Approximate decoupling of multivariate polynomials using weighted tensor decomposition

Abstract

Many scientific and engineering disciplines use multivariate polynomials.Decomposing a multivariate polynomial vector function into a sandwiched structure of univariate polynomials surrounded by linear transformations can provide useful insight into the function while reducing the number of parameters. Such a decoupled representation can be realized with techniques based on tensor decomposition methods, but these techniques have only been studied in the exact case. Generalizing the existing techniques to the noisy case is an important next step for the decoupling problem. For this extension, we have considered a weight factor during the tensor decomposition process, leading to an alternating weighted least squares scheme. In addition, we applied the proposed weighted decoupling algorithm in the area of system identification, and we observed smaller model errors with the weighted decoupling algorithm than those with the unweighted decoupling algorithm.

Decoupling multivariate polynomials: Interconnections between tensorizations

“…In both cases, an interesting open question remains how to impose the structure directly on 17 the rank-one components, without resorting to coupled tensor factorizations. Another important question is how to address the approximate decoupling problem, i.e., when we are dealing with noise (see [44] for results on the unstructured case).…”

CPD Updating Using Low-Rank Weights

“…This paper reflects only the authors' views and the Union is not liable for any use that may be made of the contained information; (3) KU Leuven Internal Funds C16/15/059. admit general weight tensors [10,11], it can be beneficial to use a low-rank weight tensor. First, low-rank structure can be exploited to make the decomposition less demanding, both in storage and computation, compared to using a full weight tensor [12].…”