By Michael Bronstein (Imperial College), Joan Bruna (NYU), Taco Cohen (Qualcomm), and Petar Veličković (DeepMind).

This blog post is based on the new “proto-book” M. M. Bronstein, J. Bruna, T. Cohen, and P. Veličković, Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges (2021), Petar’s talk at Cambridge, and Michael’s keynote talk at ICLR 2021.

 

In October 1872, the philosophy faculty of a small university in the Bavarian city of Erlangen appointed a new young professor. As customary, he was requested to deliver an inaugural research program, which he published under the somewhat long and boring title Vergleichende Betrachtungen über neuere geometrische Forschungen (“A comparative review of recent researches in geometry”). The professor was Felix Klein, only 23 years of age at that time, and his inaugural work has entered the annals of mathematics as the “Erlangen Programme” [1].

Felix Klein and his Erlangen Programme. Image: Wikipedia/University of Michigan Historical Math Collections.

The nineteenth century had been remarkably fruitful for geometry. For the first time in nearly two thousand years after Euclid, the construction of projective geometry by Poncelet, hyperbolic geometry by Gauss, Bolyai, and Lobachevsky, and elliptic geometry by Riemann showed that an entire zoo of diverse geometries was possible. However, these constructions had quickly diverged into independent and unrelated fields, with many mathematicians of that period questioning how the different geometries are related to each other and what actually defines a geometry.

The breakthrough insight of Klein was to approach the definition of geometry as the study of invariants, or in other words, structures that are preserved under a certain type of transformations (symmetries). Klein used the formalism of group theory to define such transformations and use the hierarchy of groups and their subgroups in order to classify different geometries arising from them. Thus, the group of rigid motions leads to the traditional Euclidean geometry, while affine or projective transformations produce, respectively, the affine and projective geometries. Importantly, the Erlangen Program was limited to homogeneous spaces [2] and initially excluded Riemannian geometry.

Klein’s Erlangen Program approached geometry as the study of properties remaining invariant under certain types of transformations. 2D Euclidean geometry is defined by…

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