# Smoothing splines with varying smoothing parameter

arXiv: https://arxiv.org/abs/1306.1868 · HTML: https://mlsanity.com/paper/1306.1868/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2013.

## Built on by (most cited first, 3 of 3)

- [Regularization Penalty Optimization for Addressing Data Quality Variance in OoD Algorithms](https://mlsanity.com/paper/2206.05749/) (2022)
- [A Gift from Label Smoothing: Robust Training with Adaptive Label Smoothing via Auxiliary Classifier under Label Noise](https://mlsanity.com/paper/2206.07277/) (2022)
- [Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimation](https://mlsanity.com/paper/2406.05320/) (2024)

## Abstract (arXiv)

This paper considers the development of spatially adaptive smoothing splines for the estimation of a regression function with non-homogeneous smoothness across the domain. Two challenging issues that arise in this context are the evaluation of the equivalent kernel and the determination of a local penalty. The roughness penalty is a function of the design points in order to accommodate local behavior of the regression function. It is shown that the spatially adaptive smoothing spline estimator is approximately a kernel estimator. The resulting equivalent kernel is spatially dependent. The equivalent kernels for traditional smoothing splines are a special case of this general solution. With the aid of the Green's function for a two-point boundary value problem, the explicit forms of the asymptotic mean and variance are obtained for any interior point. Thus, the optimal roughness penalty function is obtained by approximately minimizing the asymptotic integrated mean square error. Simulation results and an application illustrate the performance of the proposed estimator.