MatrixKAN: Parallelized Kolmogorov-Arnold Network

Cale Coffman; Lizhong Chen

REVIEW 3 major objections 5 minor 15 references

MatrixKAN: Parallelized Kolmogorov-Arnold Network

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read MatrixKAN claims that replacing the recursive Cox–De Boor evaluation of B-splines with a precomputed basis matrix and parallel matrix multiplications lowers effective forward-pass time from O(Lk) to O(L), with speedups up to 38.89x.

desk verdict The matrix-based B-spline trick is a valid adaptation, but the 40x speedup claim is not credible given the paper's own B-spline profiling, and the uniform-grid handling is a real gap. read the letter →

arxiv 2502.07176 v2 pith:LWS3VG74 submitted 2025-02-11 cs.LG

Cale Coffman , Lizhong Chen This is my paper · ORCID

classification cs.LG

keywords Kolmogorov-ArnoldNetworksB-splinesCox-DeBoorrecursionmatrixrepresentationofparallelizationhigh-degreeeffectivecomputationtime

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Kolmogorov-Arnold Networks put a learnable B-spline on every edge, but evaluating those splines through the Cox–De Boor recursion is sequential in the spline degree, so high-degree splines are slow. MatrixKAN rewrites the same spline evaluation as a product of a precomputed basis matrix, a power-basis vector, and the control points, turning all recursive work into parallel matrix multiplications. The paper derives that this lowers effective computation time per forward pass from O(Lk) to O(L) (L layers, spline degree k) and reports speedups up to 38.89x on a 100,000-sample dataset while reproducing KAN's loss curves exactly. If this holds, high-degree B-spline KANs, which the paper shows often fit Feynman physics functions better than low-degree ones, become practical to train.

What carries the argument

The load-bearing object is the precomputed basis matrix Ψ_k, a k × k matrix that encodes all k levels of Cox–De Boor recursion for uniform B-splines of degree k. It is derived from a Toeplitz matrix representation of B-splines and is the same for every B-spline segment of that degree, so the recursion never needs to be executed during training. With Ψ_k in hand, each spline evaluation is two matrix multiplications per control-point vector, and all spline evaluations in a layer are independent of one another.

What would settle it

Freeze a KAN and a MatrixKAN after a grid update has produced a non-uniform knot vector and compare each spline's basis-function outputs: any mismatch shows the single precomputed basis matrix no longer reproduces Cox–De Boor evaluation. Or measure seconds per step for spline degrees 2 and 30 on a large GPU batch; if effective time grows with degree, the O(1) per-spline claim fails.

Watch

Extended reading notes

Core claim

The central claim is that the Cox–De Boor recursion used to evaluate B-spline basis functions in a KAN can be replaced by one precomputed matrix Ψ_k, identical for all uniform B-splines of a given degree, so each spline output is computed as [1 u ... u^k] Ψ_k c. This removes the k sequential recursion levels from the forward pass; the only remaining sequential chain is the L layers of the network. On that basis the paper claims effective computation time drops from O(Lk) to O(L), and its experiments show MatrixKAN matching KAN's loss curves exactly while training faster, with speedups that grow as dataset size and spline degree increase.

Load-bearing premise

The method assumes one precomputed basis matrix serves every B-spline segment of a given degree, which is exact only when knots are uniformly spaced, but the paper's experiments enable grid update that can make knots non-uniform without stating or testing this restriction.

Editorial extensions

If this is right

High-degree B-spline KANs, which the paper finds often give lower loss than degree-3 splines on Feynman physics functions, become computationally affordable.
Training and inference time scales with network depth rather than spline degree, so the cost of increasing spline degree essentially disappears.
The measured speedup grows with dataset size, reaching 38.89x at 100,000 samples, implying larger real-world datasets should see even larger gains.
Because MatrixKAN and KAN compute identical spline values under identical initialization, any KAN experiment transfers unchanged to MatrixKAN.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper never states that its single precomputed basis matrix is exact only for uniformly spaced knots, yet its experiments enable grid update, which can make knot vectors non-uniform; testing non-uniform grids directly would define the method's true scope.
The O(1) per-spline effective time assumes the basis matrix is already paid for; the O(k^4) initialization is amortized over training steps, so very short training runs may not see the advertised benefit.
The same 'precompute the recursion as a matrix' recipe could apply to other spline-like recursive kernels in neural networks, not only KANs.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

MatrixKAN adapts the known matrix representation of B-splines (Cohen & Riesenfeld 1982; Qin 1998) to KAN edge activations. The math is correct for uniform B-splines, the complexity analysis is clearly presented, and the loss-curve equivalence with KAN is a useful sanity check. Code is public. That part is solid.

The headline claim does not hold up. The paper reports end-to-end speedups up to 38.89x, but Section 2.2 says B-spline calculations are only 18-50% of total KAN runtime. Under Amdahl's law, even a perfect B-spline speedup would cap at roughly 2x for degree 20. The 38.89x figure therefore cannot be explained by the matrix trick alone; it implies the baseline KAN implementation was artificially slow or the comparison is not apples-to-apples. The paper gives no profiling breakdown for the Table 1 configuration, so we cannot check. This is load-bearing because the abstract and conclusion rest on that number.

There is also a correctness gap. Section 3.1.2 states the precomputed basis matrix is valid only for uniform B-splines. Section 5.1, however, says grid update was enabled, which produces non-uniform knot vectors. The paper never explains how MatrixKAN handles that. Either the forward pass is wrong on non-uniform grids or the experiments actually disabled grid update, which would contradict the stated setup. This needs to be fixed or made explicit.

The theoretical O(L) vs O(Lk) parallel-complexity argument is fine as an asymptotic statement, but it is not a wall-clock speedup. Measured speedups are plausible only for the B-spline portion under uniform grids, not for end-to-end training.

Who this is for: people working on KAN efficiency who want a concrete optimization to try. The core idea is worth knowing, but the empirical claims need to be redone. I would send this to peer review with a request for major revision: benchmark against a properly optimized KAN, report error bars, and either drop grid update or handle non-uniform knots explicitly. The uniform-grid restriction alone is a legitimate limitation if stated clearly.

In short: the idea is real, the evidence is not. Do not quote the 40x speedup without checking the baseline.

Referee Report

3 major / 5 minor

Summary. The paper proposes MatrixKAN, an implementation of Kolmogorov-Arnold Networks that replaces the recursive Cox–de Boor B-spline evaluation with a precomputed matrix representation, aiming to parallelize B-spline calculations and reduce the effective computation time from O(Lk) to O(L). The authors provide a complexity analysis, report speedups over a KAN baseline of up to 38.89x on large datasets, and demonstrate matching learning curves between KAN and MatrixKAN on Feynman equation regression tasks. The central claim is that high-degree B-splines become computationally practical in KANs while preserving model performance.

Significance. If the speedup and the uniform-grid equivalence hold as stated, the paper would remove a key practical bottleneck for KANs and make high-degree B-splines a viable modeling choice. The authors provide a public implementation and a falsifiable complexity claim. However, the paper's internal consistency is undermined by two load-bearing concerns: the basis matrix precomputation is valid only for uniform B-splines while experiments enable adaptive grid updates, and the reported end-to-end speedups contradict the paper's own profiling of the B-spline fraction of KAN runtime. These issues must be resolved before the significance of the contribution can be assessed.

major comments (3)

[Section 3.1.2, Eq. (10); Section 5.1] The basis matrix Ψ_k is derived from a recursive formula that, as the paper states, applies only to uniform B-splines. The method therefore assumes a single knot spacing for every B-spline segment. However, all efficiency experiments in Section 5.1 enable grid update, which adapts the knot vector to the data and generally produces non-uniform knots. The paper never states that MatrixKAN is restricted to uniform grids, nor does it test whether grid update in the KAN baseline preserves uniformity. If the knots become non-uniform, the matrix representation in Eq. (9) is segment-dependent, and the single precomputed Ψ_k will not reproduce the Cox–de Boor recursion. This would invalidate both the claimed O(L) effective time and the Section 5.2 claim that MatrixKAN and KAN produce identical outputs. The authors must either restrict the method to uniform grids, prove that the grid update in their experiments keeps knots uniform, or recompute per-segment basis matrices, which would change the complexity analysis.
[Section 2.2 vs. Section 5.1.3, Table 1] The paper reports that B-spline calculations constitute 18% to over 50% of KAN execution time. If only the B-spline portion is replaced by a faster equivalent, Amdahl's law bounds the end-to-end speedup by 1/(1-p), which for p ≤ 0.5 is at most 2x. Yet Table 1 reports speedups of 2.18x, 17.59x, and 38.89x as the dataset size grows to 100,000. The explanation in Section 5.1.3 that 'the parallel fraction of MatrixKAN is greater than that of KAN' is qualitative and does not account for this discrepancy. A 38.89x end-to-end speedup requires the B-spline fraction to be at least 97.4% in the baseline under Amdahl's law, contradicting the paper's own profiling. The authors must provide a profiling breakdown for the exact configuration of Table 1, including timings of B-spline and non-B-spline components in both implementations and how grid updates are handled, to rule out an unfairly slow baseline or an implementation-level speedup unrelated to B-spline parallelization. Without that, the headline claim of 'approximately 40x' is not supported by the paper's internal evidence.
[Section 3.3.2] The effective-computation-time analysis assumes that all O(k^2 + G) per-B-spline work can be done in O(1) parallel time. This requires a precise parallel computation model. In particular, computing the power basis [1, u, ..., u^{k-1}] for a single u involves a sequential chain if done naively; with unbounded parallelism the chain length is O(log k) using repeated squaring, not O(1). Additionally, Step 3's 'zeroing' of inapplicable basis functions is a data-dependent masking operation whose cost is not accounted for in the complexity model. The O(L) claim should be stated under an explicit PRAM or equivalent model, with the power-basis step and masking cost included. This does not necessarily invalidate the practical speedup, but it makes the theoretical contribution less precise than claimed.

minor comments (5)

[Section 3.1.2 and Section 3.2, Step 1] There is a persistent notational confusion between B-spline order and degree. In Section 3.1.1, Eq. (9) is written for order k, and footnote 2 notes that order k equals degree k-1. In Section 3.2, Step 1 says 'Calculate the basis matrix Ψ_k' but then Step 2 and the surrounding text switch to degree k, and Eq. (10) uses Ψ_k for order k. The reader must reconcile these inconsistencies; the paper should use a single convention throughout.
[Section 4.2] The line 'one NVIDIA 1080ti-8MB GPU' should read '8GB' or '11GB'; 8MB is an obvious typo. Also, the dataset-generation script name 'hellokan.ipynp' appears to be a typo for 'hellokan.ipynb'.
[Section 5.1] The efficiency experiments report a single run (no seeds or error bars) and train for only 20 steps. This is sufficient as a microbenchmark, but the paper should state whether the reported seconds-per-step values are medians or means over any repetitions, and whether the 20-step window includes grid-update overhead, because that overhead may dominate at small dataset sizes and explain the low speedup at 10,000 samples.
[Section 3.3.2] The amortization argument for the basis-matrix precomputation says the cost is O(k^4/n) and approaches O(1), but the O(k^4) is a one-time cost that should be compared to the total training cost; a short sentence placing this one-time cost in perspective for typical training budgets would be helpful.
[Section 5.2] The paper states that MatrixKAN and KAN 'performed identically modeling all Feynman equations' but only shows three figures; the text should either include all nine plots or a table of final RMSE values for all equations to substantiate the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core matrix representation is imported from external prior work (Cohen & Riesenfeld 1982; Qin 1998), and the empirical speedup is measured against an external KAN baseline.

full rationale

The paper's central derivation is self-contained in the relevant sense: the generalized matrix representation of B-splines is taken from external sources, not from the authors' own prior work. Equation (9), spline_i(u) = [1 u ... u^{k-1}] Psi_k(i) c, and the recurrence in Equation (10) for Psi_k are attributed to Cohen & Riesenfeld (1982) and Qin (1998). The claim that the basis matrix can be precalculated and that all degree-k B-spline outputs reduce to matrix multiplications is a direct application of that external theorem, not a re-labeling of the paper's own assumptions. The complexity analysis in Section 3.3 compares O(N^2L(k^2+kG)) operations and O(Lk) effective time for KAN versus O(N^2L(k^2+G)) and O(L) for MatrixKAN, based on the algorithmic structure; it does not presuppose the measured speedup. The empirical efficiency comparison uses the external MultKAN implementation (Liu et al., 2024b) as the baseline, and the performance tests compare training curves against that same external implementation with identical seeds and datasets. No parameter is fitted to a subset of the target data and then reported as a prediction, and no load-bearing premise is justified exclusively by a self-citation. Possible concerns, such as the uniform-knot restriction noted in Section 3.1.2 or the tension between the reported 38.89x speedup and the B-spline runtime fractions in Section 2.2, are correctness or experimental-validity issues, not circularity. The derivation does not reduce to its inputs by construction, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted; no new physical entities are postulated. The central method rests on standard matrix representation of B-splines plus an unstated uniformity assumption and a theoretical parallel model.

assumptions (3)

standard math The generalized matrix representation of B-splines decomposes basis functions into power bases and a basis matrix (Equations 8-9).
Taken from Qin (1998) and Cohen & Riesenfeld (1982); the paper does not prove it but relies on it for Steps 1-3 of MatrixKAN.
domain assumption The basis matrix Psi_k is identical for all B-splines of a given degree, which holds only for uniform knot vectors.
Stated in Section 3.1.2 for uniform B-splines, but the experiments enable grid update (Section 5.1) that can produce non-uniform knots; the paper does not address this.
domain assumption Effective computation time is measured by the length of the longest sequential dependency, assuming unlimited parallel processors.
Used in Section 3.3 to claim O(L) parallel time for MatrixKAN; this is the PRAM-style model and may not reflect GPU scaling with fixed resources.

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Cite this review

Pith. "Pith review of MatrixKAN: Parallelized Kolmogorov-Arnold Network." pith.science (2026). https://pith.science/paper/LWS3VG74

@misc{pith2026250207176,
  author       = {Pith},
  title        = {Pith review of: MatrixKAN: Parallelized Kolmogorov-Arnold Network},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LWS3VG74}},
  note         = {Machine review of arXiv:2502.07176}
}

read the original abstract

Kolmogorov-Arnold Networks (KAN) are a new class of neural network architecture representing a promising alternative to the Multilayer Perceptron (MLP), demonstrating improved expressiveness and interpretability. However, KANs suffer from slow training and inference speeds relative to MLPs due in part to the recursive nature of the underlying B-spline calculations. This issue is particularly apparent with respect to KANs utilizing high-degree B-splines, as the number of required non-parallelizable recursions is proportional to B-spline degree. We solve this issue by proposing MatrixKAN, a novel optimization that parallelizes B-spline calculations with matrix representation and operations, thus significantly improving effective computation time for models utilizing high-degree B-splines. In this paper, we demonstrate the superior scaling of MatrixKAN's computation time relative to B-spline degree. Further, our experiments demonstrate speedups of approximately 40x relative to KAN, with significant additional speedup potential for larger datasets or higher spline degrees.

Figures

Figures reproduced from arXiv: 2502.07176 by the authors.

**Figure 1.** Diagram of a cubic B-spline curve, including applicable knots (turquoise) and control points (red). To facilitate diversity in the learned activation function on each edge of a KAN, KANs implement activation functions using B-splines. B-splines are piecewise polynomial curves whose name derives from the elastic beams (i.e., splines) used by draftsmen to construct sweeping curves in 2 [PITH_FULL_IMAGE:figures/full_… view at source ↗

**Figure 2.** Diagram of cubic B-spline basis function curves (various colors) and knots (turquoise). Given that basis functions of degree k are defined by reference to basis functions of degree k − 1, it follows that calculation of a degree k basis function requires k levels of sequential, recursive calls. This sequential dependency in the Cox-De Boor recursion algorithm prevents parallelization, resulting in a proportional in… view at source ↗

**Figure 3.** A plot comparing MatrixKAN and KAN models of shape [2,2,1] and [2,5,1] across increasing grid sizes. size, as expected from our theoretical analysis, MatrixKAN demonstrates superior performance due to the underlying B-spline degree. 5.1.2. B-SPLINE DEGREE For our test measuring computational efficiency with respect to B-spline degree, grid size was set to 2, and B-spline degree was varied across the following values… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: A plot comparing MatrixKAN and KAN models of shape [2,2,1] and [2,5,1] across increasing spline degrees. From [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: A plot comparing loss level of MatrixKAN and KAN models trained to model Feynman Equation I.6.20b: f(Θ, σ) = exp(− Θ2 2σ2 )/ √ 2πσ2 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

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work page Pith review arXiv

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Koenig, B

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URL https://arxiv.org/abs/ 2405.06721. Liu, Z., Wang, Y ., Vaidya, S., Ruehle, F., Halverson, J., Solja ˇci´c, M., Hou, T. Y ., and Tegmark, M. Kan: Kolmogorov-arnold networks, 2024a. URL https: //arxiv.org/abs/2404.19756. Liu, Z., Wang, Y ., Vaidya, S., Ruehle, F., Halver- son, J., Solja ˇci´c, M., Hou, T. Y ., and Tegmark, M. pykan. https://github.com/k...

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