InKAN: B-Spline KANs via Truncated Power Form
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× 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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BibTeX
@misc{mysore2026inkan,
title = {{InKAN: B-Spline KANs via Truncated Power Form}},
author = {Naveen Mysore},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01956},
eprint = {2609.01956},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01956},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - InKAN: B-Spline KANs via Truncated Power Form
AU - Naveen Mysore
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.01956
UR - https://arxiv.org/abs/2609.01956
AB - 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× lower forward-pass latency than the Cox-de Boor recursion. Partition-of-unity errors remain below 10^−5 for grid sizes up to 200.
ER -