# Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries

arXiv: https://arxiv.org/abs/2504.08896 · HTML: https://mlsanity.com/paper/2504.08896/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 112 of its 166 references that are in the map.
- First posted on arXiv 11 Apr 2025.

## Builds on (most cited first, 30 of 112)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [Autoregressive Model Beats Diffusion: Llama for Scalable Image Generation](https://mlsanity.com/paper/2406.06525/) (2024)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [GPT-NeoX-20B: An Open-Source Autoregressive Language Model](https://mlsanity.com/paper/2204.06745/) (2022)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)

## Built on by (most cited first, 18 of 18)

- [HELM: Hyperbolic Large Language Models via Mixture-of-Curvature Experts](https://mlsanity.com/paper/2505.24722/) (2025)
- [Hyperbolic Deep Learning for Foundation Models: A Survey](https://mlsanity.com/paper/2507.17787/) (2025)
- [The Curved Spacetime of Transformer Architectures](https://mlsanity.com/paper/2511.03060/) (2025)
- [TextBO: Bayesian Optimization in Language Space for Eval-Efficient Self-Improving AI](https://mlsanity.com/paper/2511.12063/) (2025)
- [Text Has Curvature](https://mlsanity.com/paper/2602.13418/) (2026)
- [Uncertainty-guided Compositional Alignment with Part-to-Whole Semantic Representativeness in Hyperbolic Vision-Language Models](https://mlsanity.com/paper/2603.22042/) (2026)
- [Hyperbolic Multimodal Continual Learning](https://mlsanity.com/paper/2608.09572/) (2026)
- [FlatLand: Personalized Graph Federated Learning via Tailored Lorentz Space](https://mlsanity.com/paper/2608.21096/) (2026)
- [CAT: Curvature-Adaptive Transformers for Geometry-Aware Learning](https://mlsanity.com/paper/2510.01634/) (2025)
- [HypRAG: Hyperbolic Dense Retrieval for Retrieval Augmented Generation](https://mlsanity.com/paper/2602.07739/) (2026)
- [Parameter-Efficient Fine-Tuning of LLMs with Mixture of Space Experts](https://mlsanity.com/paper/2602.14490/) (2026)
- [Memory Has Geometry: Non-Uniform Geometric Memory for Long-Horizon Personalized AI](https://mlsanity.com/paper/2609.17969/) (2026)
- [Hyper3-CLIP: Hierarchy-Conditioned Hyperbolic Vision-Language Training](https://mlsanity.com/paper/2608.29313/) (2026)
- [Hyperbolic Multimodal Continual Learning: A Closest-Admissible Solution](https://mlsanity.com/paper/2609.29329/) (2026)
- [SphMind: Towards Robust, Training-Free VLM-based Spatial Reasoning with a 360 Camera](https://mlsanity.com/paper/2609.33462/) (2026)
- [Intrinsic Associative Memory on Riemannian Manifolds: Curvature, Capacity, and Emergent Modes](https://mlsanity.com/paper/2609.35948/) (2026)
- [Riemannian Graph Tokenizer for Structural Knowledge Transfer](https://mlsanity.com/paper/2610.15427/) (2026)
- [MAGNET: Manifold-Aware Graph Diffusion Network for Connectome Generation](https://mlsanity.com/paper/2610.17014/) (2026)

## Abstract (arXiv)

In the era of foundation models and Large Language Models (LLMs), Euclidean space has been the de facto geometric setting for machine learning architectures. However, recent literature has demonstrated that this choice comes with fundamental limitations. At a large scale, real-world data often exhibits inherently non-Euclidean structures, such as multi-way relationships, hierarchies, symmetries, and non-isotropic scaling, in a variety of domains, such as languages, vision, and the natural sciences. It is challenging to effectively capture these structures within the constraints of Euclidean spaces. This position paper argues that moving beyond Euclidean geometry is not merely an optional enhancement but a necessity to maintain the scaling law for the next-generation of foundation models. By adopting these geometries, foundation models could more efficiently leverage the aforementioned structures. Task-aware adaptability that dynamically reconfigures embeddings to match the geometry of downstream applications could further enhance efficiency and expressivity. Our position is supported by a series of theoretical and empirical investigations of prevalent foundation models. Finally, we outline a roadmap for integrating non-Euclidean geometries into foundation models, including strategies for building geometric foundation models via fine-tuning, training from scratch, and hybrid approaches.