Stochastic Geometry to Generalize the Mondrian Process

The stable under iteration (STIT) tessellation process is a stochastic that produces recursive partition of space with cut directions drawn independently from distribution over the sphere. case random axis-aligned cuts known as Mondrian process. Random forests and Laplace kernel approximations built have led to efficient online learning methods Bayesian optimization. In this work, we utilize tools geometry resolve some fundamental questions concerning STIT processes in machine learning. First, show can be efficiently simulated by lifting higher-dimensional Second, characterize all possible kernels their mixtures approximate. We also give uniform convergence rate for approximation error targeted kernels, completely generalizing work Balog et al. [The kernel, 2016] case. Third, obtain consistency results density estimation regression. Finally, precise formula estimator arising forest. This allows comparisons between forest, estimation. Our paper calls further developments at novel intersection