arXiv:2609.01956 2026 Architecture 1 ideas extracted · analyzed Sep 3, 2026
The paper exposes a practical approximation-theoretic representation of cubic B-splines that removes the sequential Cox-de Boor recursion from KAN edge activations. The transferable asset is the truncated-power identity: a spline basis evaluation becomes a fixed collection of shifted ReLU-like cubic powers, which can be vectorized and fused into one GPU kernel. Clamping the normalized coordinate to the finite spline domain makes the computation numerically predictable and avoids unstable large-power cancellation outside the supported knot range. The highest-value ML use is a drop-in KAN layer whose forward pass computes all spline terms with tensorized shifted powers and torch.compile, while preserving the original spline function and gradients.
Replace Cox-de Boor evaluation of each cubic B-spline edge activation with its fixed truncated-power expansion. Normalize each scalar edge input to a bounded knot coordinate, evaluate the five shifted cubic positive-part terms in parallel, and contract them with the learned spline coefficients inside one fused kernel.