Fisher Simplicity in Kolmogorov-Arnold Networks and Multilayer Perceptrons

Abstract

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.