Deep Neural Networks are widely used for solving complex problems in several
scientific areas, such as speech recognition, machine translation, image
analysis. The strategies employed to investigate their theoretical properties
mainly rely on Euclidean geometry, but in the last years new approaches based
on Riemannian geometry have been developed. Motivated by some open problems, we
study a particular sequence of maps between manifolds, with the last manifold
of the sequence equipped with a Riemannian metric. We investigate the
structures induced trough pullbacks on the other manifolds of the sequence and
on some related quotients. In particular, we show that the pullbacks of the
final Riemannian metric to any manifolds of the sequence is a degenerate
Riemannian metric inducing a structure of pseudometric space, we show that the
Kolmogorov quotient of this pseudometric space yields a smooth manifold, which
is the base space of a particular vertical bundle. We investigate the
theoretical properties of the maps of such sequence, eventually we focus on the
case of maps between manifolds implementing neural networks of practical
interest and we present some applications of the geometric framework we
introduced in the first part of the paper.

本论文研究基于 Riemannian 几何的新方法，探索深度神经网络在流形之间的映射及其导致的结构，指出其 pullbacks 在其他流形上生成了诱导偏度量空间的退化 Riemann 度量，给出了这种映射的理论性质，并在实用神经网络中应用其几何框架