Navigation through non-compact symmetric spaces: A mathematical perspective on Cartan neural networks

Recent work has identified non-compact symmetric spaces U/H as a promising class of homogeneous manifolds to develop a geometrically consistent theory of neural networks. An initial implementation of these concepts has been presented in a parallel paper under the moniker of Cartan neural networks, showing both the feasibility and the performance of these geometric concepts in a machine learning context. This paper expands on the mathematical structures underpinning Cartan Neural Networks, detailing the geometric properties of the layers and how the maps between layers interact with such structures to make Cartan neural networks covariant and geometrically interpretable. Together, these papers constitute a first step towards a fully geometrically interpretable theory of neural networks with group-theoretic structures.