Hyperbolic space has become a popular choice of manifold for representation learning of arbitrary data, from tree-like structures and text to graphs. Building on the success of deep learning with prototypes in Euclidean and hyperspherical spaces, a few recent works have proposed hyperbolic prototypes for classification. Such approaches enable effective learning in low-dimensional output spaces and can exploit hierarchical relations amongst classes, but require privileged information about class labels to position the hyperbolic prototypes. In this work, we propose Hyperbolic Busemann Learning. The main idea behind our approach is to position prototypes on the ideal boundary of the Poincare ball, which does not require prior label knowledge. To be able to compute proximities to ideal prototypes, we introduce the penalised Busemann loss. We provide theory supporting the use of ideal prototypes and the proposed loss by proving its equivalence to logistic regression in the one-dimensional case. Empirically, we show that our approach provides a natural interpretation of classification confidence, while outperforming recent hyperspherical and hyperbolic prototype approaches.
翻译:超球空间已成为一种为从树类结构和文字到图表等任意数据进行代表性学习而广泛选择的多种方法。 根据对欧洲和超球空间原型的深层次学习的成功,最近的一些著作提出了超曲原型分类。这些方法使得能够在低维输出空间进行有效学习,并能够利用各等级之间的等级关系,但需要关于等级标签的特权信息才能定位超曲型原型。在这项工作中,我们提出了双曲型Busemann学习方案。我们方法的主要理念是将原型定位在Poincare球的理想边界上,这不需要事先的标签知识。为了能够将近似准度计算到理想原型,我们引入了惩罚性Busemann损失。我们提供理论支持使用理想原型和拟议的损失,证明它与一维案例的物流回归相当。我们巧妙地表明,我们的方法提供了对分类信任的自然解释,同时超过了最近的超球型和双曲型原型方法。