Ever wanted to do a convolution on a Klein Bottle? This paper defines CNNs over manifolds such that they are independent of which coordinate frame you choose. Amazingly, this then results in an efficient practical method to achieve state-of-the-art in several tasks!

https://youtu.be/wZWn7Hm8osA

Abstract: The principle of equivariance to symmetry transformations enables a theoretically grounded approach to neural network architecture design. Equivariant networks have shown excellent performance and data efficiency on vision and medical imaging problems that exhibit symmetries. Here we show how this principle can be extended beyond global symmetries to local gauge transformations. This enables the development of a very general class of convolutional neural networks on manifolds that depend only on the intrinsic geometry, and which includes many popular methods from equivariant and geometric deep learning. We implement gauge equivariant CNNs for signals defined on the surface of the icosahedron, which provides a reasonable approximation of the sphere. By choosing to work with this very regular manifold, we are able to implement the gauge equivariant convolution using a single conv2d call, making it a highly scalable and practical alternative to Spherical CNNs. Using this method, we demonstrate substantial improvements over previous methods on the task of segmenting omnidirectional images and global climate patterns.

Authors: Taco S. Cohen, Maurice Weiler, Berkay Kicanaoglu, Max Welling

Paper: https://arxiv.org/abs/1902.04615