PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions

Zhengyang Shen, Lingshen He, Zhouchen Lin, Jinwen Ma
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:8697-8706, 2020.

Abstract

Recent research has shown that incorporating equivariance into neural network architectures is very helpful, and there have been some works investigating the equivariance of networks under group actions. However, as digital images and feature maps are on the discrete meshgrid, corresponding equivariance-preserving transformation groups are very limited. In this work, we deal with this issue from the connection between convolutions and partial differential operators (PDOs). In theory, assuming inputs to be smooth, we transform PDOs and propose a system which is equivariant to a much more general continuous group, the $n$-dimension Euclidean group. In implementation, we discretize the system using the numerical schemes of PDOs, deriving approximately equivariant convolutions (PDO-eConvs). Theoretically, the approximation error of PDO-eConvs is of the quadratic order. It is the first time that the error analysis is provided when the equivariance is approximate. Extensive experiments on rotated MNIST and natural image classification show that PDO-eConvs perform competitively yet use parameters much more efficiently. Particularly, compared with Wide ResNets, our methods result in better results using only 12.6% parameters.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-shen20a, title = {{PDO}-e{C}onvs: Partial Differential Operator Based Equivariant Convolutions}, author = {Shen, Zhengyang and He, Lingshen and Lin, Zhouchen and Ma, Jinwen}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {8697--8706}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/shen20a/shen20a.pdf}, url = {https://proceedings.mlr.press/v119/shen20a.html}, abstract = {Recent research has shown that incorporating equivariance into neural network architectures is very helpful, and there have been some works investigating the equivariance of networks under group actions. However, as digital images and feature maps are on the discrete meshgrid, corresponding equivariance-preserving transformation groups are very limited. In this work, we deal with this issue from the connection between convolutions and partial differential operators (PDOs). In theory, assuming inputs to be smooth, we transform PDOs and propose a system which is equivariant to a much more general continuous group, the $n$-dimension Euclidean group. In implementation, we discretize the system using the numerical schemes of PDOs, deriving approximately equivariant convolutions (PDO-eConvs). Theoretically, the approximation error of PDO-eConvs is of the quadratic order. It is the first time that the error analysis is provided when the equivariance is approximate. Extensive experiments on rotated MNIST and natural image classification show that PDO-eConvs perform competitively yet use parameters much more efficiently. Particularly, compared with Wide ResNets, our methods result in better results using only 12.6% parameters.} }
Endnote
%0 Conference Paper %T PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions %A Zhengyang Shen %A Lingshen He %A Zhouchen Lin %A Jinwen Ma %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-shen20a %I PMLR %P 8697--8706 %U https://proceedings.mlr.press/v119/shen20a.html %V 119 %X Recent research has shown that incorporating equivariance into neural network architectures is very helpful, and there have been some works investigating the equivariance of networks under group actions. However, as digital images and feature maps are on the discrete meshgrid, corresponding equivariance-preserving transformation groups are very limited. In this work, we deal with this issue from the connection between convolutions and partial differential operators (PDOs). In theory, assuming inputs to be smooth, we transform PDOs and propose a system which is equivariant to a much more general continuous group, the $n$-dimension Euclidean group. In implementation, we discretize the system using the numerical schemes of PDOs, deriving approximately equivariant convolutions (PDO-eConvs). Theoretically, the approximation error of PDO-eConvs is of the quadratic order. It is the first time that the error analysis is provided when the equivariance is approximate. Extensive experiments on rotated MNIST and natural image classification show that PDO-eConvs perform competitively yet use parameters much more efficiently. Particularly, compared with Wide ResNets, our methods result in better results using only 12.6% parameters.
APA
Shen, Z., He, L., Lin, Z. & Ma, J.. (2020). PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:8697-8706 Available from https://proceedings.mlr.press/v119/shen20a.html.

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