Exponential expressivity in deep neural networks through transient chaos

TL;DR

This paper combines Riemannian geometry and chaos theory to analyze deep neural networks, revealing an order-to-chaos transition that exponentially increases expressivity with depth and formalizing how deep networks disentangle complex input manifolds.

Contribution

It introduces a novel theoretical framework linking geometry and chaos to explain deep networks' exponential expressivity and their ability to disentangle complex manifolds.

Findings

Networks in the chaotic phase compute functions with exponentially growing curvature.

Deep networks cannot be efficiently approximated by shallow ones.

Deep networks can transform highly curved input manifolds into flatter hidden representations.

Abstract

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.