  The power of multivariate functions is their ability to model a wide variety
of phenomena, but have the disadvantages that they lack an intuitive or
interpretable representation, and often require a (very) large number of
parameters. We study decoupled representations of multivariate vector
functions, which are linear combinations of univariate functions in linear
combinations of the input variables. This model structure provides a
description with fewer parameters, and reveals the internal workings in a
simpler way, as the nonlinearities are one-to-one functions. In earlier work, a
tensor-based method was developed for performing this decomposition by using
first-order derivative information. In this article, we generalize this method
and study how the use of second-order derivative information can be
incorporated. By doing this, we are able to push the method towards more
involved configurations, while preserving uniqueness of the underlying tensor
decompositions. Furthermore, even for some non-identifiable structures, the
method seems to return a valid decoupled representation. These results are a
step towards more general data-driven and noise-robust tensor-based framework
for computing decoupled function representations.
