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

arXiv: https://arxiv.org/abs/2609.17619 · HTML: https://mlsanity.com/paper/2609.17619/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 27 references that are in the map.
- First posted on arXiv 14 Sep 2026.

## Builds on (most cited first, 7 of 7)

- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Path-SGD: Path-Normalized Optimization in Deep Neural Networks](https://mlsanity.com/paper/1506.02617/) (2015)
- [Equi-normalization of Neural Networks](https://mlsanity.com/paper/1902.10416/) (2019)
- [Generalization Bounds and Model Complexity for Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2410.08026/) (2024)
- [The Perron-Frobenius theorem for multi-homogeneous mappings](https://mlsanity.com/paper/1801.05034/) (2018)
- [The Collatz-Wielandt quotient for pairs of nonnegative operators](https://mlsanity.com/paper/1710.07402/) (2017)
- [Layer-wise Lipschitz-Product Control for Deep Kolmogorov--Arnold Network Representations of Compositionally Structured Functions](https://mlsanity.com/paper/2604.26444/) (2026)

## Abstract (arXiv)

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 |ML-1cdots M0|∞→∞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.