fact that the slice rank of a trilinear form over a finite field is bounded above by a linear expression in the analytic rank. The existing proofs by Adiprasito-Kazhdan-Ziegler and…
Structural and computational understanding of tensors is the driving force behind faster matrix multiplication algorithms, the unraveling of quantum entanglement, and the breakthrough on the cap set problem. Strassen's asymptotic spectra…
In problems involving approximation, completion, denoising, dimension reduction, estimation, interpolation, modeling, order reduction, regression, etc, we argue that the near-universal practice of assuming that a function, matrix, or tensor…
For tensors of fixed order, we establish three types of upper bounds for the geometric rank in terms of the subrank. Firstly, we prove that, under a mild condition on the characteristic of the base field, the geometric rank of a tensor is…
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…
The slice-rank method, introduced by Tao as a symmetrized version of the polynomial method of Croot, Lev and Pach and Ellenberg and Gijswijt, has proved to be a useful tool in a variety of combinatorial problems. Explicit tensors have been…
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…
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 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…