  Motivated by structures that appear in deep neural networks, we investigate
nonlinear composite models alternating proximity and affine operators defined
on different spaces. We first show that a wide range of activation operators
used in neural networks are actually proximity operators. We then establish
conditions for the averagedness of the proposed composite constructs and
investigate their asymptotic properties. It is shown that the limit of the
resulting process solves a variational inequality which, in general, does not
derive from a minimization problem.
