Equivariant Networks for Pixelized Spheres

TL;DR

This paper introduces a novel group-theoretic approach to model symmetries in pixelized spherical data, leading to deep networks that outperform existing methods in climate and image tasks.

Contribution

It develops a new framework for modeling the interplay of global and local symmetries in pixelized spheres using equivariant linear maps and padding.

Findings

Achieves state-of-the-art semantic segmentation results.

Generalizes gauge equivariant CNNs for pixelized spheres.

Provides open-source code for the proposed methods.

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.