# Layer-wise Lipschitz-Product Control for Deep Kolmogorov--Arnold Network Representations of Compositionally Structured Functions

arXiv: https://arxiv.org/abs/2604.26444 · HTML: https://mlsanity.com/paper/2604.26444/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 16 references that are in the map.
- First posted on arXiv 29 Apr 2026.

## Builds on (most cited first, 2 of 2)

- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem](https://mlsanity.com/paper/1906.11945/) (2019)

## Built on by (most cited first, 1 of 1)

- [Stability-Constrained Approximation in Spline KANs: Exact Layer Balancing and Budget-Compatible Saturation](https://mlsanity.com/paper/2609.17619/) (2026)

## Abstract (arXiv)

We prove that any continuous function f from [0,1]^n to R representable by a finite computation tree with N internal nodes and compositional sparsity s = O(1) admits a deep Kolmogorov-Arnold Network (KAN) representation. Each internal node is realised by a primitive KAN block with controlled block depth and Lipschitz product. The layer-wise Lipschitz product satisfies the primary domain-sensitive bound independent of the input dimension n. It simplifies to P(KAN_f) <= max(C*,1)^L_f with L_f <= c_max * N. For the standard operations {+,-,x,sin,cos} with x nodes on [0,1]-bounded inputs we obtain P(KAN) <= 1. Layer widths satisfy n_l <= n + 2 w_max * N. The uniform approximation error is bounded by N * max(C*,1)^d(f) * epsilon_Op (simplifies when C* <=1). For f in C^m we obtain optimal B-spline rates. Range bounds are also derived (B_f <= N+1 for additive trees). This addresses the gap on Lipschitz control in deep KAN stacks noted by Liu et al. (2024). Experiments confirm P(KAN)=1.0 for several compositionally structured functions.