# Hyperbolic Neural Population Geometry Benefits Computation

arXiv: https://arxiv.org/abs/2606.10238 · HTML: https://mlsanity.com/paper/2606.10238/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 57 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Attention Networks](https://mlsanity.com/paper/1805.09786/) (2018)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [Neural population geometry: An approach for understanding biological and artificial neural networks](https://mlsanity.com/paper/2104.07059/) (2021)
- [On Sparse Modern Hopfield Model](https://mlsanity.com/paper/2309.12673/) (2023)
- [Uniform Memory Retrieval with Larger Capacity for Modern Hopfield Models](https://mlsanity.com/paper/2404.03827/) (2024)
- [Neural tuning and representational geometry](https://mlsanity.com/paper/2104.09743/) (2021)
- [Simplicial Hopfield networks](https://mlsanity.com/paper/2305.05179/) (2023)
- [Sparse and Structured Hopfield Networks](https://mlsanity.com/paper/2402.13725/) (2024)
- [Feature Learning beyond the Lazy-Rich Dichotomy: Insights from Representational Geometry](https://mlsanity.com/paper/2503.18114/) (2025)
- [Hopfield-Fenchel-Young Networks: A Unified Framework for Associative Memory Retrieval](https://mlsanity.com/paper/2411.08590/) (2024)
- [Navigable maps of structural brain networks across species](https://mlsanity.com/paper/1801.06079/) (2018)

## Abstract (arXiv)

Neural population geometry shapes downstream computation. Recent empirical findings in neurobiology suggest that a hyperbolic structure underlies population activity in the hippocampus. Here we provide a theoretical framework for this phenomenon. First, we propose a plausible construction of hippocampal tuning curves that statistically induces hyperbolic geometry. Next, we establish a connection between neural decoding and associative memory by demonstrating that the Modern Hopfield Network update rule computes the minimum mean-squared-error (MMSE) estimator. Finally, we introduce a novel associative memory model defined in hyperbolic space that yields significantly larger capacity than leading models. Our results suggest that animals encode spatial information as a latent hyperbolic cognitive map, improving both memory capacity and decoding accuracy.