Hyperbolic geometry is gaining traction in machine learning for its
effectiveness at capturing hierarchical structures in real-world data.
Hyperbolic spaces, where neighborhoods grow exponentially, offer substantial
advantages and consistently deliver state-of-the-art results across diverse
applications. However, hyperbolic classifiers often grapple with computational
challenges. Methods reliant on Riemannian optimization frequently exhibit
sluggishness, stemming from the increased computational demands of operations
on Riemannian manifolds. In response to these challenges, we present hyperDT, a
novel extension of decision tree algorithms into hyperbolic space. Crucially,
hyperDT eliminates the need for computationally intensive Riemannian
optimization, numerically unstable exponential and logarithmic maps, or
pairwise comparisons between points by leveraging inner products to adapt
Euclidean decision tree algorithms to hyperbolic space. Our approach is
conceptually straightforward and maintains constant-time decision complexity
while mitigating the scalability issues inherent in high-dimensional Euclidean
spaces. Building upon hyperDT we introduce hyperRF, a hyperbolic random forest
model. Extensive benchmarking across diverse datasets underscores the superior
performance of these models, providing a swift, precise, accurate, and
user-friendly toolkit for hyperbolic data analysis.

本研究介绍了一种基于超几何决策树算法的新方法 hyperDT，通过在超几何空间中利用内积来使得决策树算法适用于超几何空间，从而消除了计算密集型的 Riemannian 优化和数值不稳定的指数对数映射。我们的方法简单易懂，决策复杂度恒定，同时减轻了高维欧几里得空间中固有的可扩展性问题。在此基础上，我们提出了一个超几何随机森林模型 hyperRF，通过在多样化数据集上进行广泛比较，验证了这些模型的优越性能，为超几何数据分析提供了一个快速、精确、准确和易于使用的工具。