Learning representations according to the underlying geometry is of vital importance for non-Euclidean data. Studies have revealed that the hyperbolic space can effectively embed hierarchical or tree-like data. In particular, the few past years have witnessed a rapid development of hyperbolic neural networks. However, it is challenging to learn good hyperbolic representations since common Euclidean neural operations, such as convolution, do not extend to the hyperbolic space. Most hyperbolic neural networks do not embrace the convolution operation and ignore local patterns. Others either only use non-hyperbolic convolution, or miss essential properties such as equivariance to permutation. We propose HKConv, a novel trainable hyperbolic convolution which first correlates trainable local hyperbolic features with fixed kernel points placed in the hyperbolic space, then aggregates the output features within a local neighborhood. HKConv not only expressively learns local features according to the hyperbolic geometry, but also enjoys equivariance to permutation of hyperbolic points and invariance to parallel transport of a local neighborhood. We show that neural networks with HKConv layers advance state-of-the-art in various tasks.

翻译：根据底层几何结构学习表示对于非欧几里得数据至关重要。研究表明，双曲空间能够有效嵌入层次或树状结构数据。特别是，过去几年见证了双曲神经网络的快速发展。然而，由于常见的欧几里得神经操作（如卷积）无法扩展到双曲空间，学习良好的双曲表示极具挑战性。大多数双曲神经网络并未采用卷积运算，从而忽略了局部模式。其他方法要么仅使用非双曲卷积，要么缺失诸如排列等变性等核心属性。我们提出了HKConv——一种新颖的可训练双曲卷积，它首先将可训练的局部双曲特征与放置在双曲空间中的固定核点相关联，然后在局部邻域内聚合输出特征。HKConv不仅能够根据双曲几何富有表现力地学习局部特征，还具备双曲点排列等变性和局部邻域平行移动不变性。我们证明，包含HKConv层的神经网络在各项任务中均达到了最先进水平。