# Nonparametric Shape-restricted Regression

arXiv: https://arxiv.org/abs/1709.05707 · HTML: https://mlsanity.com/paper/1709.05707/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Sep 2017.

## Built on by (most cited first, 8 of 8)

- [Shape-constrained Symbolic Regression -- Improving Extrapolation with Prior Knowledge](https://mlsanity.com/paper/2103.15624/) (2021)
- [Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert](https://mlsanity.com/paper/2308.09474/) (2023)
- [Learning linear dynamical systems under convex constraints](https://mlsanity.com/paper/2303.15121/) (2023)
- [Handling Hard Affine SDP Shape Constraints in RKHSs](https://mlsanity.com/paper/2101.01519/) (2021)
- [Approximation of optimization problems with constraints through kernel Sum-Of-Squares](https://mlsanity.com/paper/2301.06339/) (2023)
- [Learning and Testing Convex Functions](https://mlsanity.com/paper/2511.11498/) (2025)
- [Adversarial Sign-Corrupted Isotonic Regression](https://mlsanity.com/paper/2207.07075/) (2022)
- [CAIRO: Decoupling Order from Scale in Regression](https://mlsanity.com/paper/2602.14440/) (2026)

## Abstract (arXiv)

We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additive shape-restricted regression, and constrained single index model. We review some of the theoretical properties of the least squares estimator (LSE) in these problems, emphasizing on the adaptive nature of the LSE. In particular, we study the behavior of the risk of the LSE, and its pointwise limiting distribution theory, with special emphasis to isotonic regression. We survey various methods for constructing pointwise confidence intervals around these shape-restricted functions. We also briefly discuss the computation of the LSE and indicate some open research problems and future directions.