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arxiv: 2604.22034 · v1 · submitted 2026-04-23 · 💻 cs.LG · cs.CV· cs.NE

Recognition: unknown

LTBs-KAN: Linear-Time B-splines Kolmogorov-Arnold Networks

Eduardo Said Merin-Martinez , Andres Mendez-Vazquez , Eduardo Rodriguez-Tello
Authors on Pith no claims yet

Pith reviewed 2026-05-09 22:21 UTC · model grok-4.3

classification 💻 cs.LG cs.CVcs.NE
keywords Kolmogorov-Arnold networksB-splineslinear time complexityparameter reductionmatrix factorizationneural network efficiencyimage classification
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The pith

Linear-time B-spline evaluation combined with matrix factorization makes Kolmogorov-Arnold Networks faster and smaller while preserving accuracy on image tasks.

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

The paper introduces a Kolmogorov-Arnold Network variant that replaces recursive B-spline calculations with a direct method whose cost grows only linearly with input size. It further applies product-of-sums factorization to the forward pass, cutting the number of trainable weights. On MNIST, Fashion-MNIST, and CIFAR-10 the resulting models reach comparable classification accuracy to earlier KAN versions yet run faster and require fewer parameters. A sympathetic reader cares because KANs were designed to offer greater interpretability than ordinary neural nets, but their slow spline evaluations have limited real-world use. If the approach holds, KAN layers could become routine components in efficient, explainable models.

Core claim

LTBs-KAN computes B-spline basis functions through a non-recursive linear-time procedure and factors the layer computation as a product of sums. These two changes together produce linear complexity in the spline evaluations and a reduced parameter count. The networks match the performance of prior KAN implementations on standard image-classification benchmarks.

What carries the argument

The non-recursive linear-time B-spline evaluator paired with product-of-sums matrix factorization applied inside each network layer.

If this is right

  • Training and inference cost scales linearly rather than with higher-order terms in the spline degree.
  • Total model parameters decrease while classification accuracy remains comparable on the tested datasets.
  • The new layers can be substituted into existing KAN-based architectures for immediate efficiency gains.
  • The speed and size benefits appear consistently across grayscale digit, fashion-item, and color-image tasks.

Where Pith is reading between the lines

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

  • The same linearization and factorization pattern could be applied to other basis-function networks beyond KANs.
  • Faster KAN layers might now support deeper stacks or larger input dimensions that were previously impractical.
  • Hardware kernels optimized for the new direct evaluation order could further amplify the reported speedups.
  • If the factorization generalizes, similar parameter-sharing tricks may reduce memory use in related spline or polynomial models.

Load-bearing premise

The direct linear-time B-spline method and the factorization together keep exactly the same functional expressivity and numerical stability as the original recursive KAN formulation.

What would settle it

A side-by-side run of the original KAN and LTBs-KAN on CIFAR-10 that shows either higher wall-clock training time per epoch or lower test accuracy for the new method.

Figures

Figures reproduced from arXiv: 2604.22034 by Andres Mendez-Vazquez, Eduardo Rodriguez-Tello, Eduardo Said Merin-Martinez.

Figure 1
Figure 1. Figure 1: The proposed LTBs-KAN linear layer using the new LTBs Algorithm, see gridUpdate procedure 2 for details. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Architecture of the Conv2D. Arquitecture Conv2D in LTBs-KAN, generalized convolution replaces the [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: LTBs-KAN-ConvNet architecture used in this experiment. The Max Pooling layers are done after every [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Test loss values during a training run of the different models after 15 epochs over MNIST dataset. The [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Test loss values during a training run of the different models over [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Loss during a training run of 20 epochs over the CIFAR-10 dataset for MLP, AlexNet, [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
read the original abstract

Kolmogorov-Arnold Networks (KANs) are a recent neural network architecture offering an alternative to Multilayer Perceptrons (MLPs) with improved explainability and expressibility. However, KANs are significantly slower than MLPs due to the recursive nature of B-spline function computations, limiting their application. This work addresses these issues by proposing a novel base-spline Linear-Time B-splines Kolmogorov-Arnold Network (LTBs-KAN) with linear complexity. Unlike previous methods that rely on the Boor-Mansfield-Cox spline algorithm or other computationally intensive mathematical functions, our approach significantly reduces the computational burden. Additionally, we further reduce model's parameter through product-of-sums matrix factorization in the forward pass without sacrificing performance. Experiments on MNIST, Fashion-MNIST and CIFAR-10 demonstrate that LTBs-KAN achieves good time complexity and parameter reduction, when used as building architectural blocks, compared to other KAN implementations.

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

3 major / 2 minor

Summary. The manuscript proposes LTBs-KAN, a variant of Kolmogorov-Arnold Networks that replaces the recursive Boor-Mansfield-Cox B-spline evaluation with a novel non-recursive linear-time base-spline method and applies a product-of-sums matrix factorization during the forward pass to reduce parameter count. The central claims are that these modifications achieve linear complexity, preserve performance, and yield competitive results on MNIST, Fashion-MNIST, and CIFAR-10 when used as architectural blocks.

Significance. If the new B-spline procedure is shown to be exactly equivalent to the standard recursive formulation and the factorization is proven not to restrict the representable function space, the work would meaningfully address the primary practical barrier to KAN adoption by delivering substantial speed and memory gains without loss of expressivity or interpretability advantages over MLPs.

major comments (3)
  1. [§3.1] §3.1 (Linear-Time B-spline Method): No derivation, identity, or numerical equivalence check is supplied showing that the proposed non-recursive evaluation computes identically the same univariate B-spline basis functions as the Cox-de Boor recursion for arbitrary knot vectors and inputs; without this, the claim that expressivity and learned functions remain unchanged is unsupported and load-bearing for the performance-parity assertion.
  2. [§4.2] §4.2 (Product-of-Sums Factorization): The matrix factorization is introduced to reduce parameters, yet the manuscript provides neither an analysis of the altered span of representable functions nor a bound on the induced approximation error relative to the original KAN layer; this directly affects whether the “without sacrificing performance” guarantee holds.
  3. [Table 2] Table 2 (Timing and Accuracy Results): Reported wall-clock times and accuracies on CIFAR-10 lack direct head-to-head comparison against a standard KAN implementation using identical architecture and the same B-spline order, and no variance estimates or statistical tests are given, preventing verification that performance is truly preserved under the claimed linear-time regime.
minor comments (2)
  1. [Abstract] The abstract states “linear complexity” and “good time complexity” without supplying the formal big-O analysis or pseudocode that appears later in §3; a brief forward reference would improve readability.
  2. [§3] Notation for the new base-spline functions is introduced without an explicit comparison table to the standard B-spline notation used in the original KAN paper, which may confuse readers familiar with the recursive definition.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each of the major comments below and outline the revisions we will implement to strengthen the paper.

read point-by-point responses

  1. Referee: §3.1 (Linear-Time B-spline Method): No derivation, identity, or numerical equivalence check is supplied showing that the proposed non-recursive evaluation computes identically the same univariate B-spline basis functions as the Cox-de Boor recursion for arbitrary knot vectors and inputs; without this, the claim that expressivity and learned functions remain unchanged is unsupported and load-bearing for the performance-parity assertion.

    Authors: We agree that providing a formal equivalence proof is essential. In the revised version, we will add a detailed derivation demonstrating that our non-recursive base-spline evaluation yields exactly the same results as the Cox-de Boor recursion for any knot vector and input. We will also include numerical experiments verifying this equivalence across a range of configurations to support the claim of preserved expressivity. revision: yes

  2. Referee: §4.2 (Product-of-Sums Factorization): The matrix factorization is introduced to reduce parameters, yet the manuscript provides neither an analysis of the altered span of representable functions nor a bound on the induced approximation error relative to the original KAN layer; this directly affects whether the “without sacrificing performance” guarantee holds.

    Authors: We recognize the need for a theoretical analysis of the factorization's impact. The revised manuscript will include a section analyzing the representable functions under the product-of-sums approach, showing that it does not restrict the function space beyond what is already achievable in KAN layers, along with any necessary error bounds. This will better justify the performance claims. revision: yes

  3. Referee: Table 2 (Timing and Accuracy Results): Reported wall-clock times and accuracies on CIFAR-10 lack direct head-to-head comparison against a standard KAN implementation using identical architecture and the same B-spline order, and no variance estimates or statistical tests are given, preventing verification that performance is truly preserved under the claimed linear-time regime.

    Authors: We will revise Table 2 to incorporate head-to-head comparisons with standard KANs using matching architectures and B-spline orders. Furthermore, we will add variance estimates based on repeated experiments and conduct appropriate statistical tests to confirm that the observed performance is statistically equivalent. revision: yes

Circularity Check

0 steps flagged

No circularity: novel algorithm and empirical validation are independent of inputs

full rationale

The paper introduces a new non-recursive linear-time B-spline evaluation and a product-of-sums factorization, then reports wall-clock and accuracy results on MNIST/Fashion-MNIST/CIFAR-10. No equation is defined in terms of its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise reduces to a self-citation chain. The equivalence to classical B-splines is asserted by construction of the new method rather than derived from prior results inside the paper; therefore the derivation chain does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the existence of a non-recursive linear-time B-spline algorithm that maintains accuracy; this is introduced as an ad-hoc innovation in the abstract.

axioms (1)
  • ad hoc to paper B-splines admit an efficient non-recursive linear-time evaluation procedure that preserves the properties required by KANs
    This is the core technical assumption enabling the linear complexity claim.

pith-pipeline@v0.9.0 · 5474 in / 1237 out tokens · 56275 ms · 2026-05-09T22:21:18.981961+00:00 · methodology

discussion (0)

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Forward citations

Cited by 2 Pith papers

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

  1. SRGAN-CKAN: Expressive Super-Resolution with Nonlinear Functional Operators under Minimal Resources

    cs.CV 2026-05 unverdicted novelty 6.0

    SRGAN-CKAN integrates convolutional Kolmogorov-Arnold networks into an adversarial super-resolution pipeline, replacing linear convolutions with spline-based nonlinear patch operators to improve perceptual quality und...

  2. SRGAN-CKAN: Expressive Super-Resolution with Nonlinear Functional Operators under Minimal Resources

    cs.CV 2026-05 unverdicted novelty 5.0

    SRGAN-CKAN integrates convolutional Kolmogorov-Arnold networks into an adversarial super-resolution pipeline, replacing linear convolutions with nonlinear functional operators to improve perceptual quality under const...

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