Hierarchical RBF-KAN and RBF-SKAN Architectures for Multidimensional Function Approximation and Random Field Learning
For researchers in neural network approximation theory, this work introduces a new architecture that partially addresses the curse of dimensionality, but results are theoretical and empirical validation is limited.
The paper proposes hierarchical RBF-KAN and RBF-SKAN architectures for function approximation and random field learning, proving universal approximation and showing potential to alleviate the curse of dimensionality by reducing effective dimensionality.
In this manuscript, we propose and analyze hierarchical Kolmogorov--Arnold neural network architectures employing radial basis functions as activation functions for approximating deterministic functions and random field models. Specifically, we develop a hierarchical radial-basis-function Kolmogorov--Arnold network (hierarchical RBF-KAN) for multidimensional deterministic function approximation and a hierarchical radial-basis-function stochastic Kolmogorov--Arnold network (hierarchical RBF-SKAN) for random field learning. From a theoretical perspective, we establish universal approximation results for both architectures. In particular, we derive quantitative approximation estimates for the hierarchical RBF-KAN, showing that the proposed framework has the potential to partially alleviate the curse of dimensionality in learning high-dimensional functions by reducing the effective dimensionality of the approximation problem. Furthermore, we show that the hierarchical RBF-SKAN can approximate random field models under the Wasserstein-2 metric. Empirically, we show that our proposed radial-basis-function-based neural network structure could effectively learn multivariate functions and random field models.