Riemann-Lebesgue Forest for Regression

We propose a novel ensemble method called Riemann-Lebesgue Forest (RLF) for regression. The core idea in RLF is to mimic the way how a measurable function can be approximated by partitioning its range into a few intervals. With this idea in mind, we develop a new tree learner named Riemann-Lebesgue Tree (RLT) which has a chance to perform Lebesgue type cutting,i.e splitting the node from response $Y$ at certain non-terminal nodes. We show that the optimal Lebesgue type cutting results in larger variance reduction in response $Y$ than ordinary CART \cite{Breiman1984ClassificationAR} cutting (an analogue of Riemann partition). Such property is beneficial to the ensemble part of RLF. We also generalize the asymptotic normality of RLF under different parameter settings. Two one-dimensional examples are provided to illustrate the flexibility of RLF. The competitive performance of RLF against original random forest \cite{Breiman2001RandomF} is demonstrated by experiments in simulation data and real world datasets.