arXiv:2511.09144v4 Announce Type: replace 
Abstract: Gaussian process regression (GPR) is a popular nonparametric Bayesian method that provides predictive uncertainty estimates and is widely used in safety-critical applications. While prior research has introduced various uncertainty bounds, most existing approaches require access to specific input features, and rely on posterior mean and variance estimates or the tuning of hyperparameters. These limitations hinder robustness and fail to capture the model's global behavior in expectation. To address these limitations, we propose a chaining-based framework for estimating upper and lower bounds on the expected extreme values over unseen data, without requiring access to specific input features. We provide kernel-specific refinements for commonly used kernels such as RBF and Mat\'ern, in which our bounds are tighter than generic constructions. We further improve numerical tightness by avoiding analytical relaxations. In addition to global estimation, we also develop a novel method for local uncertainty quantification at specified inputs. This approach leverages chaining geometry through partition diameters, adapting to local structures without relying on posterior variance scaling. Our experimental results validate the theoretical findings and demonstrate that our method outperforms existing approaches on both synthetic and real-world datasets.

تم اقتراح إطار عمل جديد لعملية الانحدار بواسطة العمليات الغاوسية (GPR)، والذي يقدر الحدود العليا والسفلى للقيم القصوى المتوقعة على بيانات غير مرئية دون الحاجة إلى ميزات إدخال محددة. تعالج هذه الطريقة القيود المفروضة على الأساليب الحالية التي تعتمد على تقديرات المتوسط والانحراف المعياري اللاحقة أو ضبط المعلمات الفائقة. يتضمن الإطار تحسينات محددة للنوى المستخدمة بشكل شائع مثل RBF وMatérn، مما يؤدي إلى حدود أكثر دقة من الإنشاءات العامة.

Se ha propuesto un nuevo marco para la regresión por procesos gaussianos (GPR), que estima los límites superior e inferior sobre los valores extremos esperados de datos no vistos sin necesidad de características de entrada específicas. Este enfoque aborda las limitaciones de los métodos existentes que dependen de estimaciones de media y varianza posteriores o del ajuste de hiperparámetros. El marco incluye refinamientos específicos para núcleos comúnmente utilizados como RBF y Matérn, resultando en límites más ajustados que las construcciones genéricas.

Un nouveau cadre pour la régression par processus gaussien (GPR) a été proposé, qui estime les bornes supérieures et inférieures sur les valeurs extrêmes attendues de données non vues sans nécessiter de caractéristiques d'entrée spécifiques. Cette approche répond aux limitations des méthodes existantes qui reposent sur des estimations de moyenne et de variance postérieures ou sur le réglage d'hyperparamètres. Le cadre comprend des améliorations spécifiques aux noyaux pour des noyaux couramment utilisés tels que RBF et Matérn, entraînant des bornes plus serrées que les constructions génériques.

A new framework for Gaussian process regression (GPR) has been proposed, which estimates upper and lower bounds on expected extreme values over unseen data without needing specific input features. This approach addresses limitations in existing methods that rely on posterior mean and variance estimates or hyperparameter tuning. The framework includes kernel-specific refinements for commonly used kernels like RBF and Matérn, resulting in tighter bounds than generic constructions.

Practical Global and Local Bounds in Gaussian Process Regression via Chaining

arXiv:2601.07760v2 Announce Type: replace 
Abstract: Kolmogorov-Arnold Networks (KANs) have shown strong potential for efficiently approximating complex nonlinear functions. However, the original KAN formulation relies on B-spline basis functions, which incur substantial computational overhead due to De Boor's algorithm. To address this limitation, recent work has explored alternative basis functions such as radial basis functions (RBFs) that can improve computational efficiency and flexibility. Yet, standard RBF-KANs often sacrifice accuracy relative to the original KAN design. In this work, we propose Free-RBF-KAN, a RBF-based KAN architecture that incorporates adaptive learning grids and trainable smoothness to close this performance gap. Our method employs freely learnable RBF shapes that dynamically align grid representations with activation patterns, enabling expressive and adaptive function approximation. Additionally, we treat smoothness as a kernel parameter optimized jointly with network weights, without increasing computational complexity. We provide a general universality proof for RBF-KANs, which encompasses our Free-RBF-KAN formulation. Through a broad set of experiments, including multiscale function approximation, physics-informed machine learning, and PDE solution operator learning, Free-RBF-KAN achieves accuracy comparable to the original B-spline-based KAN while delivering faster training and inference. These results highlight Free-RBF-KAN as a compelling balance between computational efficiency and adaptive resolution, particularly for high-dimensional structured modeling tasks.

تم تقديم بنية Free-RBF-KAN كتحسين في شبكات كولموغوروف-أرنولد (KAN)، حيث تستخدم وظائف القاعدة الشعاعية التكيفية لتعزيز كفاءة تعلم الوظائف. تتناول هذه الطريقة الجديدة التحديات الحاسوبية المرتبطة بوظائف القاعدة B-spline التقليدية، وخاصة الحمل الزائد الناتج عن خوارزمية دي بور، مما يحسن من المرونة والدقة في تقريب الوظائف.

Se ha introducido la arquitectura Free-RBF-KAN como un avance en las Redes de Kolmogorov-Arnold (KAN), utilizando funciones de base radial adaptativas para mejorar la eficiencia del aprendizaje de funciones. Este nuevo enfoque aborda los desafíos computacionales asociados con las funciones de base B-spline tradicionales, particularmente la sobrecarga del algoritmo de De Boor, mejorando así tanto la flexibilidad como la precisión en la aproximación de funciones.

L'architecture Free-RBF-KAN a été introduite comme une avancée dans les Réseaux de Kolmogorov-Arnold (KAN), utilisant des fonctions de base radiales adaptatives pour améliorer l'efficacité de l'apprentissage des fonctions. Cette nouvelle approche répond aux défis computationnels associés aux fonctions de base B-spline traditionnelles, en particulier la surcharge due à l'algorithme de De Boor, améliorant ainsi à la fois la flexibilité et la précision dans l'approximation des fonctions.

The Free-RBF-KAN architecture has been introduced as an advancement in Kolmogorov-Arnold Networks (KANs), utilizing adaptive radial basis functions to enhance function learning efficiency. This new approach addresses the computational challenges associated with traditional B-spline basis functions, particularly the overhead from De Boor's algorithm, thereby improving both flexibility and accuracy in function approximation.

Practical Global and Local Bounds in Gaussian Process Regression via Chaining

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