# Mixed-Curvature Decision Trees and Random Forests

arXiv: https://arxiv.org/abs/2406.05227 · HTML: https://mlsanity.com/paper/2406.05227/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 69 references that are in the map.
- First posted on arXiv 7 Jun 2024.

## Builds on (most cited first, 28 of 28)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Cooperative Game Theory Approaches for Network Partitioning](https://mlsanity.com/paper/1707.03587/) (2017)
- [Constant Curvature Graph Convolutional Networks](https://mlsanity.com/paper/1911.05076/) (2019)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Neural Embeddings of Graphs in Hyperbolic Space](https://mlsanity.com/paper/1705.10359/) (2017)
- [Large-Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/1806.00437/) (2018)
- [Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding](https://mlsanity.com/paper/2005.03847/) (2020)
- [A Self-supervised Mixed-curvature Graph Neural Network](https://mlsanity.com/paper/2112.05393/) (2021)
- [Heterogeneous manifolds for curvature-aware graph embedding](https://mlsanity.com/paper/2202.01185/) (2022)
- [Latent Graph Inference using Product Manifolds](https://mlsanity.com/paper/2211.16199/) (2022)
- [Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimization](https://mlsanity.com/paper/2309.04810/) (2023)
- [Product Manifold Learning](https://mlsanity.com/paper/2010.09908/) (2020)
- [Curve Your Attention: Mixed-Curvature Transformers for Graph Representation Learning](https://mlsanity.com/paper/2309.04082/) (2023)
- [Fr'echet random forests for metric space valued regression with non euclidean predictors](https://mlsanity.com/paper/1906.01741/) (2019)
- [Is rotation forest the best classifier for problems with continuous features?](https://mlsanity.com/paper/1809.06705/) (2018)
- [Linear Classifiers in Product Space Forms](https://mlsanity.com/paper/2102.10204/) (2021)
- [Hyperbolic Random Forests](https://mlsanity.com/paper/2308.13279/) (2023)
- [Fast hyperboloid decision tree algorithms](https://mlsanity.com/paper/2310.13841/) (2023)
- [Random Forest regression for manifold-valued responses](https://mlsanity.com/paper/1701.08381/) (2017)
- [Neural Snowflakes: Universal Latent Graph Inference via Trainable Latent Geometries](https://mlsanity.com/paper/2310.15003/) (2023)
- [Principal Component Analysis in Space Forms](https://mlsanity.com/paper/2301.02750/) (2023)
- [Riemannian optimization on the simplex of positive definite matrices](https://mlsanity.com/paper/1906.10436/) (2019)
- [Improving Heterogeneous Graph Learning with Weighted Mixed-Curvature Product Manifold](https://mlsanity.com/paper/2307.04514/) (2023)
- [Product Manifold Representations for Learning on Biological Pathways](https://mlsanity.com/paper/2401.15478/) (2024)
- [Manifold Oblique Random Forests: Towards Closing the Gap on Convolutional Deep Networks](https://mlsanity.com/paper/1909.11799/) (2019)

## Abstract (arXiv)

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.