The Hyperbolic space has garnered attention for its unique properties and efficient representation of hierarchical structures. Recent studies have explored hyperbolic alternatives to hyperplane-based classifiers, such as logistic regression and support vector machines. Hyperbolic methods have even been fused into random forests by constructing data splits with horosphere, which proved effective for hyperbolic datasets. However, the existing incorporation of the horosphere leads to substantial computation time, diverting attention from its application on most datasets. Against this backdrop, we introduce an extension of Xgboost, a renowned machine learning (ML) algorithm to hyperbolic space, denoted as PXgboost. This extension involves a redefinition of the node split concept using the Riemannian gradient and Riemannian Hessian. Our findings unveil the promising performance of PXgboost compared to the algorithms in the literature through comprehensive experiments conducted on 64 datasets from the UCI ML repository and 8 datasets from WordNet by fusing both their Euclidean and hyperbolic-transformed (hyperbolic UCI) representations. Furthermore, our findings suggest that the Euclidean metric-based classifier performs well even on hyperbolic data. Building upon the above finding, we propose a space fusion classifier called, EPboost. It harmonizes data processing across various spaces and integrates probability outcomes for predictive analysis. In our comparative analysis involving 19 algorithms on the UCI dataset, our EPboost outperforms others in most cases, underscoring its efficacy and potential significance in diverse ML applications. This research marks a step forward in harnessing hyperbolic geometry for ML tasks and showcases its potential to enhance algorithmic efficacy.

中文翻译：

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欧几里得和庞加莱空间系团 Xgboost


双曲空间因其独特的特性和对层次结构的有效表示而受到关注。最近的研究探索了基于超平面的分类器的双曲替代方案，例如 Logistic 回归和支持向量机。通过使用 horosphere 构建数据分片，双曲方法甚至被融合到随机森林中，这被证明对双曲数据集有效。然而，现有的星座圈的合并导致了大量的计算时间，从而转移了人们对它在大多数数据集上的应用的注意力。在此背景下，我们引入了 Xgboost 的扩展，这是一种著名的双曲空间机器学习 （ML） 算法，表示为 PXgboost。此扩展涉及使用黎曼梯度和黎曼黑森曲线重新定义节点分割概念。通过融合欧几里得和双曲变换（双曲 UCI）表示，对 UCI ML 存储库中的 64 个数据集和 WordNet 中的 8 个数据集进行了全面实验，我们的研究结果揭示了 PXgboost 与文献中的算法相比有前途的性能。此外，我们的研究结果表明，即使在双曲数据上，基于欧几里得度量的分类器也表现良好。基于上述发现，我们提出了一个名为 EPboost 的空间融合分类器。它协调了各个空间的数据处理，并集成了用于预测分析的概率结果。在我们涉及 UCI 数据集上 19 种算法的比较分析中，我们的 EPboost 在大多数情况下都优于其他算法，强调了它在不同 ML 应用中的有效性和潜在意义。 这项研究标志着在利用双曲几何进行 ML 任务方面向前迈进了一步，并展示了其增强算法功效的潜力。