aiqbench analysis
This paper argues that Kolmogorov-Arnold networks' favorable dimension scaling depends on the target function's structure, not on the architecture itself. It proves a lower bound for spline-KANs on Sobolev balls and shows that ReLU-MLPs with a cubic activation can match KANs on some targets. For practitioners, the takeaway is that architecture choice should be guided by whether the task's components depend on single or multiple input coordinates, not by claims of a general KAN advantage.
Kolmogorov-Arnold Networks: Curse of Dimensionality and Structural Bias
arXiv:2610.21480v1 Announce Type: new Abstract: Kolmogorov-Arnold networks (KANs) have a dimension-independent approximation exponent when the target admits a sufficiently smooth representation by univariate edge functions. We ask whether this favorable scaling belongs to the architecture or to the assumed target class. For arbitrary-depth standard spline-KANs, we prove a worst-case lower bound on isotropic Sobolev balls: below a dimension-independent accuracy threshold, the required number of spline coefficients grows at least exponentially with input dimension. Existing composition-matched MLP results, by contrast, show that favorable exponents on low-arity targets are not exclusive to KANs. We also revisit empirical comparisons. On four scaling-law targets, spline-KAN approximants constructed from known decompositions transfer exactly to MLPs with the cubic ReLU activation $\sigma(x)=(\max\{0,x\})^3$, although their full dense parameter counts are larger. In controlled ridge-profile scans, KANs have a strong advantage when each nonlinear component depends on one input coordinate. When a component couples multiple input coordinates, this advantage may narrow sharply. At tight budgets, interacting targets often favor MLPs. In the random-label test on MNIST, ReLU-MLPs fit the randomized training labels more efficiently than spline-only KANs. Together, these results locate favorable scaling in target structure and a compatible inductive bias, not a general KAN-specific escape from the curse of dimensionality.
Abstract:Kolmogorov-Arnold networks (KANs) have a dimension-independent approximation exponent when the target admits a sufficiently smooth representation by univariate edge functions. We ask whether this favorable scaling belongs to the architecture or to the assumed target class. For arbitrary-depth standard spline-KANs, we prove a worst-case lower bound on isotropic Sobolev balls: below a dimension-independent accuracy threshold, the required number of spline coefficients grows at least exponentially with input dimension. Existing composition-matched MLP results, by contrast, show that favorable exponents on low-arity targets are not exclusive to KANs. We also revisit empirical comparisons. On four scaling-law targets, spline-KAN approximants constructed from known decompositions transfer exactly to MLPs with the cubic ReLU activation $\sigma(x)=(\max\{0,x\})^3$, although their full dense parameter counts are larger. In controlled ridge-profile scans, KANs have a strong advantage when each nonlinear component depends on one input coordinate. When a component couples multiple input coordinates, this advantage may narrow sharply. At tight budgets, interacting targets often favor MLPs. In the random-label test on MNIST, ReLU-MLPs fit the randomized training labels more efficiently than spline-only KANs. Together, these results locate favorable scaling in target structure and a compatible inductive bias, not a general KAN-specific escape from the curse of dimensionality.
Machine Learning (cs.LG)
arXiv:2610.21480 [cs.LG]
(or arXiv:2610.21480v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.21480 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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