Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges
Abstract
The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many highdimensional 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 gradientdescent 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 predefined regularities arising from the underlying lowdimensionality 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 DOI:
 10.48550/arXiv.2104.13478
 arXiv:
 arXiv:2104.13478
 Bibcode:
 2021arXiv210413478B
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Artificial Intelligence;
 Computer Science  Computational Geometry;
 Computer Science  Computer Vision and Pattern Recognition;
 Statistics  Machine Learning
 EPrint:
 156 pages. Work in progress  comments welcome!