# Curvature Learning for Generalization of Hyperbolic Neural Networks

arXiv: https://arxiv.org/abs/2508.17232 · HTML: https://mlsanity.com/paper/2508.17232/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 86 of its 124 references that are in the map.
- First posted on arXiv 24 Aug 2025.

## Builds on (most cited first, 30 of 86)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Class-Balanced Loss Based on Effective Number of Samples](https://mlsanity.com/paper/1901.05555/) (2019)
- [Learning Imbalanced Datasets with Label-Distribution-Aware Margin Loss](https://mlsanity.com/paper/1906.07413/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [DivideMix: Learning with Noisy Labels as Semi-supervised Learning](https://mlsanity.com/paper/2002.07394/) (2020)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [Meta-Learning for Semi-Supervised Few-Shot Classification](https://mlsanity.com/paper/1803.00676/) (2018)
- [Early-Learning Regularization Prevents Memorization of Noisy Labels](https://mlsanity.com/paper/2007.00151/) (2020)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Joint Optimization Framework for Learning with Noisy Labels](https://mlsanity.com/paper/1803.11364/) (2018)
- [ASAM: Adaptive Sharpness-Aware Minimization for Scale-Invariant Learning of Deep Neural Networks](https://mlsanity.com/paper/2102.11600/) (2021)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Parametric Contrastive Learning](https://mlsanity.com/paper/2107.12028/) (2021)
- [Improving Calibration for Long-Tailed Recognition](https://mlsanity.com/paper/2104.00466/) (2021)
- [Probabilistic End-to-end Noise Correction for Learning with Noisy Labels](https://mlsanity.com/paper/1903.07788/) (2019)
- [Distribution Alignment: A Unified Framework for Long-tail Visual Recognition](https://mlsanity.com/paper/2103.16370/) (2021)
- [Meta-Baseline: Exploring Simple Meta-Learning for Few-Shot Learning](https://mlsanity.com/paper/2003.04390/) (2020)
- [Disentangling Label Distribution for Long-tailed Visual Recognition](https://mlsanity.com/paper/2012.00321/) (2020)

## Built on by (most cited first, 4 of 4)

- [Modality Alignment across Trees on Heterogeneous Hyperbolic Manifolds](https://mlsanity.com/paper/2510.27391/) (2025)
- [H4G: Unlocking Faithful Inference for Zero-Shot Graph Learning in Hyperbolic Space](https://mlsanity.com/paper/2510.12094/) (2025)
- [Learning Taxonomic Trees with Hierarchical Representation Regularization for Large Multimodal Models](https://mlsanity.com/paper/2607.02909/) (2026)
- [A Controlled Evaluation of Curvature Inspired Scalar Feature Scaling for Domain Generalization](https://mlsanity.com/paper/2610.18227/) (2026)

## Abstract (arXiv)

Hyperbolic neural networks (HNNs) have demonstrated notable efficacy in representing real-world data with hierarchical structures via exploiting the geometric properties of hyperbolic spaces characterized by negative curvatures. Curvature plays a crucial role in optimizing HNNs. Inappropriate curvatures may cause HNNs to converge to suboptimal parameters, degrading overall performance. So far, the theoretical foundation of the effect of curvatures on HNNs has not been developed. In this paper, we derive a PAC-Bayesian generalization bound of HNNs, highlighting the role of curvatures in the generalization of HNNs via their effect on the smoothness of the loss landscape. Driven by the derived bound, we propose a sharpness-aware curvature learning method to smooth the loss landscape, thereby improving the generalization of HNNs. In our method, we design a scope sharpness measure for curvatures, which is minimized through a bi-level optimization process. Then, we introduce an implicit differentiation algorithm that efficiently solves the bi-level optimization by approximating gradients of curvatures. We present the approximation error and convergence analyses of the proposed method, showing that the approximation error is upper-bounded, and the proposed method can converge by bounding gradients of HNNs. Experiments on four settings: classification, learning from long-tailed data, learning from noisy data, and few-shot learning show that our method can improve the performance of HNNs.