  Neural networks with the Rectified Linear Unit (ReLU) nonlinearity are
described by a vector of parameters $\theta$, and realized as a piecewise
linear continuous function $R_{\theta}: x \in \mathbb R^{d} \mapsto
R_{\theta}(x) \in \mathbb R^{k}$. Natural scalings and permutations operations
on the parameters $\theta$ leave the realization unchanged, leading to
equivalence classes of parameters that yield the same realization. These
considerations in turn lead to the notion of identifiability -- the ability to
recover (the equivalence class of) $\theta$ from the sole knowledge of its
realization $R_{\theta}$. The overall objective of this paper is to introduce
an embedding for ReLU neural networks of any depth, $\Phi(\theta)$, that is
invariant to scalings and that provides a locally linear parameterization of
the realization of the network. Leveraging these two key properties, we derive
some conditions under which a deep ReLU network is indeed locally identifiable
from the knowledge of the realization on a finite set of samples $x_{i} \in
\mathbb R^{d}$. We study the shallow case in more depth, establishing necessary
and sufficient conditions for the network to be identifiable from a bounded
subset $\mathcal X \subseteq \mathbb R^{d}$.
