Title: Decoupling multivariate polynomials: interconnections between tensorizations,
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.