Equivariant Networks for Pixelized Spheres

Mehran Shakerinava, Siamak Ravanbakhsh
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:9477-9488, 2021.

Abstract

Pixelizations of Platonic solids such as the cube and icosahedron have been widely used to represent spherical data, from climate records to Cosmic Microwave Background maps. Platonic solids have well-known global symmetries. Once we pixelize each face of the solid, each face also possesses its own local symmetries in the form of Euclidean isometries. One way to combine these symmetries is through a hierarchy. However, this approach does not adequately model the interplay between the two levels of symmetry transformations. We show how to model this interplay using ideas from group theory, identify the equivariant linear maps, and introduce equivariant padding that respects these symmetries. Deep networks that use these maps as their building blocks generalize gauge equivariant CNNs on pixelized spheres. These deep networks achieve state-of-the-art results on semantic segmentation for climate data and omnidirectional image processing. Code is available at https://git.io/JGiZA.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-shakerinava21a, title = {Equivariant Networks for Pixelized Spheres}, author = {Shakerinava, Mehran and Ravanbakhsh, Siamak}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {9477--9488}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/shakerinava21a/shakerinava21a.pdf}, url = {https://proceedings.mlr.press/v139/shakerinava21a.html}, abstract = {Pixelizations of Platonic solids such as the cube and icosahedron have been widely used to represent spherical data, from climate records to Cosmic Microwave Background maps. Platonic solids have well-known global symmetries. Once we pixelize each face of the solid, each face also possesses its own local symmetries in the form of Euclidean isometries. One way to combine these symmetries is through a hierarchy. However, this approach does not adequately model the interplay between the two levels of symmetry transformations. We show how to model this interplay using ideas from group theory, identify the equivariant linear maps, and introduce equivariant padding that respects these symmetries. Deep networks that use these maps as their building blocks generalize gauge equivariant CNNs on pixelized spheres. These deep networks achieve state-of-the-art results on semantic segmentation for climate data and omnidirectional image processing. Code is available at https://git.io/JGiZA.} }
Endnote
%0 Conference Paper %T Equivariant Networks for Pixelized Spheres %A Mehran Shakerinava %A Siamak Ravanbakhsh %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-shakerinava21a %I PMLR %P 9477--9488 %U https://proceedings.mlr.press/v139/shakerinava21a.html %V 139 %X Pixelizations of Platonic solids such as the cube and icosahedron have been widely used to represent spherical data, from climate records to Cosmic Microwave Background maps. Platonic solids have well-known global symmetries. Once we pixelize each face of the solid, each face also possesses its own local symmetries in the form of Euclidean isometries. One way to combine these symmetries is through a hierarchy. However, this approach does not adequately model the interplay between the two levels of symmetry transformations. We show how to model this interplay using ideas from group theory, identify the equivariant linear maps, and introduce equivariant padding that respects these symmetries. Deep networks that use these maps as their building blocks generalize gauge equivariant CNNs on pixelized spheres. These deep networks achieve state-of-the-art results on semantic segmentation for climate data and omnidirectional image processing. Code is available at https://git.io/JGiZA.
APA
Shakerinava, M. & Ravanbakhsh, S.. (2021). Equivariant Networks for Pixelized Spheres. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:9477-9488 Available from https://proceedings.mlr.press/v139/shakerinava21a.html.

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