  The stable under iterated tessellation (STIT) process is a stochastic process
that produces a recursive partition of space with cut directions drawn
independently from a distribution over the sphere. The case of random
axis-aligned cuts is known as the Mondrian process. Random forests and Laplace
kernel approximations built from the Mondrian process have led to efficient
online learning methods and Bayesian optimization. In this work, we utilize
tools from stochastic geometry to resolve some fundamental questions concerning
STIT processes in machine learning. First, we show that a STIT process with cut
directions drawn from a discrete distribution can be efficiently simulated by
lifting to a higher dimensional axis-aligned Mondrian process. Second, we
characterize all possible kernels that stationary STIT processes and their
mixtures can approximate. We also give a uniform convergence rate for the
approximation error of the STIT kernels to the targeted kernels, generalizing
the work of [3] for the Mondrian case. Third, we obtain consistency results for
STIT forests in density estimation and regression. Finally, we give a formula
for the density estimator arising from an infinite STIT random forest. This
allows for precise comparisons between the Mondrian forest, the Mondrian kernel
and the Laplace kernel in density estimation. Our paper calls for further
developments at the novel intersection of stochastic geometry and machine
learning.
