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DAREK -- Distance Aware Error for Kolmogorov Networks

Masoud Ataei, Mohammad Javad Khojasteh, Vikas Dhiman

TL;DR

This work addresses uncertainty estimation for Kolmogorov Arnold Networks (KANs) by introducing DAREK, a distance-aware worst-case error framework. It develops analytic, distance-aware bounds for spline interpolation and extends them to multi-layer spline architectures, accompanied by a practical algorithm to compute per-test-point bounds during inference. The approach is validated on a cosine interpolation task and a laser-scan–based object shape estimation problem, showing tight bounds that enclose the true shape with lower computational cost than Monte Carlo ensembles or Gaussian Process baselines. Overall, DAREK provides interpretable, scalable, distance-aware uncertainty for semi-parametric, spline-based neural networks and highlights potential for broader high-dimensional applications.

Abstract

In this paper, we provide distance-aware error bounds for Kolmogorov Arnold Networks (KANs). We call our new error bounds estimator DAREK -- Distance Aware Error for Kolmogorov networks. Z. Liu et al. provide error bounds, which may be loose, lack distance-awareness, and are defined only up to an unknown constant of proportionality. We review the error bounds for Newton's polynomial, which is then generalized to an arbitrary spline, under Lipschitz continuity assumptions. We then extend these bounds to nested compositions of splines, arriving at error bounds for KANs. We evaluate our method by estimating an object's shape from sparse laser scan points. We use KAN to fit a smooth function to the scans and provide error bounds for the fit. We find that our method is faster than Monte Carlo approaches, and that our error bounds enclose the true obstacle shape reliably.

DAREK -- Distance Aware Error for Kolmogorov Networks

TL;DR

This work addresses uncertainty estimation for Kolmogorov Arnold Networks (KANs) by introducing DAREK, a distance-aware worst-case error framework. It develops analytic, distance-aware bounds for spline interpolation and extends them to multi-layer spline architectures, accompanied by a practical algorithm to compute per-test-point bounds during inference. The approach is validated on a cosine interpolation task and a laser-scan–based object shape estimation problem, showing tight bounds that enclose the true shape with lower computational cost than Monte Carlo ensembles or Gaussian Process baselines. Overall, DAREK provides interpretable, scalable, distance-aware uncertainty for semi-parametric, spline-based neural networks and highlights potential for broader high-dimensional applications.

Abstract

In this paper, we provide distance-aware error bounds for Kolmogorov Arnold Networks (KANs). We call our new error bounds estimator DAREK -- Distance Aware Error for Kolmogorov networks. Z. Liu et al. provide error bounds, which may be loose, lack distance-awareness, and are defined only up to an unknown constant of proportionality. We review the error bounds for Newton's polynomial, which is then generalized to an arbitrary spline, under Lipschitz continuity assumptions. We then extend these bounds to nested compositions of splines, arriving at error bounds for KANs. We evaluate our method by estimating an object's shape from sparse laser scan points. We use KAN to fit a smooth function to the scans and provide error bounds for the fit. We find that our method is faster than Monte Carlo approaches, and that our error bounds enclose the true obstacle shape reliably.
Paper Structure (12 sections, 5 theorems, 20 equations, 3 figures, 1 algorithm)

This paper contains 12 sections, 5 theorems, 20 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Method
  4. Experiments
  5. Conclusion
  6. Proofs
  7. Background
  8. Proof of Theorem \ref{['thm:poly-interp-bound']}
  9. Proof of Theorem \ref{['thm:two-layer-poly-bound']}
  10. Proof of Lemma \ref{['thm:error-at-knots']}
  11. L-layer generalization
  12. Additional visualizations

Key Result

Theorem 1

Consider $f:[a,b] \to \mathbb{R}$ has $k+1$ continuous derivatives (that is $f \in C^{(k+1)}$) and is $k+1$th-order Lipschitz continuous with constant ${\cal L}^{k+1}_f$. Let ${\cal P}_{k,j} [f(\tau_{1:m})]$ be Newton's polynomial fit that passes through the knots $f(\tau_{1:m}) \coloneqq \{(\tau_i, The above bound is tight when the $k+1$th derivative is constant at the maximum amount, $f^{(k+1)}(

Figures (3)

  • Figure 1: The error bounds of KAN model on $\cos$ function. left) one-layer model, right) uncertainty estimation for a 2-layer DAREK, Ensemble, and GP on a cosine function. Ensemble and GP's uncertainty bounds are shown within the $\pm3\sigma$ range.
  • Figure 2: In the first row, the plots from left to right show the train-test dataset, the trained model's predictions, and the interpolation error of each test point. The second row depicts the prediction of distance from the object boundary ($R$) versus the laser scanner angle ($\theta$), the $x$ position of the boundary point, and the $y$ location of the boundary point.
  • Figure 3: The error bounds of two layer model on $\cos$ function.

Theorems & Definitions (10)

  • Definition 1: Piecewise polynomial
  • Definition 2: Input distance awareness
  • Theorem 1: Newton's Polynomial error bound de1978practical
  • Lemma 1: Piecewise polynomial interpolation at knots
  • Theorem 2: Two-layer KAN error bound
  • Theorem 3: Mean value theorem de1978practical
  • proof
  • proof
  • proof
  • Theorem 4: Multi-layer KAN error