Distance-Aware Error for Spline Networks: A Bottom-Up Approach to Uncertainty

ArXi:2501.04757v2 Announce Type: replace-cross We develop a new class of distance-aware error bounds that tightly characterize the approximation error of spline neural networks. Our bottom-up approach analyzes the error bound of each neuron (a spline) and then extends it to the full network. We begin with error bounds for Newton's polynomial, generalize them to arbitrary splines under higher-order Lipschitz continuity, and extend the result to function compositions, the core of deep networks such as Kolmogoro-Arnold networks.