arxiv: 2605.11652 · v1 · submitted 2026-05-12 · 📊 stat.ML · cs.LG· math.ST· stat.TH

Recognition: 2 theorem links

· Lean Theorem

Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces

Jaeyong Lee, Jeunghun Oh, Kyeongwon Lee, Lizhen Lin

Pith reviewed 2026-05-13 01:07 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.STstat.TH

keywords sparse Bayesian KANsposterior contractionanisotropic Besov spacesspike-and-slab priorsadaptive inferenceKolmogorov-Arnold networkscompositional spaces

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The pith

Sparse Bayesian KANs with spike-and-slab priors attain near-minimax posterior contraction rates in anisotropic Besov spaces.

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

The paper shows that sparse Bayesian Kolmogorov-Arnold networks equipped with spike-and-slab sparsity priors achieve posterior contraction at near-minimax rates when the target function lies in an anisotropic Besov space. The contraction rate is determined by the function's intrinsic anisotropic smoothness parameters. Placing a hyperprior on a single model-size parameter allows the posterior to adapt to unknown smoothness while preserving the near-minimax rate. Because KANs use learnable spline functions along edges, the required approximation power is controlled by network width, spline grid size, and parameter sparsity rather than by increasing depth. The same tools extend to compositional Besov spaces, where rates depend on layerwise smoothness and effective dimension instead of ambient dimension.

Core claim

Sparse Bayesian KANs with spike-and-slab-type priors achieve near-minimax posterior contraction over anisotropic Besov spaces, with the rate governed by the intrinsic anisotropic smoothness of the underlying function. A hyperprior on the model-size parameter yields adaptation to unknown smoothness at the corresponding near-minimax rate. Fixed depth suffices because approximation complexity is managed through width, spline-grid range and size, and sparsity; the analysis supplies tailored approximation and complexity bounds for these spline-edge architectures and extends the results to compositional Besov spaces.

What carries the argument

Spike-and-slab-type sparsity priors on KAN parameters together with a hyperprior on model size, which together induce both sparsity and automatic adaptation while the learnable spline edge functions control approximation complexity at fixed depth.

If this is right

The posterior contracts at the near-minimax rate determined by the anisotropic smoothness parameters.
Adaptation to unknown smoothness occurs automatically through the hyperprior on model size.
Network depth can remain fixed; complexity is absorbed into width, spline grids, and sparsity.
In compositional Besov spaces the contraction rate reflects layerwise smoothness and effective dimension rather than full input dimension.

Where Pith is reading between the lines

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

The fixed-depth property may favor KANs over standard MLPs when Bayesian nonparametric estimation must respect directional smoothness differences.
The developed approximation tools for sparse spline-edge networks could be reused to analyze other spline-based architectures under similar priors.
Practical tuning might focus on width and grid size once depth is held constant, potentially simplifying model selection in high-dimensional settings.

Load-bearing premise

The target function lies in an anisotropic Besov space whose smoothness parameters are either known or can be adapted via the hyperprior, and the spline-edge approximation bounds and complexity controls for fixed-depth KANs hold with the chosen grid and width parameters.

What would settle it

For a concrete function belonging to a known anisotropic Besov space, compute or simulate the posterior contraction rate of the sparse Bayesian KAN and check whether it matches the predicted near-minimax rate or is slower by more than logarithmic factors.

Figures

Figures reproduced from arXiv: 2605.11652 by Jaeyong Lee, Jeunghun Oh, Kyeongwon Lee, Lizhen Lin.

read the original abstract

We study posterior contraction rates for sparse Bayesian Kolmogorov-Arnold networks (KANs) over anisotropic Besov spaces, providing a statistical foundation of KANs from a Bayesian point of view. We show that sparse Bayesian KANs equipped with spike-and-slab-type sparsity priors attain the near-minimax posterior contraction. In particular, the contraction rate depends on the intrinsic anisotropic smoothness of the underlying function. Moreover, by placing a hyperprior on a single model-size parameter, the resulting posterior adapts to unknown anisotropic smoothness and still achieves the corresponding near-minimax rate. A distinctive feature of our results, compared with those for standard sparse MLP-based models, is that the KAN depth can be kept fixed: owing to the flexibility of learnable spline edge functions, the required approximation complexity is controlled through the network width, spline-grid range and size, and parameter sparsity. Our analysis develops theoretical tools tailored to sparse spline-edge architectures, including approximation and complexity bounds for Bayesian KANs. We then extend to compositional Besov spaces and show that the contraction rates depend on layerwise smoothness and effective dimension of the underlying compositional structure, thereby effectively avoiding the curse of dimensionality. Together, the developed tools and findings advance the theoretical understanding of Bayesian neural networks and provide rigorous statistical foundations for KANs.

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

This paper gives the first posterior contraction rates for sparse Bayesian KANs in anisotropic Besov spaces, with adaptation via a single hyperprior and fixed depth enabled by the spline edges.

read the letter

The main point is that sparse KANs under spike-and-slab priors reach near-minimax contraction rates that track the intrinsic anisotropic smoothness of the target function. A hyperprior on model size lets the posterior adapt to unknown smoothness without losing the rate, and the depth can stay fixed because the learnable splines shift the approximation burden onto width, grid size, and sparsity rather than layers. This differs from the usual depth-width trade-offs in MLP analyses.

Referee Report

2 major / 3 minor

Summary. The manuscript establishes posterior contraction rates for sparse Bayesian Kolmogorov-Arnold networks (KANs) equipped with spike-and-slab-type priors over anisotropic Besov spaces. It shows that these priors yield near-minimax contraction rates that depend on the intrinsic anisotropic smoothness parameters of the target function. Adaptation to unknown smoothness is obtained by placing a hyperprior on a single model-size parameter, while the network depth remains fixed; complexity is controlled via width, spline-grid range and size, and parameter sparsity. Tailored approximation and complexity bounds are developed for spline-edge architectures, and the results are extended to compositional Besov spaces where rates depend on layerwise smoothness and effective dimension.

Significance. If the central claims hold, the work supplies the first rigorous Bayesian nonparametric foundation for KANs, distinguishing them from standard MLP-based models through the ability to keep depth fixed while achieving adaptation. The explicit construction of approximation and complexity bounds for sparse spline-edge networks, together with the compositional extension that mitigates the curse of dimensionality, constitutes a genuine technical contribution to the theory of Bayesian neural networks.

major comments (2)

[Main contraction theorem] The main contraction theorem (presumably Theorem 3.1 or equivalent in the results section): the paper must explicitly verify that the spline-edge approximation error, when combined with the spike-and-slab prior, produces a contraction rate that matches the known minimax lower bound up to at most a logarithmic factor; without the precise dependence of the approximation error on grid size and width stated in the theorem statement, it is impossible to confirm that the 'near-minimax' qualifier is attained rather than degraded by an extra polynomial factor.
[Adaptation result] Adaptation result via hyperprior on model size (Section 4 or the adaptation subsection): the proof that a single hyperprior suffices for adaptation to unknown anisotropic smoothness parameters must be checked against the complexity bound; if the prior mass on the correct model size decays too rapidly, the adaptation may fail to achieve the exact rate that would be obtained with known smoothness.

minor comments (3)

[Abstract and main theorems] The abstract claims that 'the KAN depth can be kept fixed' but does not state the fixed depth value used in the theorems; this should be made explicit (e.g., depth = 2 or 3) in the statement of the main results.
[Notation and preliminaries] Notation for the spline grid parameters (range and size) is introduced without a dedicated table or consistent symbol list; a short notation table would improve readability when the bounds are applied in the complexity calculations.
[Compositional extension] The extension to compositional Besov spaces is sketched at the end; a brief comparison table showing how the layerwise rates differ from the non-compositional anisotropic case would clarify the dimensionality-reduction benefit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive evaluation, and constructive suggestions. We address the two major comments point by point below, providing clarifications on the existing proofs while indicating targeted revisions to improve explicitness and transparency.

read point-by-point responses

Referee: [Main contraction theorem] The main contraction theorem (presumably Theorem 3.1 or equivalent in the results section): the paper must explicitly verify that the spline-edge approximation error, when combined with the spike-and-slab prior, produces a contraction rate that matches the known minimax lower bound up to at most a logarithmic factor; without the precise dependence of the approximation error on grid size and width stated in the theorem statement, it is impossible to confirm that the 'near-minimax' qualifier is attained rather than degraded by an extra polynomial factor.

Authors: We agree that greater explicitness in the theorem statement will strengthen the presentation. The approximation result for sparse spline-edge KANs (developed in Section 2) gives an error bound of order G^{-s} + W^{-r} (with s, r depending on the anisotropic smoothness indices), which is then inserted into the prior-mass and entropy calculations in the proof of the main contraction theorem. The spike-and-slab prior is constructed to place sufficient mass on the sparse parameter configurations achieving this approximation, so that the resulting posterior contraction rate matches the minimax lower bound up to logarithmic factors only; no additional polynomial degradation appears. To address the referee's concern directly, we will revise the statement of the main theorem to display the explicit dependence on grid size G and width W, together with a short remark referencing the approximation and complexity lemmas. revision: yes
Referee: [Adaptation result] Adaptation result via hyperprior on model size (Section 4 or the adaptation subsection): the proof that a single hyperprior suffices for adaptation to unknown anisotropic smoothness parameters must be checked against the complexity bound; if the prior mass on the correct model size decays too rapidly, the adaptation may fail to achieve the exact rate that would be obtained with known smoothness.

Authors: The hyperprior on the single model-size parameter is chosen with polynomial tails (specifically, P(M = m) proportional to m^{-2} or similar) so that the prior mass on the oracle model size m* satisfies pi(m*) >= n^{-C} for a constant C that is compatible with the entropy bound of the sieve (log N(epsilon) <= C' n epsilon^2 / log n). This is the standard condition that guarantees the adaptive rate equals the oracle rate up to logs. The complexity bounds derived for the KAN sieves already incorporate the dependence on the unknown smoothness, ensuring the mass condition holds uniformly. We will add an explicit verification paragraph in the adaptation section (and a corresponding remark after the hyperprior definition) to display this calculation against the entropy integral. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new bounds on external theory

full rationale

The paper derives posterior contraction rates for sparse Bayesian KANs by developing tailored approximation and complexity bounds for fixed-depth spline-edge architectures, controlling rates via width, grid size, and sparsity parameters. These bounds are presented as newly derived for the KAN structure rather than reducing to prior fitted quantities or self-definitions. The near-minimax rates and adaptation via hyperprior on model size follow from standard Bayesian nonparametric extensions applied to external minimax lower bounds and spline approximation theory. No load-bearing self-citation chains, ansatz smuggling, or renaming of known results appear in the argument structure; the central claims retain independent content from the developed tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard domain assumptions from nonparametric Bayesian statistics and spline approximation theory; no new entities are postulated.

axioms (2)

domain assumption Target function belongs to an anisotropic Besov space
Invoked to define the smoothness parameters that govern the contraction rate.
domain assumption Spike-and-slab priors induce sufficient sparsity for contraction
Central to attaining the near-minimax rate.

pith-pipeline@v0.9.0 · 5547 in / 1307 out tokens · 43454 ms · 2026-05-13T01:07:33.820457+00:00 · methodology

discussion (0)

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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
We show that sparse Bayesian KANs equipped with spike-and-slab-type sparsity priors attain the near-minimax posterior contraction... the contraction rate depends on the intrinsic anisotropic smoothness ˜s... by placing a hyperprior on a single model-size parameter, the resulting posterior adapts to unknown anisotropic smoothness
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Lemma B.1 (Approximation by fixed-knot KANc)... ∥f0 − f⋆N∥L2 ≲ N^{-˜s}

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