Recent research in representation learning has shown that hierarchical data
lends itself to low-dimensional and highly informative representations in
hyperbolic space. However, even if hyperbolic embeddings have gathered
attention in image recognition, their optimization is prone to numerical
hurdles. Further, it remains unclear which applications stand to benefit the
most from the implicit bias imposed by hyperbolicity, when compared to
traditional Euclidean features. In this paper, we focus on prototypical
hyperbolic neural networks. In particular, the tendency of hyperbolic
embeddings to converge to the boundary of the Poincar\'e ball in high
dimensions and the effect this has on few-shot classification. We show that the
best few-shot results are attained for hyperbolic embeddings at a common
hyperbolic radius. In contrast to prior benchmark results, we demonstrate that
better performance can be achieved by a fixed-radius encoder equipped with the
Euclidean metric, regardless of the embedding dimension.

最近在表示学习方面的研究表明，分层数据在双曲空间中能产生低维度和高信息量的表示。然而，即使双曲嵌入在图像识别方面引起了人们的关注，其优化过程容易遇到数值障碍。此外，与传统的欧几里德特征相比，目前尚不清楚哪些应用最能受益于双曲性所施加的隐式偏差。本文主要着眼于原型双曲神经网络，特别关注双曲嵌入在高维度情况下收敛于庞加莱球边界的倾向，以及这对少样本分类的影响。我们展示了最佳的少样本结果是通过在共同的双曲半径下得到的双曲嵌入得到的。与以往的基准结果相比，我们证明了无论嵌入维度如何，通过配备欧几里德度量的固定半径编码器都可以获得更好的性能。