Group-Equivariant Poincar\'e Convolutional Networks

# Group-Equivariant Poincaré Convolutional Networks
Source: [https://arxiv.org/html/2607.00556](https://arxiv.org/html/2607.00556)
11institutetext:School of Computing Sciences, University of East Anglia, NR4 7TJ, Norwich, UK11email:aiden\.durrant@uea\.ac\.uk
22institutetext:Department of Physics and Technology, UiT The Arctic University of Norway, NO\-9037, Tromsø, Norway
22email:rahul\.baburajan@uit\.no,22email:georgios\.leontidis@uit\.no###### Abstract

While recent advancements like the Poincaré ResNet have demonstrated the potential of learning visual representations directly in hyperbolic space, their optimisation remains hampered by the computationally intensive nature of Riemannian gradients and the strict boundaries of the manifold\. Furthermore, standard hyperbolic networks treat spatial transformations of the same object as distinct hierarchical concepts, leading to redundant parameter usage and vanishing signals\. We propose Equivariant Poincaré ResNets, combining hyperbolic geometry with discrete symmetry groups \(C4C\_\{4\}andD4D\_\{4\}\)\. 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 Poincaré Midpoint Batch normalisation\. Empirically, embedding equivariance drastically reduces the optimisation space, accelerating convergence while accelerating convergence while respecting the boundary constraints of the Poincaré ball and preserving spatial\-group equivariance\.

## 1Introduction

Deep learning in hyperbolic space has demonstrated profound capabilities for embedding hierarchical visual data with minimal distortion\[krioukov2010hyperbolic,atigh2022hyperbolic,mishra2026hyperbolic\]\. The recent introduction of Poincaré residual networks has pushed this boundary further, enabling the learning of visual representations entirely within the Poincaré ball from the pixel level\[van2023poincare\]\. However, optimising deep convolutional networks on Riemannian manifolds introduces significant challenges\. The parameter space is tightly constrained, and traversing the curvature requires computationally expensive operations, such as exponential mappings and Fréchet mean estimations\.

A critical inefficiency in current hyperbolic visual networks is their inability to inherently recognise spatial symmetries\. An object rotated by 90 degrees is treated by the network as an entirely new hierarchical concept, forcing the optimiser to require expensive Riemannian gradient steps to learn redundant representations\. In Euclidean space, one approach to solve this is through group\-equivariant convolutional networks\[cohen2016group\]\. Such networks have been shown to be highly data\-efficient\[bietti2021sample\], and thus we propose that the Hyperbolic networks computational overhead can be somewhat mitigated through use of symmetric priors\.

In this paper, we investigate how to embedC4C\_\{4\}\(rotations\) andD4D\_\{4\}\(rotations and reflections\) spatial\-group equivariance over Poincaré\-valued feature fields directly into the Poincaré ResNet architecture\. Translating equivariance to hyperbolic space is highly non\-trivial due to the non\-Euclidean nature of feature concatenation and channel manipulation\. We propose three primary contributions to enable the Equivariant Poincaré ResNets: \(i\) We introduce a geometrically safeβ\\beta\-scaling formulation for flattening and unflattening Poincaré tensors, allowing orientation channels to be decoupled without violating the expected norms of the manifold\. \(ii\) We formulate hyperbolic lifting and group convolutions by projecting base tangent\-space filters through discrete symmetry transformations and left\-regular permutations\. \(iii\) We extend Poincaré Midpoint Batch normalisation to act jointly over group orientations, preventing the normalisation step from destroying the learned equivariance\.

## 2Background and Related Work

#### 2\.0\.1Hyperbolic Computer Vision\.

Hyperbolic spaces, characterised by constant negative curvature, possess an intrinsic capacity to embed complex, tree\-like structures with arbitrarily low distortion\[krioukov2010hyperbolic,nickel2017poincare\]\. As such, representing hierarchical relations in continuous spaces is naturally well fit for hyperbolic spaces\. Hyperbolic representation learning has achieved significant success in modelling continuous hierarchies across natural language processing\[tifrea2018poincar,ganea2018hyperbolic,le2019inferring\], graph networks\[chami2019hyperbolic,naddeo2026hyperbolic\], and recommender systems\[zhang2026hmamba,Tran2018HYPERML\]\.

The computer vision community has begun leveraging hyperbolic geometry more readily, establishing that both image data and labels contain latent hierarchical structures\. The introduction of hyperbolic image embeddings has thus improved uncertainty quantification and few\-shot learning\[krioukov2010hyperbolic\]\. Subsequent works have successfully scaled hyperbolic methods to metric learning, zero\-shot recognition, video action recognition\[long2020searching\], and part\-whole image segmentation\[vlasenkohyperbolic,mishra2026hyperbolic,atigh2022hyperbolic\]\. However, a critical limitation persists: the vast majority of these methods restrict hyperbolic operations to the final embedding or classifier space\. The visual representations themselves are still learned through standard Euclidean networks, discarding geometric benefits during early feature extraction\[van2023poincare\]\. While the recent introduction of the Poincaré ResNet\[van2023poincare\]has paved the way for end\-to\-end hyperbolic visual learning directly from the pixel level, optimizing these deep networks remains challenging\. They rely on computationally expensive operations, such as Fréchet mean approximations\[lou2020differentiating\], and they currently lack the structural inductive biases required to recognise spatial symmetries efficiently\. Relevant hyperbolic prerequisites are given in Appendix[0\.A\.1](https://arxiv.org/html/2607.00556#Pt0.A1.SS1)\.

#### 2\.0\.2Equivariant Deep Learning\.

Equivariant machine learning fundamentally exploits structural symmetries in data to constrain models with known priors\. This geometric inductive bias consistently leads to improved generalisation\[elesedy2021provably,petrache2023approximation\], interpretability\[bogatskiy2024explainable\], and computational efficiency\[bietti2021sample\]\. From a theoretical standpoint, enforcing these symmetries often involves parameter\-sharing mechanisms\. Works such as\[ravanbakhsh2017equivariance\]have explored designing model parameters to reflect equivariance over discrete group actions, while recent regularisation schemes directly encourage weight\-sharing to induce symmetries in low\-data regimes\[shakerinava2024weight\]\. Beyond convolutions, this ground\-up approach has been extended to Equivariant MLPs \(EMLPs\) by decomposing arbitrary discrete and Lie matrix groups into finite sets of generators\[finzi2021practical\]\.

Because designing ground\-up equivariant architectures can be laborious and computationally expensive, alternative pipelines have sought to render group\-agnostic universal approximators equivariant through techniques like frame\-averaging\[puny2021frame\], probabilistic symmetrisation\[kim2023learning\], or learnable canonicalisation\[kaba2023equivariance\]\. Additionally, real\-world data often exhibits imperfect symmetries, prompting research into symmetry breaking and relaxed equivariance\. This includes modifying EMLP constraints to handle broken symmetries\[kaba2023symmetry\]or introducing relaxed group convolutions that blend spatial invariance with a linear combination of basis filter banks\[elsayed2020revisiting,wang2022approximately\]\.

Despite these broad architectural advancements, in the visual domain, standard Euclidean convolutional neural networks still predominantly rely on innate spatial translation equivariance\. They lack robustness to other foundational geometric symmetries, forcing networks to expend parameters learning redundant filters for objects appearing in different orientations\. Group Equivariant Convolutional Networks \(G\-CNNs\)\[cohen2016group,kondor2018generalization,weiler2018learning\]resolve this by formally extending equivariance to discrete symmetry groups, such as the cyclic groupC4C\_\{4\}\(discrete rotations\) and the dihedral groupD4D\_\{4\}\(rotations and reflections\)\. By systematically transforming filters and routing feature maps through left\-regular permutations, G\-CNNs guarantee that a spatial geometric transformation of the input precisely correlates with a predictable shift in the feature space\.

However, embedding strict group equivariance into hyperbolic neural networks remains a complex, unresolved challenge\. As highlighted by\[huang2024lorentz\], symmetric representations in hyperbolic space suffer from severe geometric distortions, making strict equivariance preservation critical\. Unfortunately, existing architectures often inadvertently break these symmetries through unconstrained tangent space projections\[huang2024lorentz\]\. Furthermore, translating Euclidean group convolutions directly to the Poincaré ball is impeded by the manifold’s rigid boundaries\. Standard Euclidean mechanisms for generating equivariant feature maps—such as unconstrained tensor concatenation, decoupling, and reshaping—artificially inflate gyrovector magnitudes, inevitably pushing the resultant features outside the valid manifold\. While recent theoretical frameworks have introduced structure\-preserving operations likeβ\\beta\-concatenation to stabilise basic hyperbolic feature aggregation\[shimizu2020hyperbolic\], these mechanisms have not yet been adapted to handle the discrete algebraic routing required for symmetry preservation\. Consequently, establishing true equivariance in this domain requires a fundamental reconstruction of lifting layers, group convolutions, and batch normalisation to simultaneously satisfy both discrete group symmetries and non\-Euclidean geometric constraints\.

## 3Group\-Equivariant Poincaré Convolutions

A fundamental challenge in non\-Euclidean image classification is achieving high sample efficiency without explicit structural priors for spatial symmetries\. Without these, hyperbolic networks redundantly learn semantic concepts across multiple orientations, compounding their already high computational overhead\[van2023poincare\]\. We formalise strict group equivariance within hyperbolic neural networks as a foundational proof of concept, focusing on the cyclic \(C4C\_\{4\}\) and dihedral \(D4D\_\{4\}\) groups\. These represent the natural symmetries of the square pixel grid, enabling perfect geometric transformations without interpolation artifacts\[cohen2016group\]\.

In standard Euclidean residual networks, a weight layer is fundamentally composed of a spatial convolution followed by batch normalisation\. To formulate an Equivariant Poincaré block, we cannot apply these operations natively\. Instead, we must reconstruct lifting convolutions, group convolutions, and batch normalisation techniques to simultaneously respect the algebraic structure of the symmetry groupGGand the geometric constraints of the Poincaré manifold𝔹cn\\mathbb\{B\}\_\{c\}^\{n\}\.

### 3\.1Geometrically Safe Tensor Unflattening \(β\\beta\-Scaling\)

In Euclidean equivariant networks, the generation of group\-structured feature maps relies heavily on the unconstrained concatenation, reshaping, and decoupling of tensors \(e\.g\., transforming a flattened vector of size\|G\|×C\|G\|\\times Cback into\|G\|\|G\|orientations of sizeCC\)\. However, applying these standard Euclidean reshaping operations directly to Poincaré vectors is geometrically invalid, as naive concatenation artificially inflates the vector magnitude, inevitably pushing the resultant vector beyond the valid boundary of the manifold\.

To address this limitation,\[shimizu2020hyperbolic\]introducedβ\\beta\-concatenation, a mechanism that dynamically shrinks vectors based on a Beta function ratio during the concatenation phase, preserving the mathematical expectation of the Poincaré norm\. To establish equivariance, it is necessary that we explicitly invert this scaling process\. While a true isometric unflattening operation does not exist for hyperbolic concatenation, we propose a numerically stable inverseβ\\beta\-split\. This allows us to separate spatial and group channels while preserving the expectation of the features semantic magnitudes\.

We define a geometrically safe unflattening operation for a concatenated Poincaré output vectoryflaty\_\{\\text\{flat\}\}by first mapping the concatenated output back to the locally flat tangent space at the origin via the logarithmic map:

t=log0c⁡\(yflat\)t=\\log\_\{0\}^\{c\}\(y\_\{\\text\{flat\}\}\)
Once projected into the Euclidean tangent space, the geometric bounds are temporarily lifted, allowing us to safely decouple the dimensions from\(B,\|G\|×C,H,W\)\(B,\|G\|\\times C,H,W\)to\(B,\|G\|,C,H,W\)\(B,\|G\|,C,H,W\)\.

Because the initial forward operation natively appliedβ\\beta\-scaling to merge the channels, the true lengths of these vectors remain artificially distorted\. Therefore, we must restore the magnitude of the decoupled tangent vectors,tdecoupledt\_\{\\text\{decoupled\}\}, to accurately reflect their original semantic norm before spatial aggregation\. Whereβn\\beta\_\{n\}is computed via the log\-gamma function\[shimizu2020hyperbolic\]to prevent floating\-point underflow:

βn=B​\(n2,12\)=exp⁡\(log⁡Γ​\(n2\)\+log⁡Γ​\(0\.5\)−log⁡Γ​\(n2\+0\.5\)\)\\beta\_\{n\}=B\\left\(\\frac\{n\}\{2\},\\frac\{1\}\{2\}\\right\)=\\exp\\left\(\\log\\Gamma\\left\(\\frac\{n\}\{2\}\\right\)\+\\log\\Gamma\(0\.5\)\-\\log\\Gamma\\left\(\\frac\{n\}\{2\}\+0\.5\\right\)\\right\)
For a channel dimensionCCand a group size\|G\|\|G\|, the structurally restored tangent tensortgroupt\_\{\\text\{group\}\}is computed as:

tgroup=tdecoupled⋅βCβ\|G\|×Ct\_\{\\text\{group\}\}=t\_\{\\text\{decoupled\}\}\\cdot\\frac\{\\beta\_\{C\}\}\{\\beta\_\{\|G\|\\times C\}\}
Finally, we transpose this tensor into the target layout\(B,C,\|G\|,H,W\)\(B,C,\|G\|,H,W\)\. The structured vectors are then projected back onto the Poincaré ball using the exponential map\.

ygroup=exp0c⁡\(tgroup\)y\_\{\\text\{group\}\}=\\exp\_\{0\}^\{c\}\(t\_\{\\text\{group\}\}\)
This sequence ensures the discrete group orientations are explicitly separated\. While not a strict geometric isometry, this stable inverseβ\\beta\-split bounds the magnitude restoration to preserve the mathematical expectation of the norms, preventing manifold boundary violations during deep forward passes\.

### 3\.2Poincaré Lifting Convolutions

The initial layer of an equivariant network must perform a base\-to\-group mapping, commonly formalised as a lifting convolution in Geometric Deep Learning, to project the input data from the standard spatial domainℤ2\\mathbb\{Z\}^\{2\}into the group\-structured domainGG\. Applying rotations directly to Poincaré gyro\-vectors requires complex and computationally expensive Möbius transformations, we construct our symmetric filter banks entirely within the tangent space\.

Formally, the input to our network can be viewed as a feature fieldf:ℤ2→𝔹cdf:\\mathbb\{Z\}^\{2\}\\rightarrow\\mathbb\{B\}^\{d\}\_\{c\}\. Equivariance in this context is defined strictly as spatial\-group equivariance: the group actiong∈Gg\\in Gtransforms the spatial domain and dictates the left\-regular permutation of the orientation channels, while the feature vectors themselves remain mathematically anchored as Poincaré\-valued representations\. We define aC4C\_\{4\}orD4D\_\{4\}Poincaré lifting convolution that maps an input feature mapx∈𝔹cCinx\\in\\mathbb\{B\}\_\{c\}^\{C\_\{\\text\{in\}\}\}to an expanded output spacey∈𝔹c\|G\|×Couty\\in\\mathbb\{B\}\_\{c\}^\{\|G\|\\times C\_\{\\text\{out\}\}\}\. We begin by initialising a base spatial filter in the tangent space,w∈T0​𝔹cnw\\in T\_\{0\}\\mathbb\{B\}\_\{c\}^\{n\}\. To construct the required equivariant filter bank, we systematically apply the designated group transformations directly towwin\-line with\[cohen2016group\]\. For theD4D\_\{4\}group \(\|G\|=8\|G\|=8\), we generate the eight transformations representing all distinct rotations and horizontal flips:

wk=rot90​\(w,k\)for​k∈\{0,1,2,3\}w\_\{k\}=\\text\{rot90\}\(w,k\)\\quad\\text\{for \}k\\in\\\{0,1,2,3\\\}
wk\+4=rot90​\(flip​\(w\),k\)for​k∈\{0,1,2,3\}w\_\{k\+4\}=\\text\{rot90\}\(\\text\{flip\}\(w\),k\)\\quad\\text\{for \}k\\in\\\{0,1,2,3\\\}
These transformed tangent filters are subsequently stacked along the newly created group dimension resulting in a flat weight matrixwflatw\_\{\\text\{flat\}\}\. Crucially, the parameter\-sharing condition is strict, that the learnable weights exist solely as the single base filterw∈T0​𝔹cdw\\in T\_\{0\}\\mathbb\{B\}\_\{c\}^\{d\}\. The filter banks for all\|G\|\|G\|orientations are generated via deterministic spatial actions, guaranteeing that the parameter count remains entirely independent of the group size\|G\|\|G\|\. To perform the spatial convolution natively on the manifold, we follow the aproach of\[van2023poincare\]where we extract local spatial patches from the input feature map via a hyperbolic unfold operation\. The flat weight matrix then serves as the parameterZZwithin a standard Poincaré fully connected layerFcF\_\{c\}, which is applied to the unfolded patchesXk​lX\_\{kl\}:

hk​l=Fc​\(β∥Xk​l;wflat,r\)h\_\{kl\}=F\_\{c\}\(\\beta\\\|X\_\{kl\};w\_\{\\text\{flat\}\},r\)
The resulting output is then processed through our geometrically safe unflattening \(Section[3\.1](https://arxiv.org/html/2607.00556#S3.SS1)\) to ultimately yield the lifted, group\-equivariant representation\.

### 3\.3Poincaré Group Convolutions

Once the features have been successfully lifted into the groupGG, all subsequent convolutional layers must map fromG→GG\\to G\. A standard hyperbolic convolution is insufficient for this task, as it would blindly aggregate spatial features while ignoring the cyclical and reflective relationships intrinsic to the group channels\. For true equivariance, if an input image rotates, the feature map must correctly route its activations into the corresponding shifted orientation channel\. However, dynamically routing hyperbolic activations on the fly is computationally prohibitive\. Instead, we encode this routing directly into the expanded tangent\-space filters\.

For an input feature mapxxcharacterised by both spatial dimensions and\|G\|\|G\|orientation channels, we define the Poincaré Group Convolution\. Because the input itself now has an explicit orientation axis, applying a transformation from the groupGGto the filter requires that we not only rotate or flip the spatial dimensions but also permute the orientation channels in strict accordance with the left\-regular representation of the group\.

#### 3\.3\.1Left\-Regular Permutation\.

To encode this routing, we define a Cayley index matrixℐG\\mathcal\{I\}\_\{G\}that dictates the required permutation mapping for each transformation\. For instance, rotating aC4C\_\{4\}feature map by90∘90^\{\\circ\}cyclically shifts its orientation channels by one\. Therefore, we pre\-permute the orientation axis of the tangent\-space weights in accordance with the inverse left\-regular representation\.

#### 3\.3\.2Aligned Application\.

For each transformationi∈\{1​…​\|G\|\}i\\in\\\{1\\dots\|G\|\\\}, the base filterwspatialw\_\{\\text\{spatial\}\}is geometrically transformed, and its internal orientation axis permuted:

waligned\(i\)=wtransform\(i\)​\[:,:,ℐG​\[i\],:,:\]w\_\{\\text\{aligned\}\}^\{\(i\)\}=w\_\{\\text\{transform\}\}^\{\(i\)\}\\left\[:,:,\\mathcal\{I\}\_\{G\}\[i\],:,:\\right\]

#### 3\.3\.3Safe Flattening\.

Crucially, prior to computing the hyperbolic inner product, the input tensorxxmust be safely flattened\. We therefore merge the channel dimension and the group dimension\|G\|\|G\|utilising the nativeβ\\beta\-concatenation algorithm\. This flattening step is essential to prevent numerical instability and the propagation of NaN values during the forward pass\. The unfolded flattened input is then multiplied by the stackedwalignedw\_\{\\text\{aligned\}\}matrices within the tangent space\. As the aligned filters were stacked, maintaining a\[Cout,\|G\|\]\[C\_\{\\text\{out\}\},\|G\|\]layout, the resulting flat features are reshaped into the correct orientation channels during the subsequent magnitude restoration phase\.

By associating the weights across all symmetries and utilising the left\-regular permutation, a single Riemannian gradient update applied to the base filterwwwhich simultaneously updates the representation for all four or eight orientations\.

### 3\.4Joint\-Orientation Midpoint Batch normalisation

Residual networks rely extensively on batch normalisation to stabilise deep feature representations\. While\[lou2020differentiating\]proposed Fréchet mean normalisation for hyperbolic spaces, Poincaré Midpoint normalisation offers a computationally efficient alternative\[van2023poincare,ungar2022gyrovector\]\. However, applying either of these normalisation techniques independently to the distinct group channels of an equivariant network introduces a catastrophic failure mode\. If the0∘0^\{\\circ\}orientation detects a strong feature magnitude while the90∘90^\{\\circ\}orientation detects nothing, independent normalisation will centre and scale both to possess identical means and variances \(μ\\muandσ2\\sigma^\{2\}\)\. This removes the relative magnitude between the channels, effectively stripping the network of its explicit knowledge of the object’s true orientation\.

To maintain strict equivariance the normalisation statistics must be computed jointly over the entire group dimension\. We define the Joint\-Orientation Poincaré Batch normalisation by first reshaping the input tensorx∈𝔹cnx\\in\\mathbb\{B\}\_\{c\}^\{n\}from its structural shape of\(B,C,\|G\|,H,W\)\(B,C,\|G\|,H,W\)to a merged spatial layout of\(B,C,\|G\|×H,W\)\(B,C,\|G\|\\times H,W\)\.

Because merging the discrete rotations into the spatial dimension rearranges the vectors without altering their underlying norms or manifold positions, we can safely compute a unified Poincaré midpoint over this expanded volume:

μjoint=12⊗c∑i=1\|G\|×Hλxic​xi∑i=1\|G\|×H\(λxic−1\)\\mu\_\{\\text\{joint\}\}=\\frac\{1\}\{2\}\\otimes\_\{c\}\\frac\{\\sum\_\{i=1\}^\{\|G\|\\times H\}\\lambda\_\{x\_\{i\}\}^\{c\}x\_\{i\}\}\{\\sum\_\{i=1\}^\{\|G\|\\times H\}\(\\lambda\_\{x\_\{i\}\}^\{c\}\-1\)\}
After computing this shared midpointμjoint\\mu\_\{\\text\{joint\}\}and its corresponding varianceσjoint2\\sigma\_\{\\text\{joint\}\}^\{2\}, the vectors are transported and scaled uniformly\. Following normalisation, the tensor is reshaped back to its explicit group structure\. By calculating the variance and Poincaré midpoint over the joint spatial and group dimensions, we enforce an explicit statistical condition: all orientation channels are shifted and scaled by the exact same manifold scalar\. This condition guarantees that the relative magnitude differences between distinctC4C\_\{4\}andD4D\_\{4\}orientations are preserved, maintaining strict equivariance\.

## 4Experiments

All implementations of Hyperbolic layers, optimisation, and functions are handled by the Hypll library\[van2023hypll\], which we extend using the same convention for our proposed layers\. We evaluate our architectures primarily on the CIFAR\-10 dataset\[krizhevsky2009learning\]\. We utilise a Poincaré ResNet\-20 backbone\. To maintain manageable computation graphs and memory constraints inherent to Riemannian optimisation, we employ narrow channel widths of \(8, 16, 32\)\. Following\[van2023poincar\]all convolutional layers are initialised using an identity\-based mapping to prevent vanishing signals over deep layers\. All Poincaré models are trained using the Riemannian Adam optimiser, while Euclidean models use standard Adam with a fixed learning rate of10−310^\{\-3\}and a weight decay of10−410^\{\-4\}\. Unless otherwise specified as a learnable parameter, the negative curvature of the Poincaré ball is fixed at c=0\.1, which has been empirically shown to offer a stable Euclidean volume for floating\-point representations during optimisation\.

Crucially, standard visual representation learning typically relies on extensive data augmentation pipelines to force networks to learn approximate spatial invariance\. Because our Equivariant Poincaré ResNet enforces strict structural equivariance by design, it inherently recognises spatial transformations without this reliance\. To isolate and evaluate this architectural inductive bias, we intentionally omit all spatial data augmentations from our training pipeline\. While this strict setup accounts for the absolute performance discrepancy compared to baseline literature utilising standard augmentations\[van2023poincare\], it ensures our evaluation provides a honest assessment of the network’s intrinsic geometric capabilities and sample efficiency\.

Table 1:Top\-1 accuracy \(%\) on CIFAR\-10 for ResNet\-20\. Results are compared across Euclidean and Poincaré manifolds withC4C\_\{4\}\- andD4D\_\{4\}\-equivariant layers\.### 4\.1Classification Performance

Introducing group equivariance directly into the Poincaré manifold yields a substantial improvement in both convergence efficiency and absolute classification performance\. As detailed in Table[1](https://arxiv.org/html/2607.00556#S4.T1), we benchmarked standard,C4C\_\{4\}\-equivariant, andD4D\_\{4\}\-equivariant layers across both Euclidean and Poincaré manifolds\. The standard Poincaré ResNet\-20 achieves a top\-1 accuracy of 76\.87%\. By applying theD4D\_\{4\}\-equivariant lifting and group convolutions, the Poincaré ResNet\-20 reaches 88\.77% accuracy, a 11\.9% absolute improvement over the standard hyperbolic baseline, and outperforming the standard Euclidean ResNet\-20 \(78\.26%\)\. By tying the tangent\-space weights across all eight symmetries of the dihedral group, the optimisation space is drastically reduced\. A single gradient update to the base filter updates the representation for all spatial orientations\.

![Refer to caption](https://arxiv.org/html/2607.00556v1/x1.png)Figure 1:Data\-Efficiency Curve\.Top\-1 test accuracy evaluated across discrete fractions of the training dataset\.
![Refer to caption](https://arxiv.org/html/2607.00556v1/x2.png)Figure 2:Convergence Trajectories\.Top\-1 test accuracies evaluated during training for standard and equivariant Poincaré models\.

### 4\.2Sample Efficiency and Convergence

Embedding strict group equivariance significantly reduces the optimisation space, where without structural geometric priors, standard hyperbolic networks redundantly learn identical concepts across varying orientations\. This redundancy requires larger datasets and training iterations generalise\. To empirically validate that our architecture addresses these bottlenecks, we evaluate sample and convergence efficiency in a data\-constrained setting\.

We partition CIFAR\-10 into class\-balanced subsets representing 1%, 5%, 10%, 25%, 50%, and 100% of the total data\. To isolate the architectural inductive bias, we disable all spatial augmentations \(e\.g\., cropping, rotations\)\. Standard Euclidean, standard Poincaré, andD4D\_\{4\}\-Equivariant Poincaré ResNet\-20 models are then trained from scratch for 100 epochs per subset\.

Figure[2](https://arxiv.org/html/2607.00556#S4.F2)illustrates the comparative data\-efficiency curve\. In extreme low\-data regimes \(1% and 5%\), standard Poincaré architectures degrade severely \(achieving only 26\.92% and 33\.79% accuracy\)\. Because they treat spatial transformations as entirely distinct features, they rapidly overfit to the limited orientations in the dataset\. Conversely, theD4D\_\{4\}\-Equivariant Poincaré ResNet demonstrates vastly superior generalisation, maintaining 63\.77% accuracy on just 10% of the data\.

Furthermore, this structural prior yields marked improvements in convergence efficiency \(Figure[2](https://arxiv.org/html/2607.00556#S4.F2)\)\. The proposed equivariant models converge to peak accuracy in fewer epochs than standard Poincaré networks, helping mitigate some of the significant computational overhead\. These results confirm that hyperbolic geometry with discrete symmetry groups preserves the hierarchical embedding benefits of the Poincaré ball while reducing both the sample complexity and the epochs required\.

### 4\.3Out\-of\-Distribution Detection

Table 2:Out\-of\-distribution detection on CIFAR\-10 against Places365, SVHN, and Textures \(DTD\)\. Results are reported for ResNet\-20 architectures using the energy score\.To evaluate model robustness, we benchmarked our architectures ability to detect out\-of\-distribution \(OOD\) samples following the setup described in\[van2023poincare\]\. Models trained on CIFAR\-10 \(in\-distribution\) were evaluated against three highly distinct OOD datasets: Places\-365\[zhou2017places\], SVHN\[netzer2011reading\], and Textures \(DTD\)\[cimpoi2014describing\]\. Following standard evaluation protocols, we utilised the energy score\[liu2020energy\]to distinguish between distributions, reporting the False Positive Rate at 95% True Positive Rate \(FPR95\), Area Under the Receiver Operating Characteristic curve \(AUROC\), and Area Under the Precision\-Recall curve \(AUPR\)\.

As detailed in Table[2](https://arxiv.org/html/2607.00556#S4.T2), standard hyperbolic networks inherently demonstrate strong OOD robustness compared to their Euclidean counterparts, a direct benefit of the boundary\-aware, hierarchical structuring of the Poincaré ball\. Crucially, the introduction ofC4C\_\{4\}andD4D\_\{4\}equivariant layers maintains these strong OOD properties while simultaneously improving absolute representational performance\. For instance, on the SVHN dataset, theD4D\_\{4\}\-Equivariant Poincaré ResNet achieves an AUROC of 83\.17 compared to the standard Poincaré model of 80\.72, confirming that enforcing strict geometric symmetries does not compromise the manifold’s natural capacity for outlier rejection\.

![Refer to caption](https://arxiv.org/html/2607.00556v1/imgs/baton_pass_square_poincare_standard.png)

![Refer to caption](https://arxiv.org/html/2607.00556v1/imgs/baton_pass_square_poincare_c4.png)

\(a\)Standard convolution feature maps on the Poincaré manifold\. The activations change unpredictably as the input rotates\.\(b\)C4C\_\{4\}lifting convolution feature maps\. The activations rotate spatially and shift cyclically across the four channels\.
Figure 3:VisualizingC4C\_\{4\}equivariance\.Standard convolutions \(a\) fail to preserve the geometric relationship of the rotated inputs, whereasC4C\_\{4\}\-equivariant lifting layers \(b\) predictably rotate and permute the feature maps\. Note, visualisations are projected onto the Euclidean plane\. Features near the boundary appear distorted due to the Poincaré metric, yet the equivariant channel routing remains invariant to the distortion\.
### 4\.4Equivariant Analysis

To strictly verify that our proposed hyperbolic group convolutions preserve spatial symmetries without geometric degradation, we empirically measure the relative equivariance error across our network layers\. Importantly, we empirically measure the relative equivariance error at initialisation to ensure we are not learning the behaviour implicitly\. While our architecture is theoretically equivariant by design, the projection of vectors between the Euclidean tangent spaceT0​𝔹cnT\_\{0\}\\mathbb\{B\}\_\{c\}^\{n\}and the Poincaré manifold𝔹cn\\mathbb\{B\}\_\{c\}^\{n\}, combined with discrete tensor decoupling and floating\-point approximations, necessitates empirical validation\.

We decouple this validation from the training process by evaluating the equivariance error at initialisation\. By passing randomly sampled hyperbolic tensors through untrained layers, we isolate the structural guarantees of the architecture from any learned, approximate symmetries\.

#### 4\.4\.1The Relative Equivariance Error Metric\.

For a given layerΦ\\Phi, an input tensorx∈𝔹cnx\\in\\mathbb\{B\}\_\{c\}^\{n\}, and a spatial transformationg∈Gg\\in G, true equivariance dictates that transforming the input prior to the forward pass must yield the exact same result as applying the corresponding output transformation to the resultant feature map\.

Because the standard EuclideanL2L\_\{2\}distance is not a valid metric for comparing points directly on the coordinates of the Poincaré disk, we project the layer outputs back to the origin’s tangent space via the logarithmic maplog0c⁡\(⋅\)\\log\_\{0\}^\{c\}\(\\cdot\)\. We define the tangent\-space Relative Equivariance Error \(Δe​q\\Delta\_\{eq\}\) as:

Δe​q=‖log0c⁡\(Φ​\(ρin​\(g\)​x\)\)−log0c⁡\(ρout​\(g\)​Φ​\(x\)\)‖2‖log0c⁡\(ρout​\(g\)​Φ​\(x\)\)‖2\\Delta\_\{eq\}=\\frac\{\\\|\\log\_\{0\}^\{c\}\(\\Phi\(\\rho\_\{\\text\{in\}\}\(g\)x\)\)\-\\log\_\{0\}^\{c\}\(\\rho\_\{\\text\{out\}\}\(g\)\\Phi\(x\)\)\\\|\_\{2\}\}\{\\\|\\log\_\{0\}^\{c\}\(\\rho\_\{\\text\{out\}\}\(g\)\\Phi\(x\)\)\\\|\_\{2\}\}
where∥⋅∥2\\\|\\cdot\\\|\_\{2\}denotes the Frobenius norm,ρin\\rho\_\{\\text\{in\}\}represents the representation of the group action on the input space, andρout\\rho\_\{\\text\{out\}\}represents the representation of the group action on the output feature space\.

#### 4\.4\.2Input and Output Group Representations\.

The definition of the transformation representationsρ\\rhodepends strictly on the domain of the tensor\. For the initial Lifting Convolution, the input lacks a group dimension\. Thus,ρin​\(g\)\\rho\_\{\\text\{in\}\}\(g\)acts solely as a geometric spatial transformation \(e\.g\., rotating theHHandWWdimensions by90∘90^\{\\circ\}\)\. For Group Convolutions where the input maps fromG→GG\\to G, the features possess both spatial dimensions and a discrete orientation axis\. Therefore,ρout​\(g\)\\rho\_\{\\text\{out\}\}\(g\)must simultaneously apply the spatial geometric transformation and permute the orientation channels according to the inverse left\-regular representation\. For the cyclic groupC4C\_\{4\}, this channel permutation is a simple cyclic shift\. For the dihedral groupD4D\_\{4\}, the permutation routes features across reflections and rotations utilizing a hardcoded Cayley index table mappingg−1​hg^\{\-1\}h\.

We therefore evaluateΔe​q\\Delta\_\{eq\}across both Euclidean \(𝔼n\\mathbb\{E\}^\{n\}\) and Poincaré \(𝔹cn\\mathbb\{B\}\_\{c\}^\{n\}\) manifolds for90∘90^\{\\circ\}rotations \(C4,D4C\_\{4\},D\_\{4\}\) and horizontal reflections \(D4D\_\{4\}\)\.

Table 3:Equivariance error \(Δe​q\\Delta\_\{eq\}\) for various layer types across Euclidean and Poincaré manifolds\.Table[3](https://arxiv.org/html/2607.00556#S4.T3)shows standard hyperbolic convolutions fail the equivariance test entirely \(Δe​q≈1\.0\\Delta\_\{eq\}\\approx 1\.0\), demonstrating an inability to structurally track rotational or reflective symmetries\. Across all tested groups and manifolds, our proposed Lifting and Group convolutions yield a relative error bounded strictly by 32\-bit machine precision \(Δe​q≈10−7\\Delta\_\{eq\}\\approx 10^\{\-7\}\)\. These results mathematically confirm that our left\-regular permutation logic and geometrically safeβ\\beta\-unscaling successfully preserve absolute non\-commutative group symmetries, independent of the underlying manifold’s curvature\.

### 4\.5Ablations

To isolate the structural contributions of our architecture, we performed a combinatorial ablation study on the PoincaréD4D\_\{4\}ResNet\-20, systematically disabling our three core components\. Notably, unlike standard architectural ablations where removing a component merely degrades performance, ablating our proposed mechanisms often results in catastrophic geometric collapse \(NaNs\) or a complete loss of equivariance \(10\.0% random guessing\), proving their mathematical necessity\.

Table 4:Combinatorial ablation study of the PoincaréD4D\_\{4\}lifting layer on CIFAR\-10\. A ✓indicates the structural component is active, while a blank space indicates it was ablated\. 10% refers to the case where training failed with NaN\.Model ComponentsTop\-1 \(%\)β\\beta\-ScalingJoint\-Oriented BNRouting \(Perm\)✓✓✓88\.77✓✓85\.65✓✓10\.00✓✓81\.85✓10\.00✓79\.72✓10\.0010\.00The most critical dependency lies in theJoint\-Orientation Batch Normalisation\. While residual networks rely heavily on normalisation for deep convergence, standard techniques compute statistics independently per channel\. Ablating our joint\-orientation midpoint computation in favour of independent channel normalisation leads to catastrophic training failure \(yielding NaN values or 10\.00% random guessing\)\. Independent scaling aggressively centres and normalises each orientation on its own, which entirely erases the relative magnitudes between theD4D\_\{4\}channels and strips the network of its explicit knowledge of object orientation\.

Similarly, theLeft\-Regular Permutation Routingis fundamental to the algebraic integrity of the group convolutions\. When this routing is removed, the network blindly aggregates spatial features without respecting the cyclical relationships of the group channels\. Consequently, the model cannot correctly map shifted orientation channels as the input rotates, causing classification accuracy to degrade significantly to 81\.85%\. Finally,Geometrically Safe Magnitude Restorationis essential for preserving the manifold’s boundaries\. Standard Euclidean feature unflattening artificially inflates vector magnitudes; when our restoration step is ablated, the network attempts to map mathematically distorted tangent vectors back to the Poincaré ball\. This introduces numerical instability and reduces performance to 85\.65%\. The necessity of these components is highlighted by their combined ablation where removing both the magnitude restoration and the left\-regular routing compounds the geometric distortion, dropping accuracy even further to 79\.72%\.

#### 4\.5\.1Curvature\.

Finally, we investigated the effect of the manifold’s curvature constraint\. As shown in Figure[4](https://arxiv.org/html/2607.00556#S4.F4), extremely tight curvatures \(c=1\.0\) restrict the Euclidean volume of the Poincaré ball, making floating\-point operations more unstable and resulting in suboptimal accuracy\. However, our model remains generally robust to choice of curvature\. Conversely, allowing the network to dynamically learn the curvature parameter per layer \(Table[5](https://arxiv.org/html/2607.00556#S4.T5)\) yields a top\-1 accuracy of 88\.77%, slightly outperforming a static curvature initialised at c=0\.1 \(86\.07%\)\. The learnable curvature allows the network to relax or tighten the geometric bounds dynamically in response to the hierarchical depth of the extracted equivariant features\.

### 4\.6Limitations\.

While Equivariant Poincaré ResNets offer significant structural advantages, they inherit the computational overhead and floating\-point instability at the manifold boundaries common to deep hyperbolic networks\. This necessitates narrow channel widths and rigorous numerical clipping, limiting immediate scalability to high\-resolution datasets like ImageNet\. Specific to our equivariant formulation, the architecture is strictly bounded to discrete symmetry groups \(C4C\_\{4\}andD4D\_\{4\}\) and cannot natively generalize to continuous transformations \(e\.g\.,S​O​\(2\)SO\(2\)\) without inducing significant interpolation errors across the parameter space\. Additionally, our empirical validation is currently restricted to CIFAR\-10\. While this allows for controlled, rigorous testing of the architectural inductive biases, it leaves open the question of how these methods scale to more complex and larger\-scale datasets\. Future work would validate these findings across a broader range of benchmarks and domains with naturally occurring hierarchical structure where hyperbolic embeddings are expected to provide the greatest benefit\.

![Refer to caption](https://arxiv.org/html/2607.00556v1/x3.png)Figure 4:Effect of varying curvature magnitudes \(c∈\{0\.001,0\.01,0\.1,0\.5,1\.0\}c\\in\\\{0\.001,0\.01,0\.1,0\.5,1\.0\\\}\) on CIFAR\-10 performance within the Poincaré manifold\.Table 5:Comparison of learnable vs\. static curvature parameter initialisation on the PoincaréD4D\_\{4\}model\. Curvature is initialised to 0\.1\.

## 5Conclusion

This work introduces Equivariant Poincaré Residual Networks, the first framework to embed strict discrete group equivariance \(C4C\_\{4\}andD4D\_\{4\}\) directly into hyperbolic convolutional architectures\. We developed a suite of structure\-preserving operations: geometrically safeβ\\beta\-scaled tensor unflattening; left\-regular permutation routing; and Joint\-Orientation Poincaré Midpoint Batch Normalisation\. These mechanisms collectively ensure that discrete spatial symmetries are explicitly maintained without violating the geometric constraints of the Poincaré manifold\. We mathematically validate this by an equivariance error bounded at machine precision \(Δe​q≈10−7\\Delta\_\{eq\}\\approx 10^\{\-7\}\)\.

Empirical evaluations confirm that our approach drastically reduces the optimisation space, leading to an 88\.77% top\-1 accuracy on CIFAR\-10, which equates to a 12% improvement over standard hyperbolic baselines\. Furthermore, the embedded geometric priors significantly accelerate convergence and yield vastly superior generalisation in data\-constrained environments, maintaining 63\.77% accuracy on just 10% of the training data\. In Addition, we validate the preservation of robust out\-of\-distribution detection capabilities inherent to hyperbolic geometry\. By successfully decoupling group orientations this work establishes a foundational step toward highly efficient, geometrically grounded hyperbolic visual models\. Future research will explore extending these principles to continuous symmetry groups, such asS​O​\(2\)SO\(2\), investigating the application in Hyperbolic Graph Neural Networks, and scaling to high\-resolution, large\-scale datasets\.

## Acknowledgements

This work was funded, in part, by RCN \(the Research Council of Norway\) through Visual Intelligence, Centre for Research\-based Innovation \(grant no\. 309439\)\. The computations were performed, in part, on resources provided by Sigma2—the National Infrastructure for High\-Performance Computing and Data Storage in Norway \(Project NN8106K\)\. In addition, part of the training and evaluation was carried out on the High\-Performance Computing Cluster supported by the Research and Specialist Computing Support service at the University of East Anglia and in part by the AMD University Program\.

## References

## Appendix 0\.APrerequisites

### 0\.A\.1Hyperbolic Learning: The Poincaré Ball Model

Hyperbolic space𝔹n\\mathbb\{B\}^\{n\}is the unique simply connectednn\-dimensional Riemannian manifold of constant negative curvature, where curvatures measure the deviation from flat Euclidean geometry\. The constant negative curvature of the hyperbolic space, although analogous to the Euclidean sphere, presents some significant differences in geometric properties\. As such hyperbolic space cannot be isometrically embedded into Euclidean space, yet there exist a number of conformal models of hyperbolic geometry\[cannon1997hyperbolic\]employing hyperbolic metrics providing a subset of Euclidean space\. In this work, we employ the Poincaré ball model for hyperbolic geometry given its wide adoption in computer vision and unique properties ideal for embedding between Euclidean and hyperbolic representations\. The Poincaré ball model\(𝔹cn,g𝔹c\)\(\\mathbb\{B\}\_\{c\}^\{n\},g^\{\\mathbb\{B\}\_\{c\}\}\)is defined by the manifold𝔹cn=\{x∈ℝn:c​‖x‖2<1\}\\mathbb\{B\}\_\{c\}^\{n\}=\\\{x\\in\\mathbb\{R\}^\{n\}:c\\\|x\\\|^\{2\}<1\\\}with the Riemannian metric

g𝔹c=\(λxc\)2​gE=\(21−c​‖x‖2\)2​𝕀ng^\{\\mathbb\{B\}\_\{c\}\}=\(\\lambda^\{c\}\_\{x\}\)^\{2\}g^\{E\}=\\left\(\\frac\{2\}\{1\-c\\\|x\\\|^\{2\}\}\\right\)^\{2\}\\mathbb\{I\}^\{n\}\(1\)wheregE=𝕀ng^\{E\}=\\mathbb\{I\}^\{n\}is the Euclidean metric tensor andλxc=21−c​‖x‖2\\lambda^\{c\}\_\{x\}=\\frac\{2\}\{1\-c\\\|x\\\|^\{2\}\}is the conformal factor withcc, a hyperparameter, controlling the curvature and radius of the ball\. The conformal factor scales the local distances which approach infinity near the boundary of the ball, providing the unique property of space expansion\. Such space expansion of hyperbolic spaces makes them continuous analogues of trees, given volumes of an object with diameterrrscale exponentially withrr\. Thus, when referring to a tree with branching factorkk, there are𝒪​\(kl\)\\mathcal\{O\}\(k^\{l\}\)nodes at levelll, wherellserves as a discrete analogue of the radius\. This is the fundamental property which the advocating work\[ganea2018hyperbolic,khrulkov2020hyperbolic,yan2021unsupervised\]and ours takes advantage of, allowing for the efficient embedding of natural hierarchies\[sarkar2011low\]\.

Our approach employs encoders that operate in Euclidean space, and as such, we need to define a bijection from Euclidean embeddings of the encoder to the Poincaré ball of hyperbolic space\. To achieve this we apply an exponential mapexpvc⁡\(x\):ℝn→𝔹cn\\exp^\{c\}\_\{v\}\(x\):\\mathbb\{R\}^\{n\}\\rightarrow\\mathbb\{B\}\_\{c\}^\{n\}on Euclidean vectorxxwith some fixed base pointv∈𝔹cnv\\in\\mathbb\{B\}\_\{c\}^\{n\}which we set to be the origin, simplifying the exponential map and measures of distance which will be defined later\. The exponential map is as follows,

expvc⁡\(x\)=v⊕c\(tanh⁡\(c​λvc​‖x‖2\)​xc​‖x‖\)\\exp^\{c\}\_\{v\}\(x\)=v\{\\oplus\}\_\{c\}\\left\(\\tanh\\left\(\\sqrt\{c\}\\frac\{\\lambda^\{c\}\_\{v\}\\\|x\\\|\}\{2\}\\right\)\\frac\{x\}\{\\sqrt\{c\}\\\|x\\\|\}\\right\)\(2\)with its inverse logarithm map given by

logvc⁡\(x\)=2c​λvc​arctanh​\(c​‖−v⊕cx‖\)​−v⊕cx‖−v⊕cx‖\\log^\{c\}\_\{v\}\(x\)=\\frac\{2\}\{\\sqrt\{c\}\\lambda^\{c\}\_\{v\}\}\\text\{arctanh\}\\left\(\\sqrt\{c\}\\\|\-v\{\\oplus\}\_\{c\}x\\\|\\right\)\\frac\{\-v\{\\oplus\}\_\{c\}x\}\{\\\|\-v\{\\oplus\}\_\{c\}x\\\|\}\(3\)Given the change in geometry, hyperbolic spaces do not allow for standard vector space operations, as such we employ gyrovector formalism for standard operations such as addition, subtraction, multiplication\[ungar2008gyrovector,ganea2018hyperbolic\]\. Therefore, from Eq\.[2](https://arxiv.org/html/2607.00556#Pt0.A1.E2),⊕c\{\\oplus\}\_\{c\}is defined as the gyrovector or Möbius addition of a pair of pointsx,y∈𝔹cnx,y\\in\\mathbb\{B\}\_\{c\}^\{n\}

v⊕cw=\(1\+2​c​⟨v,w⟩\+c​‖w‖2\)​v\+\(1−c​‖v‖2\)​w1\+2​c​⟨v,w⟩\+c2​‖v‖2​‖w‖2v\{\\oplus\}\_\{c\}w=\\frac\{\(1\+2c\\langle v,w\\rangle\+c\\\|w\\\|^\{2\}\)v\+\(1\-c\\\|v\\\|^\{2\}\)w\}\{1\+2c\\langle v,w\\rangle\+c^\{2\}\\\|v\\\|^\{2\}\\\|w\\\|^\{2\}\}\(4\)Leading from gyrovector formalism is the notion of distance, vital for self\-supervised losses where typically the Euclidean cosine similarity and distance, are employed\[richemond2020byol,chen2020simple,bardes2021vicreg\]\. On the Poincaré ball of hyperbolic space we define the distance betweenx,y∈𝔹cnx,y\\in\\mathbb\{B\}\_\{c\}^\{n\}as follows:

dist𝔹​\(x,y\)=2c​arctanh​\(c​‖−x⊕cy‖\)\\text\{dist\}\_\{\\mathbb\{B\}\}\(x,y\)=\\frac\{2\}\{\\sqrt\{c\}\}\\text\{arctanh\}\\left\(\\sqrt\{c\}\\\|\-x\\oplus\_\{c\}y\\\|\\right\)\(5\)which withc=1c=1recovers the geodesic, a vital concept given cosine similarity is analogous to sphere geodesic distance, whereasc→0c\\rightarrow 0produces the Euclidean distance\.