Group-Equivariant Poincaré Convolutional Networks
2026-07-01 • Machine Learning
Machine LearningArtificial Intelligence
AI summaryⓘ
The authors address challenges in training neural networks that work in a special curved space called hyperbolic space, which is useful for representing complex data. They improve on previous methods by combining hyperbolic geometry with ways to handle symmetrical transformations of images, making the models more efficient and faster to train. To do this, they develop new mathematical tools that keep the training stable while respecting the shape constraints of hyperbolic space. Their approach helps reduce unnecessary repetition in how the model learns from transformed versions of the same object.
Hyperbolic spacePoincaré ResNetRiemannian gradientsEquivarianceDiscrete symmetry groupsGroup convolutionsBatch normalizationPoincaré ballTensor reshaping
Authors
Aiden Durrant, Rahul Baburajan, Georgios Leontidis
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 ($C_4$ and $D_4$). 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.