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arxiv: 2507.02466 · v2 · submitted 2025-07-03 · 💻 cs.LG

Variational Kolmogorov-Arnold Network

Francesco Alesiani , Henrik Christiansen , Federico Errica This is my paper

Pith reviewed 2026-05-19 05:46 UTC · model grok-4.3

classification 💻 cs.LG
keywords Kolmogorov-Arnold Networksvariational inferencebasis functionslatent variablesmodel capacityneural networkshyperparameter optimizationtruncated exponential prior
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The pith

InfinityKAN learns the number of basis functions in Kolmogorov-Arnold Networks automatically during training as a latent variable.

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

Kolmogorov-Arnold Networks represent multivariate functions through compositions of univariate basis functions but require users to manually set the number of bases per layer, a choice that controls capacity and strongly affects results. InfinityKAN removes this choice by treating the basis count itself as a latent variable equipped with a truncated exponential prior. A differentiable weighting function converts the discrete count into a continuous quantity that can be optimized by gradient descent together with the network weights. The authors prove that the resulting variational objective is Lipschitz continuous, which supports stable training. On 18 datasets spanning synthetic functions, images, tabular data, and graphs, the method reaches or surpasses the accuracy of fixed-basis KANs while eliminating the need for per-layer hyperparameter search.

Core claim

We present InfinityKAN, a variational inference framework that models the number of basis functions as a latent variable with a truncated exponential prior and introduces a differentiable weighting function that permits gradient-based optimization of this count. We establish the Lipschitz continuity of the variational objective to guarantee stable training dynamics. Experiments across 18 datasets in synthetic, image, tabular, and graph domains show that InfinityKAN matches or exceeds the performance of standard KANs without requiring manual selection of the number of bases for each layer.

What carries the argument

Variational inference treating basis count as a latent variable with truncated exponential prior, together with a differentiable weighting function that enables joint gradient optimization of capacity and weights.

If this is right

  • Model capacity becomes a learned quantity rather than a fixed hyperparameter chosen before training.
  • Gradient-based optimization can jointly adjust both network weights and the effective number of basis functions.
  • Stable training is supported by the established Lipschitz continuity of the objective.
  • Performance comparable to or better than manually tuned KANs is achieved across synthetic, image, tabular, and graph tasks without per-layer tuning.

Where Pith is reading between the lines

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

  • The same latent-variable treatment of capacity could be transferred to other architectures where layer widths or feature counts are currently chosen by hand.
  • The posterior distributions over learned basis counts may reveal how intrinsic complexity differs across data modalities.
  • Alternative priors on the latent count or task-dependent regularization of the weighting function could further improve adaptation speed or final accuracy.

Load-bearing premise

The variational approximation using the truncated exponential prior on the latent basis count faithfully captures the posterior and produces a stable optimum without uncorrectable bias from the prior or weighting function.

What would settle it

A controlled experiment on additional datasets in which InfinityKAN consistently underperforms KANs whose basis counts were chosen by exhaustive search, or in which the learned counts vary sharply across random seeds, would show that the variational mechanism fails to identify suitable capacity.

Figures

Figures reproduced from arXiv: 2507.02466 by Federico Errica, Francesco Alesiani, Henrik Christiansen.

Figure 1
Figure 1. Figure 1: The graphical model of InfinityKAN, with the observable variables (in green) xi , yi and latent variables (in blue) θ ℓn qpk, λℓ (Upper) KAN composed of two layers; (Bottom) the basis functions φ n k (x) (ReLU) used to build ϕ ℓn qp(x). and we can perform inference. For a KAN of L layers, we define θ =  θ ℓ [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Whenever we change the number of basis, to avoid storing the weights for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Training accuracy (top-left) and test accuracy (bottom-left) during training for [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The effects of the hyper-parameters of the weighting function [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The effects of the hyper-parameters of the weighting function [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p031_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p032_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p033_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p034_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Evolution of the number of basis, accuracy, and number of parameters for [PITH_FULL_IMAGE:figures/full_fig_p035_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: (Left) 2d visualization of the Spiral dataset with k = 3, on the left the ground truth, while on the right a prediction; (Right) Visualization in 3d of Spiral dataset with k = 2, left the ground truth data and right a prediction. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: From left to right, the number of basis functions per layer of the [PITH_FULL_IMAGE:figures/full_fig_p036_23.png] view at source ↗
read the original abstract

Kolmogorov-Arnold Networks (KANs) offer a theoretically grounded alternative to multi-layer perceptrons by representing multivariate functions as compositions of univariate basis functions. However, a critical limitation of KANs is the need to manually specify the number of basis functions per layer -- a hyperparameter that directly controls model capacity and substantially impacts performance, yet whose optimal value varies unpredictably across tasks. We present InfinityKAN, a variational inference framework that eliminates this design choice by learning the number of basis functions during training. Our approach models the basis count as a latent variable with a truncated exponential prior, introducing a differentiable weighting function that enables gradient-based optimization. We establish the Lipschitz continuity of the variational objective, ensuring stable training dynamics. Experiments across 18 datasets spanning synthetic, image, tabular, and graph domains demonstrate that InfinityKAN matches or exceeds the performance of KANs while requiring no manual selection of the number of bases for each layer.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces InfinityKAN, a variational inference framework for Kolmogorov-Arnold Networks that models the number of basis functions per layer as a latent variable equipped with a truncated exponential prior. A differentiable weighting function is proposed to enable gradient-based optimization of this discrete count, thereby removing the need for manual hyperparameter selection. The authors establish Lipschitz continuity of the resulting variational objective and report that the method matches or exceeds the performance of standard KANs across 18 datasets spanning synthetic, image, tabular, and graph domains.

Significance. If the variational construction reliably recovers task-appropriate basis counts without substantial bias from the prior or the continuous relaxation, the work would meaningfully reduce the practical burden of capacity tuning in KANs. The multi-domain experimental results provide preliminary support for usability, and the Lipschitz-continuity claim is a positive technical contribution that could aid stable training. However, the overall significance hinges on whether the learned counts demonstrably outperform or match carefully tuned fixed baselines rather than simply reflecting a convenient default capacity.

major comments (2)
  1. [Abstract and variational objective] Abstract and variational objective section: the claim that the truncated exponential prior together with the differentiable weighting function yields task-optimal basis counts (rather than prior-driven or degenerate solutions) is load-bearing for the central contribution. Because the count is discrete, the weighting function is necessarily an approximation; the exponential prior further biases toward smaller values. It is unclear whether the ELBO fully corrects this bias or whether gradient artifacts remain, and no direct diagnostic (e.g., posterior vs. prior comparison or ablation against oracle-tuned fixed counts) is provided to confirm unbiased recovery of optimal capacity.
  2. [Experiments] Experiments section: performance is reported to match or exceed KANs on 18 datasets, yet the manuscript provides neither error bars, details on how the variational objective is optimized in practice, nor an ablation that isolates the effect of the learned counts versus a reasonable fixed default. Without these, it is difficult to determine whether the method truly eliminates manual selection or merely substitutes one form of capacity choice for another.
minor comments (1)
  1. [Method] Notation for the differentiable weighting function and its relation to the truncated exponential prior could be made more explicit to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript introducing InfinityKAN. We address each of the major comments below and indicate the revisions we plan to make to strengthen the paper.

read point-by-point responses

  1. Referee: [Abstract and variational objective] Abstract and variational objective section: the claim that the truncated exponential prior together with the differentiable weighting function yields task-optimal basis counts (rather than prior-driven or degenerate solutions) is load-bearing for the central contribution. Because the count is discrete, the weighting function is necessarily an approximation; the exponential prior further biases toward smaller values. It is unclear whether the ELBO fully corrects this bias or whether gradient artifacts remain, and no direct diagnostic (e.g., posterior vs. prior comparison or ablation against oracle-tuned fixed counts) is provided to confirm unbiased recovery of optimal capacity.

    Authors: We agree that demonstrating the recovery of task-optimal basis counts is central to our contribution. The truncated exponential prior does bias towards smaller counts, but the variational posterior is optimized to maximize the ELBO, which incorporates the data likelihood and can thus shift the distribution away from the prior when beneficial for the task. The differentiable weighting function approximates the discrete selection in a way that allows gradients to flow, and we have proven Lipschitz continuity to ensure stable training. To directly address concerns about bias and approximation quality, we will revise the manuscript to include: (i) visualizations and quantitative comparisons of the learned posterior distributions versus the prior for representative datasets, and (ii) an ablation study comparing InfinityKAN performance to KANs with fixed basis counts tuned via oracle search on a validation set. These additions will provide evidence on whether the method recovers optimal capacities without substantial bias. revision: yes

  2. Referee: [Experiments] Experiments section: performance is reported to match or exceed KANs on 18 datasets, yet the manuscript provides neither error bars, details on how the variational objective is optimized in practice, nor an ablation that isolates the effect of the learned counts versus a reasonable fixed default. Without these, it is difficult to determine whether the method truly eliminates manual selection or merely substitutes one form of capacity choice for another.

    Authors: We acknowledge the importance of these experimental details for validating the claims. In the revised version, we will add error bars computed from multiple independent runs with different random seeds to all performance tables and figures. We will also expand the experimental section with a detailed description of the optimization procedure for the variational objective, including the choice of optimizer, learning rate schedule, number of epochs, and any techniques used to handle the continuous relaxation. Furthermore, we will include an ablation study that compares InfinityKAN to standard KANs using a fixed default number of basis functions (e.g., the value commonly used in prior KAN literature or the median across our experiments). This will help isolate the benefits of learning the counts adaptively. We believe these changes will clarify that InfinityKAN effectively removes the need for manual per-task tuning. revision: yes

Circularity Check

0 steps flagged

Minor self-citation risk but central variational construction remains independent of prior author work

full rationale

The paper introduces a new variational inference framework (InfinityKAN) that treats basis count as a latent variable with a truncated exponential prior and a differentiable weighting function. This construction is presented as original and does not reduce by definition or by self-citation chain to any fitted quantity or ansatz from the authors' prior publications. The Lipschitz continuity claim and experimental results on 18 datasets are derived from the new ELBO and weighting mechanism rather than being forced by construction. A low-level self-citation risk exists for background KAN material but is not load-bearing for the central claim of automatic basis selection.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach depends on the validity of the variational lower bound for the latent basis count and on the truncated exponential prior being a reasonable model for basis cardinality. No explicit free parameters beyond the prior are mentioned in the abstract.

axioms (2)
  • domain assumption The variational objective remains Lipschitz continuous under the introduced weighting function.
    Stated in the abstract as established; required for stable gradient-based optimization of the latent count.
  • ad hoc to paper A truncated exponential prior on the number of basis functions is appropriate for modeling capacity across tasks.
    Chosen to enable learning the count; its suitability is not justified beyond enabling differentiability.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    cs.LG 2025-10 accept novelty 3.0

    A systematic review of Kolmogorov-Arnold Networks that maps their relation to Kolmogorov superposition theory, MLPs, and kernels, examines basis-function design choices, summarizes performance advances, and supplies a...

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