We extend decision tree and random forest algorithms to product space manifolds: Cartesian products of Euclidean, hyperspherical, and hyperbolic manifolds. Such spaces have extremely expressive geometries capable of representing many arrangements of distances with low metric distortion. To date, all classifiers for product spaces fit a single linear decision boundary, and no regressor has been described. Our method enables a simple, expressive method for classification and regression in product manifolds. We demonstrate the superior accuracy of our tool compared to Euclidean methods operating in the ambient space or the tangent plane of the manifold across a range of constant-curvature and product manifolds. Code for our implementation and experiments is available at https://github.com/pchlenski/embedders.

翻译：我们将决策树与随机森林算法扩展至乘积空间流形：即欧几里得、超球面及双曲流形的笛卡尔积。此类空间具有极强的几何表达能力，能以较低的度量失真表征多种距离分布。迄今为止，所有针对乘积空间的分类器仅拟合单一线性决策边界，且尚未有回归方法被提出。我们的方法为乘积流形中的分类与回归任务提供了一种简洁而强大的解决方案。通过在多种常曲率流形与乘积流形上的实验，我们证明了该方法相较于在环境空间或流形切平面中操作的欧几里得方法具有更高的准确性。算法实现与实验代码发布于 https://github.com/pchlenski/embedders。