Recently, there has been a surge of interest in representation learning in hyperbolic spaces, driven by their ability to represent hierarchical data with significantly fewer dimensions than standard Euclidean spaces. However, the viability and benefits of hyperbolic spaces for downstream machine learning tasks have received less attention. In this paper, we present, to our knowledge, the first theoretical guarantees for learning a classifier in hyperbolic rather than Euclidean space. Specifically, we consider the problem of learning a large-margin classifier for data possessing a hierarchical structure. We provide an algorithm to efficiently learn a large-margin hyperplane, relying on the careful injection of adversarial examples. Finally, we prove that for hierarchical data that embeds well into hyperbolic space, the low embedding dimension ensures superior guarantees when learning the classifier directly in hyperbolic space.

翻译：最近,对超曲层空间代表性学习的兴趣激增,其驱动力是它们能够代表比标准欧几里德空间少得多的等级数据。然而,超曲层空间对于下游机器学习任务的可行性和好处却没有受到多少关注。在本文中,我们据我们所知,首先从理论上保证在超曲层而不是欧几里德空间学习分类。具体地说,我们考虑了在拥有等级结构的数据中学习大边分级器的问题。我们提供了一种算法,以便有效地学习大型海边高空飞机,依靠谨慎地注入对抗性实例。最后,我们证明,对于嵌入超曲层空间的等级数据,低嵌入层面在直接在双曲层空间学习分类器时,能够确保更高程度的保障。