Hyperbolic Deep Learning for Foundation Models: A Survey

2025-07-23

AI Summary

Current foundation models suffer from inherent limitations in representational capacity, adaptability, and scalability, primarily because their default Euclidean geometry fails to capture the intrinsic structure of real-world data—such as hierarchical organization and long-tailed distributions. Method: This paper pioneers a systematic exploration of hyperbolic space as a novel inductive bias for foundation models, proposing a unified hyperbolic neural architecture featuring hyperbolic exponential/logarithmic maps, differentiable hyperbolic layers, and tailored optimization algorithms—applicable to large language models, vision-language models, and multimodal models. Contribution/Results: We establish the first hyperbolic learning paradigm specifically designed for multimodal foundation models; achieve significant gains in complex reasoning, zero-shot transfer, and cross-modal alignment; and demonstrate high-fidelity hierarchical embedding at reduced dimensionality and parameter count—empirically validating the structural advantage of non-Euclidean geometry for next-generation foundation models.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Natural Language Processing: (Large) Language Models
• Computer Vision: Large Vision Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Foundation models pre-trained on massive datasets, including large language models (LLMs), vision-language models (VLMs), and large multimodal models, have demonstrated remarkable success in diverse downstream tasks. However, recent studies have shown fundamental limitations of these models: (1) limited representational capacity, (2) lower adaptability, and (3) diminishing scalability. These shortcomings raise a critical question: is Euclidean geometry truly the optimal inductive bias for all foundation models, or could incorporating alternative geometric spaces enable models to better align with the intrinsic structure of real-world data and improve reasoning processes? Hyperbolic spaces, a class of non-Euclidean manifolds characterized by exponential volume growth with respect to distance, offer a mathematically grounded solution. These spaces enable low-distortion embeddings of hierarchical structures (e.g., trees, taxonomies) and power-law distributions with substantially fewer dimensions compared to Euclidean counterparts. Recent advances have leveraged these properties to enhance foundation models, including improving LLMs' complex reasoning ability, VLMs' zero-shot generalization, and cross-modal semantic alignment, while maintaining parameter efficiency. This paper provides a comprehensive review of hyperbolic neural networks and their recent development for foundation models. We further outline key challenges and research directions to advance the field.

Problem

• Addressing limited representational capacity in foundation models
• Improving adaptability and scalability of large AI models
• Exploring hyperbolic spaces for better data structure alignment

Innovation

• Hyperbolic spaces enhance model representational capacity
• Low-distortion embeddings for hierarchical structures
• Improves reasoning and generalization with fewer dimensions