Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges

TLDR

Deep learning has revolutionized data science, enabling high‑dimensional tasks like computer vision, Go, and protein folding by learning hierarchical representations and applying gradient‑based optimization that exploits task‑specific regularities rather than generic high‑dimensional estimation. This text seeks to expose those regularities through unified geometric principles, offering a common mathematical framework for existing architectures such as CNNs, RNNs, GNNs, and Transformers, and a constructive method to embed prior physical knowledge into future neural designs.

Abstract

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer vision, playing Go, or protein folding -- are in fact feasible with appropriate computational scale. Remarkably, the essence of deep learning is built from two simple algorithmic principles: first, the notion of representation or feature learning, whereby adapted, often hierarchical, features capture the appropriate notion of regularity for each task, and second, learning by local gradient-descent type methods, typically implemented as backpropagation. While learning generic functions in high dimensions is a cursed estimation problem, most tasks of interest are not generic, and come with essential pre-defined regularities arising from the underlying low-dimensionality and structure of the physical world. This text is concerned with exposing these regularities through unified geometric principles that can be applied throughout a wide spectrum of applications. Such a 'geometric unification' endeavour, in the spirit of Felix Klein's Erlangen Program, serves a dual purpose: on one hand, it provides a common mathematical framework to study the most successful neural network architectures, such as CNNs, RNNs, GNNs, and Transformers. On the other hand, it gives a constructive procedure to incorporate prior physical knowledge into neural architectures and provide principled way to build future architectures yet to be invented.