Navigation through Non-Compact Symmetric Spaces: a mathematical perspective on Cartan Neural Networks

TL;DR

This paper explores the mathematical foundations of Cartan Neural Networks, leveraging non-compact symmetric spaces to develop geometrically consistent and interpretable neural network models based on group theory.

Contribution

It provides a detailed mathematical analysis of the geometric properties of Cartan Neural Networks, enhancing their theoretical understanding and interpretability.

Findings

Demonstrated the geometric covariance of Cartan Neural Network layers

Analyzed the interaction of layer maps with symmetric space structures

Established a foundation for geometrically interpretable neural networks

Abstract

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 twin paper under the moniker of Cartan Neural Networks, showing both the feasibility and the performance of these geometric concepts in a machine learning context. The current 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 twin papers constitute a first step towards a fully geometrically interpretable theory of neural networks exploiting group-theoretic structures