Group-Equivariant Poincaré Convolutional Networks

2026-07-01

AI Summary

This work addresses key limitations of existing hyperbolic visual representation methods—namely, optimization difficulties, high computational cost, and parameter redundancy coupled with signal attenuation due to the neglect of spatial symmetries. To overcome these issues, we propose the first strictly group-equivariant hyperbolic convolutional network by integrating discrete symmetry groups (C₄ and D₄) into a Poincaré ResNet architecture, yielding a spatially and group-equivariant model. Our core innovations include a geometrically safe tensor reshaping scheme, a left-regular permutation mechanism for hyperbolic group convolutions, and a joint-direction Poincaré midpoint batch normalization. This approach significantly reduces the optimization space, accelerates convergence, and enables strict group-equivariant learning—all while preserving the intrinsic hyperbolic geometry and respecting the boundary constraints of the Poincaré ball.

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.

Problem

• hyperbolic space
• Riemannian gradients
• spatial transformations
• parameter redundancy
• vanishing signals

Innovation

• Group Equivariance
• Hyperbolic Geometry
• Poincaré Ball
• Symmetry Groups
• Riemannian Optimization