  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.
