  We combine Riemannian geometry with the mean field theory of high dimensional
chaos to study the nature of signal propagation in generic, deep neural
networks with random weights. Our results reveal an order-to-chaos expressivity
phase transition, with networks in the chaotic phase computing nonlinear
functions whose global curvature grows exponentially with depth but not width.
We prove this generic class of deep random functions cannot be efficiently
computed by any shallow network, going beyond prior work restricted to the
analysis of single functions. Moreover, we formalize and quantitatively
demonstrate the long conjectured idea that deep networks can disentangle highly
curved manifolds in input space into flat manifolds in hidden space. Our
theoretical analysis of the expressive power of deep networks broadly applies
to arbitrary nonlinearities, and provides a quantitative underpinning for
previously abstract notions about the geometry of deep functions.
