On the Expressive Power of Deep Learning: A Tensor Analysis

TL;DR

This paper provides a theoretical foundation showing that deep hierarchical networks can efficiently represent complex functions that shallow networks require exponential size to approximate, highlighting the expressive power of deep learning architectures.

Contribution

The paper establishes a connection between deep networks and hierarchical tensor factorizations, proving the exponential advantage of deep networks over shallow ones for certain functions.

Findings

Deep networks correspond to Hierarchical Tucker tensor decompositions.

Shallow networks require exponential size to approximate functions realizable by deep networks.

Theoretical results apply to architectures like SimNets, linking theory with empirical deep learning practices.

Abstract

It has long been conjectured that hypotheses spaces suitable for data that is compositional in nature, such as text or images, may be more efficiently represented with deep hierarchical networks than with shallow ones. Despite the vast empirical evidence supporting this belief, theoretical justifications to date are limited. In particular, they do not account for the locality, sharing and pooling constructs of convolutional networks, the most successful deep learning architecture to date. In this work we derive a deep network architecture based on arithmetic circuits that inherently employs locality, sharing and pooling. An equivalence between the networks and hierarchical tensor factorizations is established. We show that a shallow network corresponds to CP (rank-1) decomposition, whereas a deep network corresponds to Hierarchical Tucker decomposition. Using tools from measure theory and matrix algebra, we prove that besides a negligible set, all functions that can be implemented by a deep network of polynomial size, require exponential size in order to be realized (or even approximated) by a shallow network. Since log-space computation transforms our networks into SimNets, the result applies directly to a deep learning architecture demonstrating promising empirical performance. The construction and theory developed in this paper shed new light on various practices and ideas employed by the deep learning community.