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Deep Neural Networks are widely used for solving complex problems in severalscientific areas, such as speech recognition, machine translation, imageanalysis. The strategies employed to investigate their theoretical propertiesmainly rely on Euclidean geometry, but in the last years new approaches basedon Riemannian geometry have been developed. Motivated by some open problems, westudy a particular sequence of maps between manifolds, with the last manifoldof the sequence equipped with a Riemannian metric. We investigate thestructures induced trough pullbacks on the other manifolds of the sequence andon some related quotients. In particular, we show that the pullbacks of thefinal Riemannian metric to any manifolds of the sequence is a degenerateRiemannian metric inducing a structure of pseudometric space, we show that theKolmogorov quotient of this pseudometric space yields a smooth manifold, whichis the base space of a particular vertical bundle. We investigate thetheoretical properties of the maps of such sequence, eventually we focus on thecase of maps between manifolds implementing neural networks of practicalinterest and we present some applications of the geometric framework weintroduced in the first part of the paper.

A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations