# The Binary Space Partitioning-Tree Process

arXiv: https://arxiv.org/abs/1903.09343 · HTML: https://mlsanity.com/paper/1903.09343/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Mar 2019.

## Built on by (most cited first, 5 of 5)

- [Minimax Rates for High-Dimensional Random Tessellation Forests](https://mlsanity.com/paper/2109.10541/) (2021)
- [Hierarchical Infinite Relational Model](https://mlsanity.com/paper/2108.07208/) (2021)
- [Shape Modeling with Spline Partitions](https://mlsanity.com/paper/2108.02507/) (2021)
- [Statistical Advantages of Oblique Randomized Decision Trees and Forests](https://mlsanity.com/paper/2407.02458/) (2024)
- [Proper decision trees: An axiomatic framework for solving optimal decision tree problems with arbitrary splitting rules](https://mlsanity.com/paper/2503.01455/) (2025)

## Abstract (arXiv)

The Mondrian process represents an elegant and powerful approach for space partition modelling. However, as it restricts the partitions to be axis-aligned, its modelling flexibility is limited. In this work, we propose a self-consistent Binary Space Partitioning (BSP)-Tree process to generalize the Mondrian process. The BSP-Tree process is an almost surely right continuous Markov jump process that allows uniformly distributed oblique cuts in a two-dimensional convex polygon. The BSP-Tree process can also be extended using a non-uniform probability measure to generate direction differentiated cuts. The process is also self-consistent, maintaining distributional invariance under a restricted subdomain. We use Conditional-Sequential Monte Carlo for inference using the tree structure as the high-dimensional variable. The BSP-Tree process's performance on synthetic data partitioning and relational modelling demonstrates clear inferential improvements over the standard Mondrian process and other related methods.