InKAN: B-Spline KANs via Truncated Power Form

• URL: http://arxiv.org/abs/2609.01956v3
• Date: Sat, 05 Sep 2026 17:54:21 GMT
• Title: InKAN: B-Spline KANs via Truncated Power Form
• Abstract summary: Kolmogorov-Arnold Networks place learnable B-spline activations on network edges rather than fixed activations on nodes.<n>InKAN replaces the Cox-de Boor recursion with the truncated power form.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Kolmogorov-Arnold Networks (KANs) place learnable B-spline activations on network edges rather than fixed activations on nodes. The standard Cox-de Boor recursion evaluates these activations through $k$ sequential passes for degree-$k$ splines, consuming over 90% of forward-pass time. InKAN replaces this recursion with the truncated power form, a classical result from approximation theory that expresses each uniform cubic B-spline as five $(x)_+^3$ terms at shifted knot positions. The resulting expression computes exact B-spline basis values: the same mathematical function as the Cox-de Boor recursion, evaluated without sequential passes. This paper documents three contributions: (1) an implementation structured for torch . compile fusion, eliminating recursion, span lookup, and scatter-gather operations; (2) a bounded-coordinate evaluation that clamps the normalized input to $[0, k{+}1]$, preventing the growth of cancellation error at large off-support coordinates; and (3) an open-source package (pip install inkan). In the tested configurations, InKAN has 2.8--3.5$\times$ lower forward-pass latency than the Cox-de Boor recursion. Partition-of-unity errors remain below $10^{-5}$ for grid sizes up to 200.

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