Mathematical Foundations of Equivariant Neural Networks

Sammanfattning: Deep learning has revolutionized industry and academic research. Over the past decade, 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. However, training a neural network typically requires large amounts of data and computational resources. This is not only costly, it also prevents deep learning from being used for applications in which data is scarce. It is therefore important to simplify the learning task by incorporating inductive biases - prior knowledge and assumptions - into the neural network design. 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 symmetries to reduce the complexity of a learning task. Symmetries are properties that do not change under certain transformations. For example, rotation-equivariant neural networks trained to identify tumors in medical images are not sensitive to the orientation of a tumor within an image. Another example is graph neural networks, i.e., permutation-equivariant neural networks that operate on graphs, such as molecules or social networks. Permuting the ordering of vertices and edges either transforms the output of a graph neural network in a predictable way (equivariance), or has no effect on the output (invariance). In this thesis we study a fiber bundle theoretic framework for equivariant neural networks. Fiber bundles are often used in mathematics and theoretical physics to model nontrivial geometries, and offer a geometric approach to symmetry. This framework connects to many different areas of mathematics, including Fourier analysis, representation theory, and gauge theory, thus providing a large set of tools for analyzing equivariant neural networks.