On the expressivity of the ExSpliNet KAN model

ExSpliNet is a neural network model that combines ideas of Kolmogorov networks, ensembles of probabilistic trees, and multivariate B-spline representations. In this paper, we study the expressivity of the ExSpliNet model and present two constructive approximation results that mitigate the curse of dimensionality. More precisely, we prove new error bounds for the ExSpliNet approximation of a subset of multivariate continuous functions and also of multivariate generalized bandlimited functions. The main ingredients of the proofs are a constructive version of the Kolmogorov superposition theorem, Maurey's theorem, and spline approximation results. The curse of dimensionality is lessened in the first case, while it is completely overcome in the second case. Since the considered ExSpliNet model can be regarded as a particular version of the recently introduced neural network architecture called Kolmogorov-Arnold network (KAN), our results also provide insights into the analysis of the expressivity of KANs.