G-equivariant convolutional neural networks

Abstract: Over the past decade, deep learning has revolutionized industry and academic research. Neural networks have been used to solve a multitude of previously unsolved problems and to significantly improve the state-of-the-art on other tasks, in some cases reaching superhuman levels of performance. However, most neural networks have to be carefully adapted to each application and often require large amounts of data and computational resources. Geometric deep learning aims to reduce the amount of information that neural networks have to learn, by taking advantage of geometric properties in data. In particular, equivariant neural networks use (local or global) symmetry to reduce the complexity of a learning task. In this thesis, we investigate a popular deep learning model for tasks exhibiting global symmetry: G-equivariant convolutional neural networks (GCNNs). We analyze the mathematical foundations of GCNNs and discuss where this model fits in the broader scheme of equivariant learning. More specifically, we discuss a general framework for equivariant neural networks using notions from gauge theory, and then show how GCNNs arise from this framework in the presence of global symmetry. We also characterize convolutional layers, the main building blocks of GCNNs, in terms of more general G-equivariant layers that preserve the underlying global symmetry.