From Isotonic to Lipschitz Regression: A New Interpolative Perspective on Shape-restricted Estimation
Abstract
This manuscript bridges nonparametric smoothness-based and shape-restricted estimation, which may appear as two disjoint paradigms in the field. The proposed approach is motivated by a conceptually simple observation: every Lipschitz function is a sum of a monotonic and a linear function. This principle is further generalized to the higher-order monotonicity and multivariate settings. A family of estimators is proposed based on a sample-splitting procedure, inheriting desirable methodological, theoretical, and computational properties of shape-restricted estimators. The theoretical analysis provides convergence guarantees of the estimator under heteroscedastic and heavy-tailed errors, as well as adaptivity to the unknown ``complexity" of the true regression function. The generality of the proposed decomposition framework is demonstrated through new approximation results and numerical studies.
Cite
@article{arxiv.2307.05732,
title = {From Isotonic to Lipschitz Regression: A New Interpolative Perspective on Shape-restricted Estimation},
author = {Kenta Takatsu and Tianyu Zhang and Arun Kumar Kuchibhotla},
journal= {arXiv preprint arXiv:2307.05732},
year = {2026}
}