Convolutional rectifier networks as generalized tensor decompositions

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 realizable by a deep network of polynomial size, require exponential size in order to be realized (or 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.

Original languageEnglish
Title of host publication33rd International Conference on Machine Learning, ICML 2016
EditorsMaria Florina Balcan, Kilian Q. Weinberger
Number of pages20
ISBN (Electronic)9781510829008
StatePublished - 2016
Event33rd International Conference on Machine Learning, ICML 2016 - New York City, United States
Duration: 19 Jun 201624 Jun 2016

Publication series

Name33rd International Conference on Machine Learning, ICML 2016


Conference33rd International Conference on Machine Learning, ICML 2016
Country/TerritoryUnited States
CityNew York City

All Science Journal Classification (ASJC) codes

  • Artificial Intelligence
  • Software
  • Computer Networks and Communications


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