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

TL;DR

This paper introduces a novel distance-aware error bounding method for spline neural networks, enabling deterministic, tight error estimates that improve safety and efficiency in applications like object shape estimation and navigation.

Contribution

It develops a bottom-up approach to derive deterministic error bounds for spline networks, extending classical bounds to deep compositions and introducing a distance-awareness metric.

Findings

Error bounds are tight and deterministic, applicable to deep spline networks.

Our method outperforms Gaussian process and Monte Carlo approaches in speed.

Distance-aware uncertainty estimates are more regionally comprehensive than baselines.

Abstract

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 Kolmogorov-Arnold networks. By analyzing error propagation through composed spline layers, we obtain error bounds for the entire network. These bounds are deterministic, do not rely on sampling or probabilistic assumptions, and hold under mild regularity conditions. We evaluate our method on object shape estimation from sparse laser scans and safe navigation in unstructured environments. Our method is faster than the Gaussian process and Monte Carlo approaches, and our bounds reliably enclose the true error. We also develop a metric for the distance-awareness of an uncertainty estimator and show that distance-aware uncertainty for Kolmogorov networks (DAREK) is distance-aware in more regions than the baselines.