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arxiv: 2512.12448 · v2 · submitted 2025-12-13 · 💻 cs.LG · cs.NE· physics.data-an· stat.ML

Optimized Architectures for Kolmogorov-Arnold Networks

James Bagrow , Josh Bongard This is my paper

Pith reviewed 2026-05-16 22:23 UTC · model grok-4.3

classification 💻 cs.LG cs.NEphysics.data-anstat.ML
keywords Kolmogorov-Arnold networkssparsificationdepth selectionminimum description lengthmodel optimizationinterpretable machine learningfunction approximation
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The pith

Combining sparsification with depth selection in overprovisioned KANs yields smaller models with competitive accuracy.

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

Kolmogorov-Arnold networks offer interpretable alternatives to standard neural nets but enhancements often reduce that benefit. This work shows that starting with overprovisioned architectures and applying sparsification along with depth selection under a minimum description length objective allows end-to-end differentiable optimization of the model structure. Experiments indicate that while sparsification by itself is not enough, adding depth selection produces models that are substantially smaller yet match or beat accuracy on function approximation, dynamical systems, and real-world tasks. This matters for scientific machine learning where both accuracy and the ability to inspect the model are needed.

Core claim

Overprovisioned KAN architectures combined with sparsification, deep supervision, and depth selection, optimized differentiably under a minimum description length objective, allow learning compact interpretable networks that achieve competitive or superior accuracy across benchmarks without the complexity of other enhancements.

What carries the argument

Differentiable joint optimization of activations, structure, and depth under a minimum description length objective applied to overprovisioned KANs

If this is right

  • Substantially smaller models are discovered while accuracy remains competitive or better.
  • Interpretability is preserved through the principled optimization process.
  • The approach outperforms sparsification alone on multiple task types.
  • End-to-end optimization of model depth becomes practical for KANs.

Where Pith is reading between the lines

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

  • Similar optimization strategies could be tested on other network types to balance size and performance.
  • This may enable wider adoption of KANs in domains requiring model inspection such as physics-informed modeling.
  • The method suggests a general template for making interpretable models more practical by overprovisioning then pruning.

Load-bearing premise

Differentiable mechanisms under the minimum description length objective can jointly optimize activations, structure, and depth end-to-end while preserving interpretability.

What would settle it

Demonstrating on the paper's benchmarks that the full method does not produce smaller models with accuracy at least as good as sparsification alone would falsify the central result.

Figures

Figures reproduced from arXiv: 2512.12448 by James Bagrow, Josh Bongard.

Figure 1
Figure 1. Figure 1: Learning the example function 𝑧 = sin 𝑥 + 𝑦 2  with Kolmogorov–Arnold Networks (KANs). Forward connections are highlighted in blue. 2 Background 2.1 Kolmogorov–Arnold Networks Kolmogorov–Arnold Networks (KANs), motivated by the Kolmogorov–Arnold Representation Theorem [17, 18, 19], consist of 𝐿 layers with shapes [𝑛0, 𝑛1, . . . , 𝑛𝐿]. The layer update is given by: 𝑥 (ℓ+1) 𝑗 = ∑︁𝑛ℓ 𝑖=1 𝜙ℓ𝑖 𝑗  𝑥 (ℓ ) 𝑖  (… view at source ↗
read the original abstract

Efforts to improve Kolmogorov--Arnold networks (KANs) with architectural enhancements have been stymied by the complexity those enhancements bring, undermining the interpretability that makes KANs attractive in the first place. Here we study overprovisioned architectures combined with sparsification, deep supervision, and depth selection, to learn compact, interpretable KANs without sacrificing accuracy. Crucially, we focus on differentiable mechanisms under a principled minimum description length objective, jointly optimizing activations, structure, and depth end-to-end. Experiments across function approximation benchmarks, dynamical systems forecasting, and real-world prediction tasks demonstrate that sparsification alone is insufficient, but the combination with depth selection achieves competitive or superior accuracy while discovering substantially smaller models. The result is a principled path toward models that are both more expressive and more interpretable, addressing a key tension in scientific machine learning.

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 / 2 minor

Summary. The manuscript presents an approach to optimize Kolmogorov-Arnold Networks (KANs) using overprovisioned architectures sparsified via differentiable mechanisms under a minimum description length (MDL) objective, combined with deep supervision and depth selection. This is claimed to yield compact, interpretable models with competitive accuracy on function approximation benchmarks, dynamical systems forecasting, and real-world prediction tasks, where sparsification alone is insufficient but the full combination succeeds.

Significance. If the central claims hold, this work offers a significant contribution to scientific machine learning by providing a principled, end-to-end differentiable method to balance expressiveness and interpretability in KANs. The emphasis on MDL for joint optimization of structure and depth is a strength, potentially leading to more reliable models in applications requiring interpretability.

major comments (2)
  1. Abstract: The abstract reports competitive results on multiple benchmarks but provides no visible error bars, ablation details, or data exclusion criteria; this makes the central claim of 'substantially smaller models' with competitive accuracy difficult to verify from the given information.
  2. Abstract/Experiments narrative: The MDL objective is presented as principled for jointly optimizing activations, structure, and depth, yet the description leaves open whether structure and depth selection use the same data as the accuracy evaluation; without explicit train/validation separation this risks circularity in the 'substantially smaller models' claim.
minor comments (2)
  1. Abstract: The role of 'deep supervision' in the overall pipeline is mentioned but not elaborated; a brief description of its differentiable implementation would improve clarity.
  2. Abstract: The assumption that the differentiable MDL mechanisms preserve the original interpretability motivation of KANs is stated but would benefit from a short supporting discussion or example in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and will revise the manuscript to strengthen the abstract and clarify experimental protocols. All changes will be incorporated in the next version.

read point-by-point responses

  1. Referee: Abstract: The abstract reports competitive results on multiple benchmarks but provides no visible error bars, ablation details, or data exclusion criteria; this makes the central claim of 'substantially smaller models' with competitive accuracy difficult to verify from the given information.

    Authors: We agree that the abstract would benefit from greater specificity. In the revised version we will add a brief statement on error bars (computed over multiple random seeds), summarize the key ablation outcomes that isolate the contribution of depth selection, and note the data exclusion criteria used for the real-world tasks. These additions will be kept concise while making the 'substantially smaller models' claim directly verifiable from the abstract. revision: yes

  2. Referee: Abstract/Experiments narrative: The MDL objective is presented as principled for jointly optimizing activations, structure, and depth, yet the description leaves open whether structure and depth selection use the same data as the accuracy evaluation; without explicit train/validation separation this risks circularity in the 'substantially smaller models' claim.

    Authors: We appreciate the referee highlighting this potential ambiguity. In the full experimental protocol, structure and depth selection are performed on a held-out validation split that is disjoint from both the training data and the final test sets used for accuracy reporting. We will explicitly state this separation in the revised abstract and in the experimental setup section to remove any possibility of circularity. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents an approach using differentiable mechanisms under a minimum description length objective to jointly optimize activations, structure, and depth in overprovisioned KANs, with experimental results across benchmarks showing that combining sparsification and depth selection yields smaller models with competitive accuracy. No load-bearing derivation step in the abstract or described claims reduces by construction to a self-definition, fitted input renamed as prediction, or self-citation chain. The experimental narrative relies on ablation-style comparisons that remain independent of the optimization inputs, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are detailed in the provided text.

axioms (1)
  • domain assumption Sparsification and depth selection under MDL preserve KAN interpretability advantages
    Central to the claim that the resulting models remain more interpretable than alternatives.

pith-pipeline@v0.9.0 · 5442 in / 1169 out tokens · 26219 ms · 2026-05-16T22:23:39.192556+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    LMDL = Lmodel + Lmodel|data ... Lmodel = (log n / n) ||θ||0 ... with ||θ||0 approximated via E[z] gate expectations under differentiable L0 relaxation

  • IndisputableMonolith/Foundation/BranchSelection.lean branch_selection echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Overprovisioning and sparsification are synergistic... combination with depth selection achieves competitive or superior accuracy while discovering substantially smaller models

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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  1. KANs need curvature: penalties for compositional smoothness

    cs.LG 2026-05 unverdicted novelty 7.0

    A curvature penalty for KANs, derived to respect compositional effects and equipped with a proven upper bound on full-model curvature, produces smoother activations while preserving accuracy.

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