AI 中文总结
针对样条KAN，研究硬Lipschitz预算下的逼近，精确求解层平衡，给出保持预算的离散化定理及极小极大下界，并证明层误差不抵消。
AI 中文摘要
深度样条叠加网络在逼近阶与跨深度稳定性之间存在张力。我们研究了在硬逐层Lipschitz预算下的逼近问题，并围绕两个量来组织：给定深度因子分解的因子分解稳定性复杂度，以及离散化算子的预算兼容逼近复杂度。首先，我们精确求解了固定非负包络矩阵链的有限深度对角平衡问题：最优均匀层预算等于$\\|M_{L-1}\cdots M_0\\|_{\infty\to\infty}^{1/L}$，由显式单遍最小化器实现，适用于矩形层，并完整处理了退化与不可达情况。最优值可以任意大于网络本身的Lipschitz常数，因为传递到包络会破坏符号抵消。其次，我们给出了一个构造性样条离散化定理，在受控松弛内保持预算，并具有显式网格阈值。相反，对于精确保持预算的线性样条值算子，我们证明了在同时受限于一阶和三阶导数范数的类别上的预算兼容极小极大下界——这一约束对由问题本身强制，排除了通常的缩放逃逸。最后，我们展示了相应的层误差在复合下不必抵消：对于该类的每个算子，存在一个稳定的深度$L$塔实现累积误差的恒定比例，因此上界的线性深度累积并非证明伪影。
英文摘要
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-pass minimiser, for rectangular layers, with a complete treatment of degeneracies and non-attainment. The optimum can be arbitrarily larger than the Lipschitz constant of the network itself, because passing to envelopes destroys sign cancellation. Second, we give a constructive spline discretisation theorem preserving the budget up to a controlled slack, with an explicit grid threshold. Conversely, for linear spline-valued operators that preserve the budget exactly, we prove budget-compatible minimax lower bounds on classes constrained simultaneously in the first and third derivative norms -- a constraint pair that is forced by the problem and that rules out the usual scaling escapes. Finally, we show that the corresponding layer errors need not cancel under composition: for every operator of the class there is a stable depth-$L$ tower realising a constant fraction of the accumulated error, so the linear-in-depth accumulation of the upper bound is not a proof artefact.
Comments33 pages. Ancillary code reproduces the numerical constants of Appendix B. Complements arXiv:2604.26444, which constructs deep KAN representations with controlled layer-wise Lipschitz product; here the allocation of that product across layers is solved exactly, and the cost of preserving it under spline discretisation is bounded from both sides
