  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.
