发表机构
University of Maine; Rochester Institute of Technology（缅因大学; 罗切斯特理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对高维样条KAN中虚构节点破坏距离感知保证的问题，提出排水机制，通过构造单调递减不确定性路径恢复保证，在2D和100维实验中将SDA从85%提升至98-99%。
AI 中文摘要
具有样条激活函数的Kolmogorov-Arnold网络（KANs）近来在可解释函数逼近方面展现出潜力。Kolmogorov网络的距离感知误差（DAREK）通过为KANs配备距离感知误差界，引入了一种计算高效的、自底向上的不确定性量化方法；然而，在高维设置中，理论保证可能因虚构节点的出现而被削弱。受Kolmogorov-Arnold表示定理的启发，DAREK采用逐分量公式，其中每个输入维度被单独处理；结果，诱导的节点位置可能出现在组合输入空间中，而并不对应实际的训练数据。这些虚构节点误导DAREK不确定性估计器，使其在远离任何真实观测处报告低不确定性，从而违反了距离感知保证。我们精确识别了这一失效模式，刻画了其几何结构，并提出了一种排水不确定性机制，通过从任何虚构节点区域向最近真实节点构造单调递减的不确定性路径来恢复距离感知。所提出的排水方法提供了一种实用的启发式修正，在恢复高维设置中理论距离感知的同时，缓解了虚构节点失效模式。在2D合成基准和100维人脸数据集上的实验表明，排水将采样距离感知（SDA）从85%提升至98-99%，以较低的计算成本匹配高斯过程。
英文摘要
Kolmogorov-Arnold Networks (KANs) with spline activations have recently shown promise for interpretable function approximation. Distance-Aware Error for Kolmogorov Networks (DAREK) introduces a computationally efficient bottom-up approach to uncertainty quantification by equipping KANs with distance-aware error bounds; yet, in high-dimensional settings, the theoretical guarantees can be weakened by the emergence of fictitious knots. Inspired by the Kolmogorov-Arnold representation theorem, DAREK adopts a componentwise formulation in which each input dimension is treated separately; as a result, induced knot locations may appear in the combined input space without corresponding to actual training data. These fictitious knots mislead the DAREK uncertainty estimator into reporting low uncertainty far from any real observation, violating the distance-awareness guarantee. We identify this failure mode precisely, characterize its geometric structure, and propose a drainage uncertainty mechanism that restores distance-awareness by constructing a monotonically decreasing uncertainty path from any fictitious knot region toward the nearest real knot. The proposed drainage method provides a practical heuristic correction that mitigates the fictitious-knot failure mode while restoring theoretical distance-awareness in high-dimensional settings. Experiments on a 2D synthetic benchmark and a 100-dimensional face dataset show that drainage raises sampled distance-awareness (SDA) from 85% to 98-99%, matching Gaussian processes at lower computational cost.
