Hyperbolic space is becoming a popular choice for representing data due to
the hierarchical structure - whether implicit or explicit - of many real-world
datasets. Along with it comes a need for algorithms capable of solving
fundamental tasks, such as classification, in hyperbolic space. Recently,
multiple papers have investigated hyperbolic alternatives to hyperplane-based
classifiers, such as logistic regression and SVMs. While effective, these
approaches struggle with more complex hierarchical data. We, therefore, propose
to generalize the well-known random forests to hyperbolic space. We do this by
redefining the notion of a split using horospheres. Since finding the globally
optimal split is computationally intractable, we find candidate horospheres
through a large-margin classifier. To make hyperbolic random forests work on
multi-class data and imbalanced experiments, we furthermore outline a new
method for combining classes based on their lowest common ancestor and a
class-balanced version of the large-margin loss. Experiments on standard and
new benchmarks show that our approach outperforms both conventional random
forest algorithms and recent hyperbolic classifiers.

通过使用水平面将随机森林推广到双曲空间，使用大边际分类器找到候选分割，再结合子类最低共同祖先和类平衡的大边际损失方法，可以解决多类数据和不平衡实验问题，并且在标准和新的基准测试中表现优于传统随机森林算法和最近双曲分类器。

双曲随机森林

Hyperbolic Random Forests

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.

本文针对数据具有分层结构的情况，提出了一个算法，通过对抗性示例注入的方式，有效地学习了一个能够在如下情况下使用的大边际超平面：数据具有分层结构，嵌入到双曲空间的性能不劣，且直接在双曲空间中学习分类器时的维度较低。