Introduction

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 data and 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 sec-

ond, learning by gradient descent-type optimisation, 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 prede-

ﬁned regularities arising from the underlying low-dimensionality and structure

of the physical world. This book is concerned with exposing these regulari-

ties through uniﬁed geometric principles that can be applied throughout a wide

spectrum of applications.

Exploiting the known symmetries of a large system is a powerful and clas-

sical remedy against the curse of dimensionality, and forms the basis of most

physical theories. Deep learning systems are no exception, and since the early

days researchers have adapted neural networks to exploit the low-dimensional

geometry arising from physical measurements, e.g. grids in images, sequences

in time-series, or position and momentum in molecules, and their associated

symmetries, such as translation or rotation. Throughout our exposition, we will

describe these models, as well as many others, as natural instances of the same

underlying principle of geometric regularity.

Geometric Deep Learning is a ‘geometric uniﬁcation’ endeavour in the spirit

of Klein’s Erlangen Programme that serves a dual purpose. On one hand, it

provides a common mathematical framework to study the classical successful

4 Chapter 1

neural network architectures, such as CNNs, RNNs, GNNs, and Transform-

ers. We will show that all of the above can be obtained by the choice of a

geometric domain, its symmetry group, and appropriately constructed invari-

ant and equivariant neural network layers—what we refer to as the ‘Geometric

Deep Learning blueprint.’ We will exemplify instances of this blueprint on the

‘5G of Geometric Deep Learning’: Graphs, Grids, Groups, Geometric Graphs,

and Gauges. On the other hand, Geometric Deep Learning gives a construc-

tive procedure to incorporate prior physical knowledge into neural networks

and provide a principled way to build new architectures. With this premise,

we shall now explore how the ideas central to Geometric Deep Learning have

crystallised through time.

1.1 On the Shoulders of Giants

“Symmetry, as wide or as narrow as you may deﬁne its meaning, is one idea

by which man through the ages has tried to comprehend and create order,

beauty, and perfection.” This somewhat poetic deﬁnition of symmetry is given

in the eponymous book of the great mathematician Hermann Weyl (1952),

his Schwanengesang on the eve of retirement from the Institute for Advanced

Study in Princeton. Weyl traces the special place symmetry has occupied in

science and art to the ancient times, from Sumerian symmetric designs to the

Pythagoreans who believed the circle to be perfect due to its rotational sym-

metry. Plato considered the ﬁve regular polyhedra bearing his name today so

fundamental that they must be the basic building blocks shaping the material

world.

Yet, though Plato is credited with coining the term σνµµϵτρ´ια, which lit-

erally translates as ‘same measure’, he used it only vaguely to convey the

beauty of proportion in art and harmony in music. It was the astronomer and

mathematician Johannes Kepler (1611) to attempt the ﬁrst rigorous analysis of

the symmetric shape of water crystals. In his treatise ‘On the Six-Cornered

Snowﬂake’

he attributed the six-fold dihedral structure of snowﬂakes to

hexagonal packing of particles – an idea that though preceded the clear

understanding of how matter is formed, still holds today as the basis of

crystallography (Ball 2011).

In modern mathematics, symmetry is almost univocally expressed in the

language of group theory. The origins of this theory are attributed to Évariste

Galois, who coined the term and used it to study the solvability of polynomial

equations in the 1830s.

Two other names associated with group theory are

those of Sophus Lie and Felix Klein, who met and worked fruitfully together

for a period of time (Tobies 2019). The former would develop the theory of

Introduction 5

Plato

(ca. 370 BC)

Johannes Kepler

(1571–1630)

Figure 1.1

Plato believed that symmetric polyhedra (“Platonic solids”) were the fundamental

building blocks of nature. Johannes Kepler attributed for the ﬁrst time the six-fold

symmetry of water crystals to the hexagonal packing of particles, antedating modern

crystallography.

continuous symmetries that today bears his name (Lie groups); the latter pro-

claimed group theory to be the organising principle of geometry in his Erlangen

Programme, which we already mentioned in the Preface. Given that Klein’s

Programme is the inspiration for our book, it is worthwhile to spend more time

on its historical context and revolutionary impact.

1.1.1 A Strange New Universe out of Nothing

The foundations of modern geometry were formalised in ancient Greece nearly

2300 years ago by Euclid in a treatise named the Elements (Στ oιχϵ`ια).

Euclidean geometry (which is still taught at school as ‘the geometry’) was

a set of results built upon ﬁve intuitive axioms or postulates. The Fifth Pos-

tulate – stating that it is possible to pass only one line parallel to a given line

through a point outside it – appeared less obvious and an illustrious row of

mathematicians broke their teeth trying to prove it since antiquity, to no avail.

An early approach to the problem of the parallels appears in the eleventh

century Persian treatise A commentary on the difﬁculties concerning the postu-

lates of Euclid’s Elements by Omar Khayyam.

The eighteenth-century Italian

Jesuit priest Giovanni Saccheri (1733) was likely aware of this previous work

judging by the title of his own work Euclides ab omni nævo vindicatus (‘Euclid

cleared of every stain’, see Figure 1.2). Like Khayyam, he considered the

summit angles of a quadrilateral with sides perpendicular to the base. The con-

clusion that acute angles lead to inﬁnitely many non-intersecting lines that can

be passed through a point not on a straight line seemed so counter-intuitive

6 Chapter 1

that he rejected it as ‘repugnatis naturæ linæ rectæ’ (‘repugnant to the nature

of straight lines’).

Euclid

(ca. 300 BC)

Omar Khayyam

(1048–1131)

Figure 1.2

The “Father of Geometry”, Euclid, laid the foundations of modern geometry in the

Elements. Omar Khayyam made early attempts to prove Euclid’s ﬁfth postulate, which

were also referenced in Saccheri’s work (Euclides ab omni nævo vindicatus).

The nineteenth century has brought the realisation that the Fifth Postulate is

not essential and one can construct alternative geometries based on different

notions of parallelism. One such early example is projective geometry, arising,

as the name suggests, in perspective drawing and architecture. In this geom-

etry, points and lines are interchangeable, and there are no parallel lines in

the usual sense: any lines meet in a ‘point at inﬁnity.’ While results in projec-

tive geometry are known since antiquity, it was ﬁrst systematically studied by

Jean-Victor Poncelet (1822)

; see Figure 1.3.

Gérard Desargues

(1591–1661)

Jean-Victor

Poncelet

(1788–1867)

Figure 1.3

Based on the prior work of Gérard Desargues, Jean-Victor Poncelet revived the interest

in projective geometry, one of the earliest examples of geometries not requiring the

parallel postulate.