index.md

TensorDec

Package for the decomposition of tensors and polynomial-exponential series.

Introduction

The package TensorDec.jl provides tools for the following decomposition problems:

Symmetric tensor decomposition

For symmetric tensors or multivariate homogeneous polynomials \sigma(\mathbf{x}) = \sum_{|\alpha|=d} \sigma_{\alpha} {d \choose \alpha} \mathbf{x}^{\alpha}, we consider their Waring decomposition:

$$\sigma(\mathbf{x}) = \sum_{i=1}^r \omega_i\, (\xi_{i,1} x_1+ \cdots + \xi_{i,n} x_n)^d$$

with r minimal.

Multilinear tensor decomposition

For multilinear tensors, \sigma=(\sigma_{i,j,k})\in E_1 \otimes E_2 \otimes E_3 we consider the decomposition:

$$\sigma = \sum_{i=1}^r \omega_i\, U_i^1 \otimes U_i^2 \otimes U_i^3$$

with U_i^j \in E_j vectors and r minimal.

[Tutorials](@id sec_examples)

Pages = map(file -> joinpath("expl", file), filter(x ->endswith(x, "md"), readdir("expl")))

Manual

Pages = map(file -> joinpath("code", file), filter(x ->endswith(x, "md"), readdir("code"))) 

[Installation](@id sec_installation)

The package is available at https://github.com/AlgebraicGeometricModeling/TensorDec.jl.

To install it from Julia:

] add https://github.com/AlgebraicGeometricModeling/TensorDec.jl

It can then be used as follows:

using TensorDec

For more details, see the [tutorials](@ref sec_examples).

Dependencies

The package TensorDec depends on the following packages:

• LinearAlgebra standard package for linear algebra.
• DynamicPolynomials package on multivariate polynomials represented as lists of monomials.
• MultivariatePolynomials generic interface package for multivariate polynomials.
• MultivariateSeries for duality on multivariate polynomials.

These packages are installed with TensorDec (see [installation](@ref sec_installation)).