Geometric foundations of Deep Learning

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 programme, 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]. 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.