# Decoupling multivariate polynomials: interconnections between tensorizations

## TL;DR

This paper explores the relationships between different tensor-based methods for decoupling multivariate polynomials, revealing their connections and how considering intrinsic tensor structures enhances decomposition uniqueness and applicability.

## Contribution

It uncovers the linear relationships between tensor representations in polynomial decoupling and improves decomposition uniqueness by leveraging intrinsic tensor structures.

## Key findings

- Tensors involved are linearly related with interconnected CP decompositions.
- Considering intrinsic tensor structure enhances uniqueness of decompositions.
- The approach broadens the applicability of tensor-based polynomial decoupling methods.

## Abstract

Decoupling multivariate polynomials is useful for obtaining an insight into the workings of a nonlinear mapping, performing parameter reduction, or approximating nonlinear functions. Several different tensor-based approaches have been proposed independently for this task, involving different tensor representations of the functions, and ultimately leading to a canonical polyadic decomposition. We first show that the involved tensors are related by a linear transformation, and that their CP decompositions and uniqueness properties are closely related. This connection provides a way to better assess which of the methods should be favored in certain problem settings, and may be a starting point to unify the two approaches. Second, we show that taking into account the previously ignored intrinsic structure in the tensor decompositions improves the uniqueness properties of the decompositions and thus enlarges the applicability range of the methods.