AI 中文总结
研究 KAN 与 MLP 中架构简单性与 Fisher 简单性的一致性，发现死亡 ReLU 一致而固定基 KAN 不一致，并提出有效边度量以识别 Fisher 零方向，表明系数大小不能作为 KAN 的剪枝标准。
AI 中文摘要
Kolmogorov-Arnold 网络（KAN）的动机部分源于可解释性：其学习到的边缘函数可以被检查、剪枝并简化为符号结构。在固定基 KAN 中，这使得一个较小或为零的基系数看起来像是简单性的证明，正如多层感知机（MLP）中一个死亡的修正线性单元（ReLU）标志着未使用的计算。Fisher 零度给出了一种精确的统计概念：当且仅当在任务分布下扰动某个参数方向不可见时，该方向是 Fisher 简单的。我们研究这些架构概念与 Fisher 概念何时一致。对于死亡的 ReLU 单元，它们是一致的：闭合的激活区域使得相关的得分方向消失。对于固定基 KAN，它们并不一致。在单层高斯情形下，系数 Fisher 矩阵是输入分布下的基 Gram 矩阵，并且与拟合系数无关。在多层 KAN 中，Fisher 简单性是基于图路径的：数据必须到达一个基原子，并且其扰动必须通过下游网络传播。我们将这两个条件编码为一种有效边度量，并在局部字典独立性和有效度量非退化条件下，证明零有效暴露恰好识别出边内的 Fisher 零方向。受控诊断确认，零系数可以保持秩，而有效路径断开则移除预测方向。因此，系数大小本身并非 KAN 的基于 Fisher 的剪枝标准。
英文摘要
Kolmogorov-Arnold Networks (KANs) are motivated in part by interpretability: their learned edge functions can be inspected, pruned, and reduced to symbolic structure. In a fixed-basis KAN, this makes a small or zero basis coefficient look like a certificate of simplicity, much as a dead rectified linear unit (ReLU) marks unused computation in a multilayer perceptron (MLP). Fisher nullity gives a precise statistical notion: a parameter direction is Fisher-simple exactly when perturbing it is invisible under the task distribution. We study when these architectural and Fisher notions agree. For a dead ReLU unit, they agree: the closed activation region makes the associated score directions vanish. For a fixed-basis KAN, they do not. In the single-layer Gaussian case, the coefficient Fisher matrix is a basis Gram matrix under the input distribution and is independent of the fitted coefficients. In a multilayer KAN, Fisher simplicity is graph-path based: the data must reach a basis atom and its perturbation must propagate through the downstream network. We encode these two conditions in an effective edge measure and, under local dictionary independence and effective-measure nondegeneracy, show that zero effective exposure exactly identifies Fisher-null directions within an edge. Controlled diagnostics confirm that zero coefficients can preserve rank while effective path disconnections remove the predicted directions. Coefficient magnitude alone is therefore not a Fisher-based pruning criterion for KANs.
Comments8 pages, 2 figures. Presented at Allerton 2026
