  We address the structure identification and the uniform approximation of sums
of ridge functions $f(x)=\sum_{i=1}^m g_i(a_i\cdot x)$ on ${\mathbb R}^d$,
representing a general form of a shallow feed-forward neural network, from a
small number of query samples. Higher order differentiation, as used in our
constructive approximations, of sums of ridge functions or of their
compositions, as in deeper neural network, yields a natural connection between
neural network weight identification and tensor product decomposition
identification. In the case of the shallowest feed-forward neural network,
second order differentiation and tensors of order two (i.e., matrices) suffice
as we prove in this paper. We use two sampling schemes to perform approximate
differentiation - active sampling, where the sampling points are universal,
actively, and randomly designed, and passive sampling, where sampling points
were preselected at random from a distribution with known density. Based on
multiple gathered approximated first and second order differentials, our
general approximation strategy is developed as a sequence of algorithms to
perform individual sub-tasks. We first perform an active subspace search by
approximating the span of the weight vectors $a_1,\dots,a_m$. Then we use a
straightforward substitution, which reduces the dimensionality of the problem
from $d$ to $m$. The core of the construction is then the stable and efficient
approximation of weights expressed in terms of rank-$1$ matrices $a_i \otimes
a_i$, realized by formulating their individual identification as a suitable
nonlinear program. We prove the successful identification by this program of
weight vectors being close to orthonormal and we also show how we can
costructively reduce to this case by a whitening procedure, without loss of any
generality.
