On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups

Risi Kondor, Shubhendu Trivedi
; Proceedings of the 35th International Conference on Machine Learning, PMLR 80:2747-2755, 2018.

Abstract

Convolutional neural networks have been extremely successful in the image recognition domain because they ensure equivariance with respect to translations. There have been many recent attempts to generalize this framework to other domains, including graphs and data lying on manifolds. In this paper we give a rigorous, theoretical treatment of convolution and equivariance in neural networks with respect to not just translations, but the action of any compact group. Our main result is to prove that (given some natural constraints) convolutional structure is not just a sufficient, but also a necessary condition for equivariance to the action of a compact group. Our exposition makes use of concepts from representation theory and noncommutative harmonic analysis and derives new generalized convolution formulae.

Cite this Paper


BibTeX
@InProceedings{pmlr-v80-kondor18a, title = {On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups}, author = {Kondor, Risi and Trivedi, Shubhendu}, booktitle = {Proceedings of the 35th International Conference on Machine Learning}, pages = {2747--2755}, year = {2018}, editor = {Jennifer Dy and Andreas Krause}, volume = {80}, series = {Proceedings of Machine Learning Research}, address = {Stockholmsmässan, Stockholm Sweden}, month = {10--15 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v80/kondor18a/kondor18a.pdf}, url = {http://proceedings.mlr.press/v80/kondor18a.html}, abstract = {Convolutional neural networks have been extremely successful in the image recognition domain because they ensure equivariance with respect to translations. There have been many recent attempts to generalize this framework to other domains, including graphs and data lying on manifolds. In this paper we give a rigorous, theoretical treatment of convolution and equivariance in neural networks with respect to not just translations, but the action of any compact group. Our main result is to prove that (given some natural constraints) convolutional structure is not just a sufficient, but also a necessary condition for equivariance to the action of a compact group. Our exposition makes use of concepts from representation theory and noncommutative harmonic analysis and derives new generalized convolution formulae.} }
Endnote
%0 Conference Paper %T On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups %A Risi Kondor %A Shubhendu Trivedi %B Proceedings of the 35th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2018 %E Jennifer Dy %E Andreas Krause %F pmlr-v80-kondor18a %I PMLR %J Proceedings of Machine Learning Research %P 2747--2755 %U http://proceedings.mlr.press %V 80 %W PMLR %X Convolutional neural networks have been extremely successful in the image recognition domain because they ensure equivariance with respect to translations. There have been many recent attempts to generalize this framework to other domains, including graphs and data lying on manifolds. In this paper we give a rigorous, theoretical treatment of convolution and equivariance in neural networks with respect to not just translations, but the action of any compact group. Our main result is to prove that (given some natural constraints) convolutional structure is not just a sufficient, but also a necessary condition for equivariance to the action of a compact group. Our exposition makes use of concepts from representation theory and noncommutative harmonic analysis and derives new generalized convolution formulae.
APA
Kondor, R. & Trivedi, S.. (2018). On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups. Proceedings of the 35th International Conference on Machine Learning, in PMLR 80:2747-2755

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