Convolutional Rectifier Networks as Generalized Tensor Decompositions

TL;DR

This paper explores the expressive power of convolutional rectifier networks, revealing their limitations compared to convolutional arithmetic circuits, and suggests that training arithmetic circuits could lead to more powerful deep learning models.

Contribution

It introduces a novel connection between convolutional rectifier networks and tensor decompositions, providing new theoretical insights into their expressive capabilities and limitations.

Findings

Rectifier networks are universal with max pooling but not with average pooling.

Depth efficiency is weaker in rectifier networks than in arithmetic circuits.

Training arithmetic circuits may lead to more powerful deep learning architectures.

Abstract

Convolutional rectifier networks, i.e. convolutional neural networks with rectified linear activation and max or average pooling, are the cornerstone of modern deep learning. However, despite their wide use and success, our theoretical understanding of the expressive properties that drive these networks is partial at best. On the other hand, we have a much firmer grasp of these issues in the world of arithmetic circuits. Specifically, it is known that convolutional arithmetic circuits possess the property of "complete depth efficiency", meaning 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 implemented (or even approximated) by a shallow network. In this paper we describe a construction based on generalized tensor decompositions, that transforms convolutional arithmetic circuits into convolutional rectifier networks. We then use mathematical tools available from the world of arithmetic circuits to prove new results. First, we show that convolutional rectifier networks are universal with max pooling but not with average pooling. Second, and more importantly, we show that depth efficiency is weaker with convolutional rectifier networks than it is with convolutional arithmetic circuits. This leads us to believe that developing effective methods for training convolutional arithmetic circuits, thereby fulfilling their expressive potential, may give rise to a deep learning architecture that is provably superior to convolutional rectifier networks but has so far been overlooked by practitioners.