Minimax Rates for STIT and Poisson Hyperplane Random Forests

In [12], Mourtada, Ga\"{i}ffas and Scornet showed that, under proper tuning of the complexity parameters, random trees and forests built from the Mondrian process in $\mathbb{R}^d$ achieve the minimax rate for $\beta$-H\"{o}lder continuous functions, and random forests achieve the minimax rate for $(1+\beta)$-H\"{o}lder functions in arbitrary dimension, where $\beta \in (0,1]$. In this work, we show that a much larger class of random forests built from random partitions of $\mathbb{R}^d$ also achieve these minimax rates. This class includes STIT random forests, the most general class of random forests built from a self-similar and stationary partition of $\mathbb{R}^d$ by hyperplane cuts possible, as well as forests derived from Poisson hyperplane tessellations. Our proof technique relies on classical results as well as recent advances on stationary random tessellations in stochastic geometry.