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

arXiv: https://arxiv.org/abs/2307.05732 · HTML: https://mlsanity.com/paper/2307.05732/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jul 2023.

## Built on by (most cited first, 1 of 1)

- [Fitting scattered data with optional monotonicity constraints on GPU: LipFit package](https://mlsanity.com/paper/2606.04670/) (2026)

## Abstract (arXiv)

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.