用于回归的 Riemann-Lebesgue Forest
Riemann-Lebesgue Forest for Regression
AI总结：
本文提出 RLF 回归集成方法，通过 RLT 引入 Lebesgue 型响应分裂以获得更大方差缩减，并证明其渐近正态性，实验显示其性能可与原始 random forest 竞争。
摘要：
我们提出了一种名为 Riemann-Lebesgue Forest（RLF）的新型集成回归方法。RLF 的核心思想是模拟可测函数通过将其值域划分为若干区间而被逼近的方式。基于这一思想，我们开发了一种名为 Riemann-Lebesgue Tree（RLT）的新型树学习器，它有机会执行 Lebesgue 型切割，即在某些非叶节点处从响应变量 Y 出发对节点进行分裂。我们证明，最优的 Lebesgue 型切割相比普通 CART 切割（类似于 Riemann 划分）能够在响应变量 Y 上带来更大的方差缩减。这一性质有利于 RLF 的集成部分。我们还在不同参数设置下推广了 RLF 的渐近正态性。文中给出了两个一维示例，以说明 RLF 的灵活性。在模拟数据和真实世界数据集上的实验表明，RLF 相对于原始 random forest 具有具有竞争力的性能。
英文摘要：
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.
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