arXiv subjects

Sunwoo Lim

Publications and source records attributed to Sunwoo Lim.

2 recordsLinked to original sources

arXiv2026

Bayesian Triangulation Splines: Spatial Adaptation on Irregular Domains

Conventional nonparametric regression methods for two-dimensional non-rectangular domains often overlook domain geometry and allow smoothing across boundaries. In spatial and geostatistical applications, this assumption is frequently invalid because domain boundaries typically constrain interactions among observations. Accommodating spatially varying smoothness is also substantially more challenging than in the univariate setting, and most existing methods do not adequately capture this local structure of the target function. To address these challenges, we propose Bayesian triangulation splines, which constructs locally adaptive splines over a polygonal domain. The method employs constrained Delaunay triangulations to respect boundary geometry and adapt to heterogeneous smoothness. A carefully designed prior further improves empirical performance. Under a global Sobolev smoothness assumption, we show that the proposed method achieves the optimal posterior contraction rate and adapts to unknown smoothness. We also show that the method exhibits ideal spatial adaptation in the sense that it achieves the oracle rate for inhomogeneous or locally varying structural features. Crucially, this oracle guarantee is not specific to constrained Delaunay triangulations, but holds over any triangulation satisfying weak shape-regularity conditions. Simulation studies confirm that the proposed method outperforms existing approaches by achieving higher estimation accuracy while maintaining low model complexity.

stat.ME↗

arXiv2023

Penalty-Induced Basis Exploration for Bayesian Splines

Spline basis exploration via Bayesian model selection is a widely employed strategy for determining the optimal set of basis terms in nonparametric regression. However, despite its widespread use, this approach often encounters performance limitations owing to the finite approximation of infinite-dimensional parameters. This limitation arises because Bayesian model selection tends to favor simpler models over more complex ones when the true model is not among the candidates. Drawing inspiration from penalized splines, one potential remedy is to incorporate an additional roughness penalty that directly regulates the smoothness of functions. This strategy mitigates underfitting by allowing the inclusion of more basis terms while preventing overfitting through explicit smoothness control. Motivated by this insight, we propose a novel penalty-induced prior distribution for Bayesian basis exploration. The proposed prior evaluates the complexity of spline functions based on a convex combination of a roughness penalty and a ridge-type penalty for model selection. Our method adapts to the unknown level of smoothness and achieves the minimax-optimal posterior contraction rate up to a logarithmic factor. We also provide an efficient Markov chain Monte Carlo algorithm for its implementation. Extensive simulation studies demonstrate that our method outperforms competing approaches in terms of performance metrics and model complexity. An application to real datasets further substantiates the validity of our proposed approach.

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