This paper introduces an end-to-end residual network that operates entirely on the Poincar\'e ball model of hyperbolic space. Hyperbolic learning has recently shown great potential for visual understanding, but is currently only performed in the penultimate layer(s) of deep networks. All visual representations are still learned through standard Euclidean networks. In this paper we investigate how to learn hyperbolic representations of visual data directly from the pixel-level. We propose Poincar\'e ResNet, a hyperbolic counterpart of the celebrated residual network, starting from Poincar\'e 2D convolutions up to Poincar\'e residual connections. We identify three roadblocks for training convolutional networks entirely in hyperbolic space and propose a solution for each: (i) Current hyperbolic network initializations collapse to the origin, limiting their applicability in deeper networks. We provide an identity-based initialization that preserves norms over many layers. (ii) Residual networks rely heavily on batch normalization, which comes with expensive Fr\'echet mean calculations in hyperbolic space. We introduce Poincar\'e midpoint batch normalization as a faster and equally effective alternative. (iii) Due to the many intermediate operations in Poincar\'e layers, we lastly find that the computation graphs of deep learning libraries blow up, limiting our ability to train on deep hyperbolic networks. We provide manual backward derivations of core hyperbolic operations to maintain manageable computation graphs.

翻译：本文提出了一种完全基于双曲空间庞加莱球模型的端到端残差网络。双曲学习最近在视觉理解方面展现出巨大潜力，但目前仅应用于深度网络的倒数第二层。所有视觉表示仍通过标准欧几里得网络学习。本文探讨如何直接从像素级学习视觉数据的双曲表示。我们提出庞加莱残差网络（Poincaré ResNet），作为著名残差网络的双曲对应物，涵盖从庞加莱二维卷积到庞加莱残差连接的完整架构。我们识别出在双曲空间中训练全卷积网络的三个障碍，并为每个问题提出解决方案：(i)当前双曲网络初始化会导致参数坍缩至原点，限制其在深层网络中的适用性。我们提出基于恒等映射的初始化方法，可在多层间保持范数不变。(ii)残差网络严重依赖批归一化，但双曲空间中的弗雷歇均值计算代价高昂。我们引入庞加莱中点批归一化作为更快速且同等有效的替代方案。(iii)由于庞加莱层中存在大量中间运算，深度学习库的计算图会过度膨胀，限制我们训练深层双曲网络的能力。我们为核心双曲运算提供手动反向传播推导，以维持可控的计算图。