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Stability-Constrained Approximation in Spline KANs: Exact Layer Balancing and Budget-Compatible Saturation

arXiv · AI, language, vision and robotics · article · Sep 14, 2026 · UTC

Deep spline superposition networks face a tension between approximation order and stability across depth. We study approximation under a hard layerwise Lipschitz budget, and organise it around two quantities: the factorisation stability complexity of a given deep factorisation, and the budget-compatible approximation complexity of a discretisation operator. First, we solve exactly the finite-depth diagonal balancing problem for a fixed chain of nonnegative envelope matrices: the optimal uniform layer budget equals $\|M_{L-1}\cdots M_0\|_{\infty\to\infty}^{1/L}$, attained by an explicit one-pas

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