1
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 ve 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’
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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.
2
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 (Στoιξϵ`ια).
Euclidean geometry (which is still taught at school as ‘the geometry’) was
a set of results built upon ve 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.
3
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’).
4
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)
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; 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.