Approximation error from discretizations and its applications
离散化的近似误差及其应用AI 解读 ↓
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离散化的近似误差及其应用
- 研究问题
- 将连续变量离散化时，不同切分策略造成的近似偏差如何比较，其大小与诱导分布之间的距离及函数光滑性有什么关系？
- 方法与思路
- 作者建立一元离散化的 Poincaré 型不等式并扩展到多元等情形，分析降维矩阵误差，同时结合随机森林与响应局部离散化。
- 材料陈述的贡献
- 摘要称，离散化偏差可由特定分布距离和函数光滑性控制；所提出回归算法在所述模拟中相较经典随机森林显示优势。
- 阅读关注点
- 应核对两种诱导分布与距离的定义、不等式条件、切分策略的比较标准，以及随机森林模拟中的调参、误差指标和基准公平性。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要解读。摘要未给出误差界常数、模拟数值或优势范围，不能认定该算法在所有回归问题中更优。
原文摘要
Converting a continuous variable into a discrete one is a commonly used technique for solving various problems in both statistics and machine learning. It is well known that discretizations result in biases. However, this issue has not been studied systematically. In this paper, a general framework is proposed to understand and compare the approximation errors of different slicing strategies. Poincaré-type inequalities are first established for univariate discretizations and then generalized to the multivariate and other settings. It is shown that the bias is controlled by two factors: the distance between two specific distributions that are generated with and without discretizations respectively, and the smoothness of the functions involved. Several important applications are considered to illustrate the usefulness of the results. Our results help to understand the approximation error of some matrix used in the literature of dimension reduction. Furthermore, as an illustration of the usefulness of discretizations, we propose an algorithm for regression problems, by combining random forest with partial discretizations of responses. Simulation results confirm the advantages of this algorithm over the classical random forest.