Expressive power of outer product manifolds on feed-forward neural networks

TL;DR

This paper introduces a Riemannian geometric framework to analyze and optimize the expressive power of hierarchical feedforward neural networks, enabling efficient training and potential performance improvements.

Contribution

It develops a reparametrization invariant Riemannian metric to understand hierarchical structures, allowing early switching to shallow networks and improving training efficiency.

Findings

Approximate metric improves performance after few training epochs

Sparse representations enable switching to shallow networks

Method sometimes surpasses original network performance

Abstract

Hierarchical neural networks are exponentially more efficient than their corresponding "shallow" counterpart with the same expressive power, but involve huge number of parameters and require tedious amounts of training. Our main idea is to mathematically understand and describe the hierarchical structure of feedforward neural networks by reparametrization invariant Riemannian metrics. By computing or approximating the tangent subspace, we better utilize the original network via sparse representations that enables switching to shallow networks after a very early training stage. Our experiments show that the proposed approximation of the metric improves and sometimes even surpasses the achievable performance of the original network significantly even after a few epochs of training the original feedforward network.