发表机构
School of Computing Sciences, University of East Anglia; Department of Physics and Technology, UiT The Arctic University of Norway（东安格利亚大学计算机科学学院; 北极大学挪威分校物理与技术系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对庞加莱ResNet优化困难及标准双曲网络无法处理空间变换等变性的问题，提出结合双曲几何与离散对称群的等变庞加莱ResNet，通过几何安全的张量重塑、左正则置换和联合定向批归一化，实现边界约束下的快速收敛和空间群等变性。
AI 中文摘要
尽管最近的进展如庞加莱ResNet已经展示了直接在双曲空间中学习视觉表示的潜力，但它们的优化仍然受到黎曼梯度的计算密集性和流形严格边界的阻碍。此外，标准双曲网络将同一物体的空间变换视为不同的层次概念，导致参数冗余和信号消失。我们提出了等变庞加莱ResNet，将双曲几何与离散对称群（$C_4$和$D_4$）相结合。我们识别了将欧几里得等变性应用于双曲空间的关键障碍，并提出了几何安全的张量重塑、用于双曲群卷积的左正则置换以及联合定向庞加莱中点批归一化。实验表明，嵌入等变性大幅减少了优化空间，在加速收敛的同时尊重庞加莱球的边界约束并保持空间群等变性。
英文摘要
While recent methods like that of the Poincaré 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 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. 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 Poincaré ball and retaining spatial group equivariance.
Comments20 Pages, 5 figures
