  Critical jamming transitions are characterized by an astonishing degree of
universality. Analytic and numerical evidence points to the existence of a
large universality class that encompasses finite and infinite dimensional
spheres and continuous constraint satisfaction problems (CCSP) such as the
non-convex perceptron and related models. In this paper we investigate
multilayer neural networks (MLNN) learning random associations as models for
CCSP which could potentially define different jamming universality classes. As
opposed to simple perceptrons and infinite dimensional spheres, which are
described by a single effective field in terms of which the constraints appear
to be one-dimensional, the description of MLNN, involves multiple fields, and
the constraints acquire a multidimensional character. We first study the models
numerically and show that similarly to the perceptron, whenever jamming is
isostatic, the sphere universality class is recovered, we then write the exact
mean-field equations for the models and identify a dimensional reduction
mechanism that leads to a scaling regime identical to one of the infinite
dimensional spheres. We suggest that this mechanism could be general enough to
explain finite dimensional universality.
