# Group-Equivariant Poincar'e Convolutional Networks

arXiv: https://arxiv.org/abs/2607.00556 · HTML: https://mlsanity.com/paper/2607.00556/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 44 references that are in the map.
- First posted on arXiv 1 Jul 2026.

## Builds on (most cited first, 30 of 31)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Describing Textures in the Wild](https://mlsanity.com/paper/1311.3618/) (2013)
- [Energy-based Out-of-distribution Detection](https://mlsanity.com/paper/2010.03759/) (2020)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Hyperbolic Image Segmentation](https://mlsanity.com/paper/2203.05898/) (2022)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [BYOL works even without batch statistics](https://mlsanity.com/paper/2010.10241/) (2020)
- [Provably Strict Generalisation Benefit for Equivariant Models](https://mlsanity.com/paper/2102.10333/) (2021)
- [Differentiating through the Fr'echet Mean](https://mlsanity.com/paper/2003.00335/) (2020)
- [Inferring Concept Hierarchies from Text Corpora via Hyperbolic Embeddings](https://mlsanity.com/paper/1902.00913/) (2019)
- [Approximation-Generalization Trade-offs under (Approximate) Group Equivariance](https://mlsanity.com/paper/2305.17592/) (2023)
- [Revisiting Spatial Invariance with Low-Rank Local Connectivity](https://mlsanity.com/paper/2002.02959/) (2020)
- [Learning Probabilistic Symmetrization for Architecture Agnostic Equivariance](https://mlsanity.com/paper/2306.02866/) (2023)
- [Poincar'e ResNet](https://mlsanity.com/paper/2303.14027/) (2023)
- [Symmetry Breaking and Equivariant Neural Networks](https://mlsanity.com/paper/2312.09016/) (2023)
- [Explainable Equivariant Neural Networks for Particle Physics: PELICAN](https://mlsanity.com/paper/2307.16506/) (2023)
- [HMamba: Hyperbolic Mamba for Sequential Recommendation](https://mlsanity.com/paper/2505.09205/) (2025)
- [HypLL: The Hyperbolic Learning Library](https://mlsanity.com/paper/2306.06154/) (2023)
- [Weight-Sharing Regularization](https://mlsanity.com/paper/2311.03096/) (2023)
- [Hyperbolic Graph Neural Networks Under the Microscope: The Role of Geometry-Task Alignment](https://mlsanity.com/paper/2602.01828/) (2026)

## Abstract (arXiv)

While recent methods like that of the Poincar'e ResNet have demonstrated the ability to learning visual representations directly in hyperbolic space, their optimisation remains a challenge, primarily due to the parameter redundancy of learning distinct orientation filters. In addition, hyperbolic learning exhibits distinct computational overheads that limit their wide use, where efforts to improve their efficiency via optimisation have seen good success, there has been limited exploration into structural priors that enable stronger sample efficiency at training. To address this, we propose Equivariant Poincar'e ResNets, combining hyperbolic geometry with discrete symmetry groups (C4 and D4). We identify critical roadblocks in applying Euclidean equivariance to hyperbolic space and propose geometrically safe tensor reshaping, left-regular permutations for hyperbolic group convolutions, and joint-orientation Poincar'e Midpoint Batch normalisation. Empirical evaluations show that embedding equivariance significantly improves the sample efficiency during training which in-turn accelerates convergence while respecting the boundary constraints of the Poincar'e ball and retaining spatial group equivariance.