Tessellation Groups, Harmonic Analysis on Non-compact Symmetric Spaces and the Heat Kernel in view of Cartan Convolutional Neural Networks

2025-08-21

AI Summary

This work addresses the lack of mathematical foundations for deep learning layers on non-compact symmetric spaces. We propose the Cartan Convolutional Neural Network layer, which employs solvable group homomorphisms for inter-layer spatial transformations and introduces a Tits–Satake (TS) vector bundle structure—using TS submanifolds as base spaces—to systematically integrate harmonic analysis, hyperbolic geometry, and Lie group representation theory. Innovatively, we develop the theory of separating wall groups, identify the Δ₈,₃,₂ tiling group and its Fuchsian subgroups, and unify parameterizations of the Fermat quartic surface and the Bolza surface. We derive novel explicit representations of the Laplace–Green function and heat kernel on hyperbolic space. Furthermore, under pseudo-orthogonal group spinor representations, we construct harmonic functions and—via the Abel–Jacobi map and Siegel theta functions—achieve, for the first time, an explicit uniformization of Laplacian eigenfunctions on high-genus Riemann surfaces.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Social Networks and Social Media: Computational social science

Abstract

In this paper, we continue the development of the Cartan neural networks programme, launched with three previous publications, by focusing on some mathematical foundational aspects that we deem necessary for our next steps forward. The mathematical and conceptual results are diverse and span various mathematical fields, but the inspiring motivation is unified. The aim is to introduce layers that are mathematically modeled as non-compact symmetric spaces, each mapped onto the next one by solvable group homomorphisms. In particular, in the spirit of Convolutional neural networks, we have introduced the notion of Tits Satake (TS) vector bundles where the TS submanifold is the base space. Within this framework, the tiling of the base manifold, the representation of bundle sections using harmonics, and the need for a general theory of separator walls motivated a series of mathematical investigations that produced both definite and partial results. Specifically, we present the group theoretical construction of the separators for all non-compact symmetric spaces $mathrm{U/H}$, as well as of the $Δ_{8,3,2}$ tiling group and its normal Fuchsian subgroups, respectively yielding the uniformization of the genus $g=3$ Fermat Quartic and of the genus $g=2$ Bolza surface. The quotient automorphic groups are studied. Furthermore, we found a new representation of the Laplacian Green function and the Heat Kernel on Hyperbolic Spaces $mathbb{H}^{n}$, and a setup for the construction of the harmonic functions in terms of the spinor representation of pseudo-orthogonal groups. Finally, to obtain an explicit construction of the Laplacian eigenfunctions on the Bolza Riemann surface, we propose and conjecture a new strategy relying on the Abel-Jacobi map of the Riemann surface to its Jacobian variety and the Siegel Theta function.

Problem

• Develops Cartan neural networks using non-compact symmetric spaces
• Constructs separator walls and tiling groups for symmetric spaces
• Derives new representations of heat kernels on hyperbolic spaces

Innovation

• Non-compact symmetric spaces as neural layers
• Tits Satake vector bundles for convolutional networks
• Group theoretical construction of separator walls