Optimal test loss at the interpolation threshold for grid-extended KANs

Prove that for Kolmogorov–Arnold Networks trained via grid extension on supervised function-fitting tasks, the test loss is minimized near the interpolation threshold where the number of trainable parameters equals the number of training samples.

Background

The authors observe U-shaped test-loss behavior when progressively fine-graining spline grids and hypothesize that the minimum occurs at the interpolation threshold determined by matching parameter count to sample count.

They provide a concrete example estimating the threshold from the relationship between grid intervals and total parameters, motivating a general claim requiring proof.