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arXiv · 2307.05732

From Isotonic to Lipschitz Regression: A New Interpolative Perspective on Shape-restricted Estimation

Kenta Takatsu·Tianyu Zhang·Arun Kumar Kuchibhotla
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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.

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Kenta Takatsu, Tianyu Zhang, Arun Kumar Kuchibhotla. 2026-05-20. From Isotonic to Lipschitz Regression: A New Interpolative Perspective on Shape-restricted Estimation. https://arxiv.org/abs/2307.05732

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