# Adaptive Hyperbolic Kernels: Modulated Embedding in de Branges-Rovnyak Spaces

arXiv: https://arxiv.org/abs/2511.09921 · HTML: https://mlsanity.com/paper/2511.09921/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 40 references that are in the map.
- First posted on arXiv 13 Nov 2025.

## Builds on (most cited first, 17 of 17)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [A Broad-Coverage Challenge Corpus for Sentence Understanding through Inference](https://mlsanity.com/paper/1704.05426/) (2017)
- [SimCSE: Simple Contrastive Learning of Sentence Embeddings](https://mlsanity.com/paper/2104.08821/) (2021)
- [A large annotated corpus for learning natural language inference](https://mlsanity.com/paper/1508.05326/) (2015)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Label-Embedding for Image Classification](https://mlsanity.com/paper/1503.08677/) (2015)
- [Learning a Deep Embedding Model for Zero-Shot Learning](https://mlsanity.com/paper/1611.05088/) (2016)
- [Latent Embeddings for Zero-shot Classification](https://mlsanity.com/paper/1603.08895/) (2016)
- [Hyperbolic Attention Networks](https://mlsanity.com/paper/1805.09786/) (2018)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [Zero-Shot Visual Recognition using Semantics-Preserving Adversarial Embedding Networks](https://mlsanity.com/paper/1712.01928/) (2017)
- [Clipped Hyperbolic Classifiers Are Super-Hyperbolic Classifiers](https://mlsanity.com/paper/2107.11472/) (2021)
- [Rethinking Zero-Shot Learning: A Conditional Visual Classification Perspective](https://mlsanity.com/paper/1909.05995/) (2019)
- [Large-Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/1806.00437/) (2018)
- [Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels](https://mlsanity.com/paper/1412.0265/) (2014)

## Abstract (arXiv)

Hierarchical data pervades diverse machine learning applications, including natural language processing, computer vision, and social network analysis. Hyperbolic space, characterized by its negative curvature, has demonstrated strong potential in such tasks due to its capacity to embed hierarchical structures with minimal distortion. Previous evidence indicates that the hyperbolic representation capacity can be further enhanced through kernel methods. However, existing hyperbolic kernels still suffer from mild geometric distortion or lack adaptability. This paper addresses these issues by introducing a curvature-aware de Branges-Rovnyak space, a reproducing kernel Hilbert space (RKHS) that is isometric to a Poincare ball. We design an adjustable multiplier to select the appropriate RKHS corresponding to the hyperbolic space with any curvature adaptively. Building on this foundation, we further construct a family of adaptive hyperbolic kernels, including the novel adaptive hyperbolic radial kernel, whose learnable parameters modulate hyperbolic features in a task-aware manner. Extensive experiments on visual and language benchmarks demonstrate that our proposed kernels outperform existing hyperbolic kernels in modeling hierarchical dependencies.