# HYDRA: Hyperbolic Dynamic Representation Architecture for Kolmogorov-Arnold Networks

arXiv: https://arxiv.org/abs/2608.12194 · HTML: https://mlsanity.com/paper/2608.12194/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 18 references that are in the map.
- First posted on arXiv 12 Aug 2026.

## Builds on (most cited first, 11 of 11)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [OpenML: networked science in machine learning](https://mlsanity.com/paper/1407.7722/) (2014)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation](https://mlsanity.com/paper/2405.07200/) (2024)
- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [GAMI-Net: An Explainable Neural Network based on Generalized Additive Models with Structured Interactions](https://mlsanity.com/paper/2003.07132/) (2020)
- [Hypformer: Exploring Efficient Transformer Fully in Hyperbolic Space](https://mlsanity.com/paper/2407.01290/) (2024)
- [Hyperbolic Deep Learning for Foundation Models: A Survey](https://mlsanity.com/paper/2507.17787/) (2025)

## Abstract (arXiv)

Kolmogorov-Arnold Networks (KANs) enhance nonlinear function approximation by replacing scalar weights with learnable univariate functions. However, assigning an independent function to every connection results in substantial parameter redundancy, limiting their scalability and efficiency. To reduce this redundancy, we introduce HYperbolic Dynamic Representation Architecture (HYDRA), a parameter-efficient hyperbolic extension of KAN that combines spline-based functional learning with representations in the Poincar'e ball. HYDRA maps vector-valued inputs into a bounded hyperbolic latent space, performs KAN-style updates in tangent space, and employs a low-rank prototype block to share functional transformations across hidden dimensions. The resulting hyperbolic representations provide a structured radial coordinate for interpretation, while radius control improves training stability by preventing boundary saturation. Extensive experiments across eight benchmark datasets demonstrate that HYDRA consistently achieves competitive or superior predictive performance while improving parameter efficiency and representation interpretability.