Reproduce “Cartan Networks: Group theoretical Hyperbolic Deep Learning” on OpenResearch

About this paper

Federico Milanesio, Matteo Santoro, Pietro G. Fré, Guido Sanguinetti

Researchers from the University of Turin and SISSA, in collaboration with Additati&Partners, developed Cartan Networks, a class of deep learning models that leverage the solvable Lie group structure underlying hyperbolic spaces, achieving competitive or superior performance on regression and classification tasks, notably demonstrating enhanced expressivity with increased network depth. This approach provides a geometrically consistent alternative to existing hyperbolic deep learning methods, which primarily focus on metric properties.

Key insights

• Hyperbolic spaces, as non-compact symmetric spaces, possess a dual nature as both Riemannian manifolds and metrically equivalent solvable Lie groups, which can be leveraged for more intrinsic deep learning architectures.
• The inherent structure of Cartan Networks, particularly the fiber rotation isometry, naturally introduces non-linearities across layers, contributing to expressivity without always requiring explicit pointwise activation functions.
• The proposed group-theoretic framework is generalizable to any non-compact symmetric space, suggesting a unified approach for deep learning on a wider array of complex manifolds beyond just hyperbolic spaces.

Summary by alphaXiv.