# Mixed-curvature decision trees and random forests

arXiv: https://arxiv.org/abs/2410.13879 · HTML: https://mlsanity.com/paper/2410.13879/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 69 references that are in the map.
- First posted on arXiv 3 Oct 2024.

## Builds on (most cited first, 29 of 29)

- [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)
- [Neural Spacetimes for DAG Representation Learning](https://mlsanity.com/paper/2408.13885/) (2024)
- [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)

## Built on by (most cited first, 2 of 2)

- [Manify: A Python Library for Learning Non-Euclidean Representations](https://mlsanity.com/paper/2503.09576/) (2025)
- [Even Faster Hyperbolic Random Forests: A Beltrami-Klein Wrapper Approach](https://mlsanity.com/paper/2506.04360/) (2025)

## Abstract (arXiv)

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.