Posterior Contraction of Lévy Adaptive B-spline Regression in Besov Spaces

2026-05-19

AI Summary

This work addresses the lack of posterior contraction guarantees for nonparametric regression in Besov spaces under LARK-type models. To overcome this limitation, we propose the LABS model, which integrates Lévy adaptive regression kernels (LARK) with multi-resolution B-spline basis functions. By employing independent knot placement and Bayesian nonparametric inference, LABS flexibly captures local irregularities of the true regression function. We establish, for the first time, posterior contraction rates for LABS in Besov spaces, achieving near-minimax optimality and automatic adaptation to unknown smoothness levels. Numerical experiments on benchmark functions such as Blocks and Bumps demonstrate the model’s effectiveness in practice.

Abstract

We investigate the asymptotic properties of the Lévy Adaptive B-spline (LABS) regression model, a Bayesian nonparametric method that incorporates B-spline kernels into the Lévy Adaptive Regression Kernel (LARK) model. LABS applies splines of varying degrees with independently defined knots, yielding a flexible model class capable of adapting to irregular and locally structured features of the true function. Within the nonparametric regression framework with univariate random design and Gaussian errors, we establish that the LABS posterior contracts around the true function in Besov classes at nearly minimax-optimal rates, up to a logarithmic factor, while adapting automatically to unknown smoothness. This study contributes to filling a gap in the literature, where theoretical results on posterior contraction of the LARK model in Besov spaces remain scarce. Simulation experiments on standard test functions in Besov spaces, including Blocks, Bumps, HeaviSine, and Doppler, complement the theoretical results and demonstrate the practical utility of LABS.

Problem

• posterior contraction
• Lévy Adaptive B-spline
• Besov spaces
• nonparametric regression
• Bayesian nonparametrics

Innovation

• Lévy Adaptive B-spline
• posterior contraction
• Besov spaces
• Bayesian nonparametrics
• adaptive smoothing