将决策树拓展到混合曲率空间，提升复杂数据建模能力。
Mixed-curvature decision trees and random forests
- 基于流形几何设计角度重参数化，支持多种曲率组合的决策树
- 在57项任务中41项进入前2，29项排名第一，性能显著优于传统方法
- 适合处理具有异质结构的数据，如社交网络、知识图谱等复杂关系
决策树（DT）及其随机森林（RF）扩展是欧几里得空间中分类与回归的常用工具。然而，非欧几里得空间的学习算法仍有限。本文将DT和RF算法扩展至乘积流形：由多个双曲、超球或欧氏分量构成的笛卡尔积。此类流形可处理异质曲率，同时保持组件分解的简洁性，是复杂数据的理想嵌入空间。我们提出的新角度重参数化方法尊重流形几何，同时保留决策树的有效算法特性。在单分量流形情况下，该方法退化为欧氏或双曲版本，或引入超球面决策树算法。在涵盖57个分类、回归及链接预测任务的基准测试中，我们的乘积随机森林在29项任务中排名第一，在41项任务中进入前两名，凸显其作为产品流形中简单而强大的数据分析工具的价值。代码已开源：https://github.com/pchlenski/manify。
原文摘要 · Abstract (English)
Decision trees (DTs) and their random forest (RF) extensions are workhorses of classification and regression in Euclidean spaces. However, algorithms for learning in non-Euclidean spaces are still limited. We extend DT and RF algorithms to product manifolds: Cartesian products of several hyperbolic, hyperspherical, or Euclidean components. Such manifolds handle heterogeneous curvature while still factorizing neatly into simpler components, making them compelling embedding spaces for complex datasets. Our novel angular reformulation respects manifold geometry while preserving the algorithmic properties that make decision trees effective. In the special cases of single-component manifolds, our method simplifies to its Euclidean or hyperbolic counterparts, or introduces hyperspherical DT algorithms, depending on the curvature. In benchmarks on a diverse suite of 57 classification, regression, and link prediction tasks, our product RFs ranked first on 29 tasks and came in the top 2 for 41. This highlights the value of product RFs as straightforward yet powerful new tools for data analysis in product manifolds. Code for our method is available at https://github.com/pchlenski/manify.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。