# Explaining Hyperbolic Neural Networks via Geometry-Aware Relevance Propagation

arXiv: https://arxiv.org/abs/2609.35128 · HTML: https://mlsanity.com/paper/2609.35128/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 34 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 24 of 24)

- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [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)
- [Explaining NonLinear Classification Decisions with Deep Taylor Decomposition](https://mlsanity.com/paper/1512.02479/) (2015)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Poincar'e GloVe: Hyperbolic Word Embeddings](https://mlsanity.com/paper/1810.06546/) (2018)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Hyperbolic Image-Text Representations](https://mlsanity.com/paper/2304.09172/) (2023)
- [AttnLRP: Attention-Aware Layer-Wise Relevance Propagation for Transformers](https://mlsanity.com/paper/2402.05602/) (2024)
- [XAI for Transformers: Better Explanations through Conservative Propagation](https://mlsanity.com/paper/2202.07304/) (2022)
- [Explaining Recurrent Neural Network Predictions in Sentiment Analysis](https://mlsanity.com/paper/1706.07206/) (2017)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [Fully Hyperbolic Convolutional Neural Networks for Computer Vision](https://mlsanity.com/paper/2303.15919/) (2023)
- [Diffeomorphic Counterfactuals with Generative Models](https://mlsanity.com/paper/2206.05075/) (2022)
- [HELM: Hyperbolic Large Language Models via Mixture-of-Curvature Experts](https://mlsanity.com/paper/2505.24722/) (2025)
- [Explaining and Interpreting LSTMs](https://mlsanity.com/paper/1909.12114/) (2019)
- [Explainable AI for Time Series via Virtual Inspection Layers](https://mlsanity.com/paper/2303.06365/) (2023)
- [Manifold Integrated Gradients: Riemannian Geometry for Feature Attribution](https://mlsanity.com/paper/2405.09800/) (2024)
- [Hyperbolic Deep Learning for Foundation Models: A Survey](https://mlsanity.com/paper/2507.17787/) (2025)
- [HEEGNet: Hyperbolic Embeddings for EEG](https://mlsanity.com/paper/2601.03322/) (2026)
- [EEG-Based Multimodal Learning via Hyperbolic Mixture-of-Curvature Experts](https://mlsanity.com/paper/2604.12579/) (2026)
- [Normalized Relevance Measure as a Unifying Framework to Explain Neural Network Latent Structures](https://mlsanity.com/paper/2606.00557/) (2026)
- [Hyperbolic embedding of brain networks as a tool for epileptic seizures forecasting](https://mlsanity.com/paper/2406.10184/) (2024)

## Abstract (arXiv)

Hyperbolic neural networks introduce geometric operations that require explicit treatment in relevance propagation. Equivalent geometric realizations can produce different feature attributions, even when local relevance is conserved. We study this problem through Geometric Representation Invariance (GRI), a specialization of Implementation Invariance, and zero-curvature consistency, which requires identity relevance propagation when a geometric module approaches the identity. We propose LRP-radial-all for origin-centered radial modules, treating geometric scaling as modulation and assigning relevance entirely to the signal branch. The rule conserves relevance, is invariant to equivalent radial factorizations, and satisfies zero-curvature consistency, yielding GRI for a specified Poincar'e-Lorentz logarithmic-map construction. In contrast, a conservative LRP-half baseline can violate both consistency criteria. Experiments on hyperbolic MNIST, sEEG, and CIFAR-10 classifiers assess attribution fidelity, qualitative explanations, and runtime. LRP-radial-all achieves competitive attribution fidelity across datasets with runtime comparable to Gradient×Input and substantially lower than Integrated Gradients. These findings motivate geometry-aware propagation rules that distinguish relevance conservation from consistency across equivalent computations.