提出可衡量距离的误差界，精准评估样条网络的逼近误差。
Distance-Aware Error for Spline Networks: A Bottom-Up Approach to Uncertainty
- 从单个样条神经元出发，逐层推导网络整体误差界。
- 误差界不依赖采样，且在稀疏激光扫描任务中覆盖真实误差。
- 新指标证明DAREK模型在更多区域具备距离感知能力。
我们提出一类新的距离感知误差界，精确刻画样条神经网络的近似误差。采用自下而上的方法，先分析每个神经元（样条）的误差界，再推广至整个网络。从牛顿多项式的误差界出发，推广到高阶Lipschitz连续下的任意样条，并扩展至函数复合结构，即Kolmogorov-Arnold网络的核心机制。通过分析样条层间的误差传播，获得全网络的误差界。这些界为确定性结果，无需采样或概率假设，在较弱正则性条件下成立。我们在稀疏激光扫描下的物体形状估计和非结构化环境中的安全导航任务上进行了评估。该方法比高斯过程和蒙特卡洛方法更快，且误差界能可靠包络真实误差。此外，我们还提出了衡量不确定性估计器距离感知性的指标，并证明距离感知型Kolmogorov网络（DAREK）在更广泛区域内具有距离感知特性。
原文摘要 · Abstract (English)
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.
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