Dimension dependence of the constant in KAN approximation bound

Determine the dependence on input dimension of the constant C appearing in the approximation bound of Theorem [Approximation theory, KAT] (Equation (2.13)), which states that Kolmogorov–Arnold Network spline approximations satisfy ||f − (Φ^G_{L−1} ∘ … ∘ Φ^G_{0}) x||_{C^m} ≤ C G^{−k−1+m}. Provide explicit bounds or characterizations of C as a function of the input dimension and representation parameters.

Background

The paper establishes an approximation theorem for KANs using k-th order B-spline activations, proving a dimension-independent convergence rate with respect to grid size G. While the convergence exponent does not depend on dimension, the constant C in the bound may depend on the representation and dimension.

The authors explicitly defer analyzing how C scales with the dimension, indicating a concrete unresolved question about the quantitative dependence of C on dimensionality.