Equivariant And Coordinate Independent Convolutional Networks: A Gauge Field Theory Of Neural Networks

What is the appropriate geometric structure for neural networks for processing spatial signals like images, point clouds, or tensor fields? This question takes us on a journey towards a gauge field theory of convolutional neural networks.A considerable part of machine learning applications is concerned with analyzing fields of feature vectors on Euclidean spaces or more general manifolds. Model predictions should thereby remain consistent when applying geometric transformations to the fields or the coordinate systems describing them. This book derives and characterizes the implied symmetry constraints on thus defined equivariant convolutional neural networks. Instead of focusing on one specific setting, it develops a general representation theoretic formulation for arbitrary symmetry groups and spaces. The theory is made concrete in several chapters discussing implementations of equivariant convolutional networks on various manifolds.This monograph is essential reading for anyone interested in signal processing in the presence of symmetries. It is relevant for any applications where patterns appear in various geometric poses, including, for instance, medical and satellite imaging, molecule generation, or climate modeling.