Mixed-Curvature Decision Trees and Random Forests

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.