Hyperbolic Busemann Neural Networks

2026-02-21

AI Summary

This work addresses the core challenge of constructing efficient neural networks for hierarchical data in hyperbolic space by proposing an intrinsic design based on Busemann functions. The authors introduce Busemann-based multinomial logistic regression (MLR) and fully connected (FC) layers that operate directly in hyperbolic space without requiring embedding into Euclidean space. The proposed architecture features compact parameters, strong interpretability, and supports efficient batch computation, while naturally reducing to the Euclidean case as a limiting scenario. Empirical evaluations across diverse tasks—including image classification, genomic sequence modeling, node classification, and link prediction—demonstrate that the proposed modules significantly outperform existing hyperbolic neural network approaches, achieving substantial gains in both performance and computational efficiency.

Technology Category

• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP
• Humans and AI: Human-in-the-loop Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Hyperbolic spaces provide a natural geometry for representing hierarchical and tree-structured data due to their exponential volume growth. To leverage these benefits, neural networks require intrinsic and efficient components that operate directly in hyperbolic space. In this work, we lift two core components of neural networks, Multinomial Logistic Regression (MLR) and Fully Connected (FC) layers, into hyperbolic space via Busemann functions, resulting in Busemann MLR (BMLR) and Busemann FC (BFC) layers with a unified mathematical interpretation. BMLR provides compact parameters, a point-to-horosphere distance interpretation, batch-efficient computation, and a Euclidean limit, while BFC generalizes FC and activation layers with comparable complexity. Experiments on image classification, genome sequence learning, node classification, and link prediction demonstrate improvements in effectiveness and efficiency over prior hyperbolic layers. The code is available at https://github.com/GitZH-Chen/HBNN.

Problem

• hyperbolic space
• neural networks
• hierarchical data
• Busemann functions
• tree-structured data

Innovation

• Hyperbolic Neural Networks
• Busemann functions
• Multinomial Logistic Regression
• Fully Connected Layers
• Hierarchical Representation