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

• URL: http://arxiv.org/abs/2604.26444v1
• Date: Wed, 29 Apr 2026 08:55:49 GMT
• Title: Layer-wise Lipschitz-Product Control for Deep Kolmogorov--Arnold Network Representations of Compositionally Structured Functions
• Abstract summary: Lipschitz product satisfies the primary domain-sensitive bound independent of the input dimension n.<n>Experiments confirm P(KAN)=1.0 for several compositionally structured functions.
• Score: 51.56484100374058
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• On Explicit Super-Expressive Approximation for Neural Networks [14.904203072235434]
  We investigate the fixed-architecture neural network approximation with explicit parameter bounds and elementary activations.<n>While prior work demonstrated super-expressive approximation using fixed-size networks, they lack quantitative and non-asymptotic characterizations of parameter magnitude with respect to the approximation error.
  arXiv  Detail & Related papers  (2026-07-07T20:21:57Z)
• Topological DeepONets and a generalization of the Chen-Chen operator approximation theorem [0.0]
  We provide a branch-trunk neural architecture for approxing nonlinear operators acting between function spaces.<n>Our main theorem shows that continuous operators $G:Vto C(K;mathbbRm)$, where $Vsubset X$ and $KsubsetmathbbRd$ are compact, can be uniformly approximated by such topological DeepONets.
  arXiv  Detail & Related papers  (2026-03-12T14:23:35Z)
• Tripartite information of free fermions: a universal entanglement coefficient from the sine kernel [51.56484100374058]
  We study the tripartite information I_3 of free fermions on two-dimensional lattices partitioned into three adjacent strips of width w.<n>We find that g(z) has a unique zero at z* = 1.329 +/- 0.001: modes with k_F w z* violate monogamy of mutual information.<n>Two exact cancellations -- of the z ln z area-law terms and of the z2 terms -- are intrinsic to the I_3 combination.
  arXiv  Detail & Related papers  (2026-03-03T15:39:35Z)
• Optimal Convergence Rates of Deep Neural Network Classifiers [25.56187933090708]
  We study the binary classification problem on $[0,1]d$ under the Tsybakov noise condition.<n>We prove that the optimal convergence rate for the excess 0-1 risk of classifiers is $$ left( frac1n right)fracbetacdot (1wedgebeta)qfracd_*s+1+ (1+frac1s+1)cdotbetacdot (1wedgebeta)q;;, $$ which is independent of the input
  arXiv  Detail & Related papers  (2025-06-17T18:13:09Z)
• Enhancing Convergence of Decentralized Gradient Tracking under the KL Property [10.925931212031692]
  We establish convergence of the same type for the notorious decentralized gradient-tracking-based algorithm SONATA.<n>This matches the convergence behavior of centralized-gradient algorithms except when.<n>thetain (1/2,1)$, sub rate is certified; and.<n>textbf(iii)$ when $thetain (1/2,1)$, sub rate is certified; and.<n>textbf(iii)$ when $thetain (1/2,1)$, sub rate is certified.
  arXiv  Detail & Related papers  (2024-12-12T18:44:36Z)
• Polynomial Width is Sufficient for Set Representation with High-dimensional Features [69.65698500919869]
  DeepSets is the most widely used neural network architecture for set representation. We present two set-element embedding layers: (a) linear + power activation (LP) and (b) linear + exponential activations (LE)
  arXiv  Detail & Related papers  (2023-07-08T16:00:59Z)
• A Law of Robustness beyond Isoperimetry [84.33752026418045]
  We prove a Lipschitzness lower bound $Omega(sqrtn/p)$ of robustness of interpolating neural network parameters on arbitrary distributions. We then show the potential benefit of overparametrization for smooth data when $n=mathrmpoly(d)$. We disprove the potential existence of an $O(1)$-Lipschitz robust interpolating function when $n=exp(omega(d))$.
  arXiv  Detail & Related papers  (2022-02-23T16:10:23Z)
• Submodular + Concave [53.208470310734825]
  It has been well established that first order optimization methods can converge to the maximal objective value of concave functions. In this work, we initiate the determinant of the smooth functions convex body $$F(x) = G(x) +C(x)$. This class of functions is an extension of both concave and continuous DR-submodular functions for which no guarantee is known.
  arXiv  Detail & Related papers  (2021-06-09T01:59:55Z)
• Deep neural network approximation of analytic functions [91.3755431537592]
  entropy bound for the spaces of neural networks with piecewise linear activation functions. We derive an oracle inequality for the expected error of the considered penalized deep neural network estimators.
  arXiv  Detail & Related papers  (2021-04-05T18:02:04Z)
• Optimal Approximation Rate of ReLU Networks in terms of Width and Depth [5.37133760455631]
  This paper concentrates on the approximation power of deep feed-forward neural networks in terms of width and depth. It is proved that ReLU networks with width $mathcalObig(maxdlfloor N1/drfloor,, N+2big)$ and depth $mathcalO(L)$ can approximate a H"older continuous function on $[0,1]d$ with an approximation rate $mathcalObig(lambdasqrtd (N2L2ln
  arXiv  Detail & Related papers  (2021-02-28T13:15:55Z)
• Deep regularization and direct training of the inner layers of Neural Networks with Kernel Flows [0.609170287691728]
  We introduce a new regularization method for Artificial Neural Networks (ANNs) based on Kernel Flows (KFs) KFs were introduced as a method for kernel selection in regression/kriging based on the minimization of the loss of accuracy incurred by halving the number of points in random batches of the dataset.
  arXiv  Detail & Related papers  (2020-02-19T18:20:36Z)
• Deep Network Approximation for Smooth Functions [9.305095040004156]
  We show that deep ReLU networks of width $mathcalO(Nln N)$ and depth $mathcalO(L L)$ can approximate $fin Cs([0,1]d)$ with a nearly optimal approximation error. Our estimate is non-asymptotic in the sense that it is valid for arbitrary width and depth specified by $NinmathbbN+$ and $LinmathbbN+$, respectively.
  arXiv  Detail & Related papers  (2020-01-09T15:06:10Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.