Approximation error from discretizations and its applications
Junlong Zhao, Xiumin Liu, Bin Du, Yufeng Liu
The Annals of Statistics · 2025 · Volume 53 · Issue 2
Abstract
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.
Keywords
62-08, 62G08, bias, discretization, Nonparametric regression, sliced inverse regression
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