Distance-Aware Error for Spline Networks: A Bottom-Up Approach to Uncertainty

Masoud Ataei; Mohammad Javad Khojasteh; Vikas Dhiman

arxiv: 2501.04757 · v2 · submitted 2025-01-08 · 📡 eess.SP · cs.LG

Distance-Aware Error for Spline Networks: A Bottom-Up Approach to Uncertainty

Masoud Ataei , Mohammad Javad Khojasteh , Vikas Dhiman This is my paper

Pith reviewed 2026-05-23 06:16 UTC · model grok-4.3

classification 📡 eess.SP cs.LG

keywords spline neural networkserror boundsdistance-aware uncertaintyKolmogorov-Arnold networksdeterministic boundsapproximation errorspline layersuncertainty quantification

0 comments

The pith

Spline neural networks admit deterministic distance-aware error bounds obtained by propagating single-neuron bounds through layer compositions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a bottom-up procedure that first bounds the approximation error of each spline neuron and then composes those bounds across layers to produce network-wide guarantees. The bounds depend on input distance from observed data, remain fully deterministic, and rest only on mild regularity conditions rather than sampling or probability. The construction begins with classical error formulas for Newton polynomials, extends them to general splines via higher-order Lipschitz continuity, and carries the result through the function compositions that define deep spline architectures such as Kolmogorov-Arnold networks. Experiments on laser-scan shape estimation and unstructured navigation show the bounds enclose observed error while running faster than Gaussian-process or Monte-Carlo alternatives. A distance-awareness metric further indicates that the resulting estimator remains informative over larger input regions than the baselines.

Core claim

By analyzing error bounds for Newton's polynomials and generalizing them to arbitrary splines under higher-order Lipschitz continuity, then extending the result to function compositions, the paper derives deterministic error bounds for entire spline networks that characterize approximation error in a distance-aware manner.

What carries the argument

Error propagation through composed spline layers, obtained by generalizing Newton's polynomial error bounds to arbitrary splines under higher-order Lipschitz continuity.

If this is right

Network-level bounds follow directly from propagating per-spline bounds through arbitrary compositions.
The resulting guarantees are deterministic and require no probabilistic assumptions or sampling.
The bounds enclose observed error on object-shape estimation and safe-navigation tasks while computing faster than Gaussian-process or Monte-Carlo methods.
The DAREK metric registers distance-awareness over more input regions than the tested baselines.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same composition rule could be checked on spline networks trained for tasks other than shape estimation or navigation.
Distance-aware bounds might allow tighter safety margins in real-time control loops that currently rely on looser probabilistic envelopes.
If analogous propagation rules can be derived for mixed spline-and-standard activations, the method would apply to a wider class of hybrid architectures.

Load-bearing premise

The generalization of error bounds from Newton's polynomials to arbitrary splines holds under higher-order Lipschitz continuity.

What would settle it

A concrete counter-example in which the actual approximation error of a spline network exceeds the derived bound for an input that satisfies the stated regularity conditions.

Figures

Figures reproduced from arXiv: 2501.04757 by Masoud Ataei, Mohammad Javad Khojasteh, Vikas Dhiman.

**Figure 1.** 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 ±3σ range. f, h, g ∈ C (k+1), with known Lipschitz constants, and let hˆ, gˆ be piecewise polynomials. Then the error bound for the twolayer approximation is given by: |f(x) − ˆf(x)| ≤ uh(1 … view at source ↗

**Figure 2.** Figure 2: In the first row, the plots from left to right show the train-test dataset, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: The error bounds of two layer model on cos function. APPENDIX C ADDITIONAL VISUALIZATIONS [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

read the original abstract

We develop a new class of distance-aware error bounds that tightly characterize the approximation error of spline neural networks. Our bottom-up approach analyzes the error bound of each neuron (a spline) and then extends it to the full network. We begin with error bounds for Newton's polynomial, generalize them to arbitrary splines under higher-order Lipschitz continuity, and extend the result to function compositions, the core of deep networks such as Kolmogorov-Arnold networks. By analyzing error propagation through composed spline layers, we obtain error bounds for the entire network. These bounds are deterministic, do not rely on sampling or probabilistic assumptions, and hold under mild regularity conditions. We evaluate our method on object shape estimation from sparse laser scans and safe navigation in unstructured environments. Our method is faster than the Gaussian process and Monte Carlo approaches, and our bounds reliably enclose the true error. We also develop a metric for the distance-awareness of an uncertainty estimator and show that distance-aware uncertainty for Kolmogorov networks (DAREK) is distance-aware in more regions than the baselines.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper builds deterministic error bounds for spline networks bottom-up from per-spline Lipschitz conditions, which is new, but the abstract leaves the key steps on constants and tightness unshown.

read the letter

The main claim is a bottom-up derivation of deterministic error bounds for spline networks like KANs. They start from error bounds on Newton's polynomials, extend to general splines via higher-order Lipschitz continuity, then propagate the bounds through layer compositions to cover the full network. This avoids sampling or probabilistic assumptions and aims for distance-aware uncertainty. They also introduce a metric to check how distance-aware an estimator actually is. That composition approach and the metric are the clearest new pieces relative to prior spline and KAN work. The experiments on sparse laser scan shape estimation and safe navigation show the method runs faster than Gaussian processes or Monte Carlo sampling while the bounds enclose the true error in the tested cases. That practical speed comparison is useful for real-time settings. The soft spots sit in the missing details. The abstract states the generalization from Newton's polynomials to arbitrary splines but supplies no equations, proof sketch, or explicit construction for the Lipschitz constants. If those constants grow large or become hard to compute from the spline coefficients after composition, the network-level bounds could lose tightness or practicality even under the stated mild conditions. The evaluation reports enclosure but gives no quantitative tightness numbers, no ablation on the Lipschitz assumption, and no check on how often the bounds stay non-vacuous. Without those, it is difficult to judge whether the deterministic guarantee delivers a real advantage over existing methods. This work is aimed at researchers already using or extending KAN-style spline networks who want sampling-free uncertainty for safety or speed reasons. A reader focused on uncertainty quantification in structured function approximators could extract the composition rules and the distance-awareness metric. The paper deserves peer review because the bottom-up construction is distinct and the speed results are concrete, even though the referee will need to see the full derivations and tighter empirical checks on the constants.

Referee Report

2 major / 2 minor

Summary. The paper claims to develop distance-aware deterministic error bounds for spline neural networks (including KANs) via a bottom-up analysis: starting from Newton's polynomial error, generalizing to arbitrary splines under higher-order Lipschitz continuity, then propagating the bounds through function compositions to obtain network-level guarantees. These bounds require no sampling or probabilistic assumptions and are evaluated on object shape estimation from laser scans and safe navigation, where they enclose true error, run faster than GP/MC baselines, and score higher on a proposed distance-awareness metric.

Significance. If the central derivation is sound and the Lipschitz constants remain finite and computable from spline coefficients, the approach would supply a practical, sampling-free uncertainty tool for spline networks with direct relevance to safety-critical robotics and control; the distance-awareness metric is a useful addition for comparing estimators.

major comments (2)

[Abstract; error bounds for splines] Abstract and the section on error bounds for splines: the generalization from Newton's polynomial error bounds to arbitrary splines under higher-order Lipschitz continuity is stated without an explicit construction, bound on the resulting Lipschitz constants, or verification that the constants remain finite and obtainable from the spline coefficients; this step is load-bearing for the deterministic, sampling-free claim for the full network.
[Evaluation] Evaluation section: the two-task experiments demonstrate enclosure of true error but report no quantitative tightness metrics (e.g., bound-to-error ratio) or ablation on the Lipschitz assumption, so it is unclear whether the composed bounds remain non-vacuous in practice.

minor comments (2)

[Metric definition] The definition and computation of the proposed distance-awareness metric should be given explicitly (including any hyperparameters) to allow reproduction.
[Assumptions] Clarify whether the spline coefficients are assumed known exactly or estimated, as this affects how the Lipschitz constants are obtained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the error bounds and evaluation.

read point-by-point responses

Referee: [Abstract; error bounds for splines] Abstract and the section on error bounds for splines: the generalization from Newton's polynomial error bounds to arbitrary splines under higher-order Lipschitz continuity is stated without an explicit construction, bound on the resulting Lipschitz constants, or verification that the constants remain finite and obtainable from the spline coefficients; this step is load-bearing for the deterministic, sampling-free claim for the full network.

Authors: We agree that the generalization step is load-bearing and that the current presentation would benefit from greater explicitness. In the revised manuscript we will add an explicit construction of the spline error bound under higher-order Lipschitz continuity, derive explicit upper bounds on the resulting Lipschitz constants from the spline coefficients, and include a short verification that these constants remain finite under the regularity conditions already stated in the paper. These additions will be placed in the section on error bounds for splines and referenced from the abstract. revision: yes
Referee: [Evaluation] Evaluation section: the two-task experiments demonstrate enclosure of true error but report no quantitative tightness metrics (e.g., bound-to-error ratio) or ablation on the Lipschitz assumption, so it is unclear whether the composed bounds remain non-vacuous in practice.

Authors: We accept that quantitative tightness metrics and an ablation on the Lipschitz-order assumption would make the practical non-vacuousness of the bounds clearer. In the revised evaluation section we will report bound-to-error ratios for both tasks and add an ablation that varies the assumed Lipschitz order while keeping all other factors fixed, thereby showing that the composed bounds remain informative rather than vacuous under the conditions used in the experiments. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from external polynomial bounds via stated Lipschitz assumptions

full rationale

The paper's core chain begins with Newton's polynomial error bounds (a standard external result), generalizes them to splines under higher-order Lipschitz continuity, and propagates through compositions. None of these steps reduce by construction to fitted parameters from the target data, self-citations, or renamed inputs; the bounds are presented as deterministic consequences of the regularity conditions. No load-bearing self-citation, ansatz smuggling, or uniqueness theorem from the authors appears in the provided text. The distance-aware property follows directly from the bottom-up propagation rather than being presupposed.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on higher-order Lipschitz continuity of the target functions and the validity of error propagation rules through function composition; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Higher-order Lipschitz continuity of the splines and target functions
Invoked to generalize Newton's polynomial bounds to arbitrary splines and to control error propagation through compositions.
domain assumption Error bounds compose under function composition for spline layers
Central to extending per-neuron bounds to the full network.

pith-pipeline@v0.9.0 · 5717 in / 1347 out tokens · 30988 ms · 2026-05-23T06:16:44.555344+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

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KAN: Kolmogorov-Arnold Networks

Z. Liu, Y . Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Solja ˇci´c, T. Y . Hou, and M. Tegmark, “Kan: Kolmogorov-arnold networks,”arXiv preprint arXiv:2404.19756, 2024

work page internal anchor Pith review Pith/arXiv arXiv 2024

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work page arXiv 2024

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work page arXiv 2024

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work page arXiv 2024

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Kolmogorov-arnold networks (kans) for time series analysis

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work page arXiv 2024

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work page arXiv 2024

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B. Lakshminarayanan, A. Pritzel, and C. Blundell, “Simple and scalable predictive uncertainty estimation using deep ensembles,” NeuRIPS, vol. 30, 2017

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Deep gaussian processes,

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Novel architecture of deep feature- based gaussian processes with an ensemble of kernels,

Y . Song, Y . Liu, and P. M. Djuri ´c, “Novel architecture of deep feature- based gaussian processes with an ensemble of kernels,” inICASSP 2024- 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2024, pp. 6750–6754

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Scalable model-based gaussian process clustering,

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work page 2024

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Limits of probabilistic safety guarantees when considering human uncertainty,

R. Cheng, R. M. Murray, and J. W. Burdick, “Limits of probabilistic safety guarantees when considering human uncertainty,” in 2021 IEEE International Conference on Robotics and Automation (ICRA) . IEEE, 2021, pp. 3182–3189

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Jaulin, M

L. Jaulin, M. Kieffer, O. Didrit, E. Walter, L. Jaulin, M. Kieffer, O. Didrit, and ´E. Walter, Interval analysis. Springer, 2001

work page 2001

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Simple and principled uncertainty estimation with deterministic deep learning via distance awareness,

J. Liu, Z. Lin, S. Padhy, D. Tran, T. Bedrax Weiss, and B. Lak- shminarayanan, “Simple and principled uncertainty estimation with deterministic deep learning via distance awareness,” Advances in neural information processing systems , vol. 33, pp. 7498–7512, 2020

work page 2020

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Deep deterministic uncertainty: A new simple baseline,

J. Mukhoti, A. Kirsch, J. van Amersfoort, P. H. Torr, and Y . Gal, “Deep deterministic uncertainty: A new simple baseline,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , 2023, pp. 24 384–24 394

work page 2023

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Uncertainty estima- tion using a single deep deterministic neural network,

J. Van Amersfoort, L. Smith, Y . W. Teh, and Y . Gal, “Uncertainty estima- tion using a single deep deterministic neural network,” in International conference on machine learning . PMLR, 2020, pp. 9690–9700

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G. M. Phillips, Interpolation and approximation by polynomials . Springer Science & Business Media, 2003, vol. 14

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T. M. Apostol, Calculus, Volume 1. John Wiley & Sons, 1967. DAREK - DISTANCE A W ARE ERROR FOR KOLMOGOROV NETWORKS (Supplementary Material) Masoud Ataei ⋆ Mohammad Javad Khojasteh † Vikas Dhiman⋆ ⋆Electrical and Computer Engg. Dept., University of Maine, Orono, ME, USA †Electrical and Microelectronic Engg. Dept., Rochester Institute of Technology, Rochest...

work page 1967