Decoupling multivariate polynomials: interconnections between tensorizations
For researchers working on nonlinear function approximation and parameter reduction, this work provides theoretical unification and practical guidance for choosing between tensor methods, though the results are incremental.
The paper establishes a linear transformation linking two tensor-based approaches for decoupling multivariate polynomials, showing their CP decompositions and uniqueness properties are closely related. It also demonstrates that exploiting intrinsic tensor structure improves uniqueness, extending the methods' applicability.
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.