The Binary Space Partitioning-Tree Process

TL;DR

The paper introduces a generalized Binary Space Partitioning (BSP)-Tree process that extends the Mondrian process by allowing oblique cuts, improving space partition modeling flexibility and inference performance.

Contribution

It proposes a self-consistent BSP-Tree process with oblique cuts and non-uniform direction measures, enhancing space partition modeling beyond axis-aligned constraints.

Findings

Demonstrates improved partitioning accuracy on synthetic data

Shows better relational modeling performance

Maintains distributional invariance under subdomains

Abstract

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.