Universal Equivariant Multilayer Perceptrons
Abstract
Group invariant and equivariant Multilayer Perceptrons (MLP), also known as Equivariant Networks and Group Group Convolutional Neural Networks (G-CNN) have achieved remarkable success in learning on a variety of data structures, such as sequences, images, sets, and graphs. This paper proves the universality of a broad class of equivariant MLPs with a single hidden layer. In particular, it is shown that having a hidden layer on which the group acts regularly is sufficient for universal equivariance (invariance). For example, some types of steerable-CNN’s become universal. Another corollary is the unconditional universality of equivariant MLPs for all Abelian groups. A third corollary is the universality of equivariant MLPs with a high-order hidden layer, where we give both group-agnostic bounds and group-specific bounds on the order of the hidden layer that guarantees universal equivariance.
BibTeX
@InProceedings{pmlr-v119-ravanbakhsh20a,
title = {Universal Equivariant Multilayer Perceptrons},
author = {Ravanbakhsh, Siamak},
booktitle = {Proceedings of the 37th International Conference on Machine Learning},
pages = {7996--8006},
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/ravanbakhsh20a/ravanbakhsh20a.pdf},
url = {https://proceedings.mlr.press/v119/ravanbakhsh20a.html},
abstract = {Group invariant and equivariant Multilayer Perceptrons (MLP), also known as Equivariant Networks and Group Group Convolutional Neural Networks (G-CNN) have achieved remarkable success in learning on a variety of data structures, such as sequences, images, sets, and graphs. This paper proves the universality of a broad class of equivariant MLPs with a single hidden layer. In particular, it is shown that having a hidden layer on which the group acts regularly is sufficient for universal equivariance (invariance). For example, some types of steerable-CNN’s become universal. Another corollary is the unconditional universality of equivariant MLPs for all Abelian groups. A third corollary is the universality of equivariant MLPs with a high-order hidden layer, where we give both group-agnostic bounds and group-specific bounds on the order of the hidden layer that guarantees universal equivariance.}
}