Minimax Rates for STIT and Poisson Hyperplane Random Forests
2021
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. 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.
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