  In this paper we approach the problem of unique and stable identifiability of
generic deep artificial neural networks with pyramidal shape and smooth
activation functions from a finite number of input-output samples. More
specifically we introduce the so-called entangled weights, which compose
weights of successive layers intertwined with suitable diagonal and invertible
matrices depending on the activation functions and their shifts. We prove that
entangled weights are completely and stably approximated by an efficient and
robust algorithm as soon as $\mathcal O(D^2 \times m)$ nonadaptive input-output
samples of the network are collected, where $D$ is the input dimension and $m$
is the number of neurons of the network. Moreover, we empirically observe that
the approach applies to networks with up to $\mathcal O(D \times m_L)$ neurons,
where $m_L$ is the number of output neurons at layer $L$. Provided knowledge of
layer assignments of entangled weights and of remaining scaling and shift
parameters, which may be further heuristically obtained by least squares, the
entangled weights identify the network completely and uniquely. To highlight
the relevance of the theoretical result of stable recovery of entangled
weights, we present numerical experiments, which demonstrate that multilayered
networks with generic weights can be robustly identified and therefore
uniformly approximated by the presented algorithmic pipeline. In contrast
backpropagation cannot generalize stably very well in this setting, being
always limited by relatively large uniform error. In terms of practical impact,
our study shows that we can relate input-output information uniquely and stably
to network parameters, providing a form of explainability. Moreover, our method
paves the way for compression of overparametrized networks and for the training
of minimal complexity networks.
