Navigation through Non-Compact Symmetric Spaces: a mathematical perspective on Cartan Neural Networks
This work provides a foundational step towards a geometrically interpretable theory of neural networks for researchers in mathematical machine learning, though it is incremental as it expands on prior concepts.
The paper tackles the mathematical foundations of Cartan Neural Networks by detailing the geometric properties of layers and their interactions to ensure covariance and interpretability, building on initial implementations that demonstrated feasibility and performance in machine learning.
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