Granger Causality Detection with Kolmogorov-Arnold Networks

Hongyu Lin; Mohan Ren; Paolo Barucca; Tomaso Aste

REVIEW 3 major objections 4 minor 1 cited by

Granger Causality Detection with Kolmogorov-Arnold Networks

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Kolmogorov-Arnold networks, with learnable spline edge functions and a first-layer proximal threshold, can match or slightly beat multilayer perceptrons at detecting sparse Granger causal relationships in time series.

desk verdict GC-KAN is a sensible combination of KANs and component-wise neural Granger causality, with a clean proximal sparsity trick that does produce exact zeros, but the paper's central claim that KANs outperform MLPs is not supported by the current evaluation—the cMLP baselines are borrowed from another paper under different conditions, and Eq. 18 is underspecified. read the letter →

arxiv 2412.15373 v1 pith:V5C2GW6A submitted 2024-12-19 cs.LG cs.AIstat.ML

Hongyu Lin , Mohan Ren , Paolo Barucca , Tomaso Aste This is my paper · ORCID

classification cs.LGcs.AIstat.ML MSC 62M1068T07

keywords GrangercausalityKolmogorov-ArnoldNetworksnonlineartimeseriessparsity-inducingregularizationproximalgradientneuralcausaldiscoveryLorenz-96interpretablemachinelearning

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

This paper argues that Kolmogorov-Arnold Networks (KANs), which place learnable spline functions on network edges, can serve as the engine for nonlinear Granger causality detection instead of the multilayer perceptrons used in current neural Granger causality models. It introduces GC-KAN, a framework that trains one small KAN per target time series with an L1-plus-entropy sparsity penalty and a proximal soft-threshold step on the first layer, so that lagged inputs that do not help predict the target are driven to exactly zero weight. On synthetic vector autoregressive data and on the chaotic Lorenz-96 system, GC-KAN matches the component-wise multilayer perceptron baseline and edges ahead at high sample sizes for VAR and at low sample sizes for Lorenz-96. The intended payoff is a parameter-efficient, interpretable way to discover which time series cause which others in high-dimensional nonlinear settings.

What carries the argument

The carrying object is the KAN's first edge layer, $\Phi_0 = W_{\mathrm{base}}\sigma(z) + W_{\mathrm{spline}}\,\mathrm{spline}(z)$, where each connection is a learnable B-spline function rather than a scalar weight. Causality is extracted from this layer by summing the magnitudes of the first-layer functions that map each lagged input to each hidden neuron, and the proximal update $w \leftarrow \operatorname{sign}(w)\max(|w|-\eta\lambda_{\mathrm{prox}},0)$ is applied only to $W_{\mathrm{base}}$ and $W_{\mathrm{spline}}$ so that weak functions become exactly zero. The L1 and entropy penalties on the whole layer encourage sparse and balanced use of the splines, and the deliberately small architectures (one or ten hidden neurons) keep the resulting causal map readable. The mechanism converts the question 'does series $j$ Granger-cause series $i$?' into 'does the corresponding first-layer spline contribution survive proximal thresholding?'

What would settle it

Train a GC-KAN on the paper's VAR(1) data and recompute $C_{j,k}$ with two natural definitions of $|\varphi|$, such as the $L^2$ norm over the training inputs and the $\ell^1$ norm of the B-spline coefficients; if the two rankings of candidate parents disagree, the causal extraction rule is not well-defined.

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Extended reading notes

Core claim

The central claim is that replacing the scalar input weights of a component-wise neural Granger model with KAN's learnable spline functions, and adding a proximal operator that soft-thresholds only the first-layer functions, yields a Granger causality detector that is at least as accurate as the established MLP baseline and more directly interpretable. Concretely, the paper defines GC-KAN, in which each target series is modeled by a KAN fed with all lagged series, and after training the Granger contribution of lag $k$ of series $j$ is read off as $C_{j,k}=\sum_{i=1}^{n_{\text{hidden}}}|\varphi_{j,k,i}|$. Because the proximal step forces irrelevant first-layer functions to exact zero, causal parents can be identified from the nonzero contributions without manual thresholding. In experiments, GC-KAN's AUROC is comparable to cMLP with hierarchical group lasso, with the largest reported margins on VAR(1) and VAR(2) at $T=1000$ and on Lorenz-96 with forcing $F=10$ at $T=250$.

Load-bearing premise

The load-bearing premise is that every first-layer spline function has a well-defined magnitude so that the contribution score $C_{j,k}$ in Eq.~18 can be computed; the paper never states how the magnitude of a spline function is measured.

Editorial extensions

If this is right

GC-KAN with a single hidden neuron matches or beats cMLP with 100 hidden neurons on VAR data at $T=500$ and $T=1000$, so KAN-based detection can be much more parameter-efficient.
Because the proximal step zeros out irrelevant inputs, GC-KAN can produce a Granger causality matrix directly from the trained network, without a manual pruning threshold.
On Lorenz-96 with $F=10$ and $T=250$, GC-KAN slightly outperforms cMLP, suggesting spline activations help in low-sample nonlinear regimes.
If the pattern holds, KAN-based frameworks are a viable route for causal discovery in high-dimensional physical and economic time series.

Reading between the lines

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

The paper's contribution score assumes a well-defined magnitude for each spline function $\varphi_{j,k,i}$, but no norm is specified; choosing different natural norms, such as the $L^2$ norm over inputs or the $\ell^1$ norm of B-spline coefficients, could change the ranking of candidate causal parents.
A natural follow-up is to group lags of the same variable inside GC-KAN, mirroring the hierarchical group lasso, which would add automatic lag-order selection to the spline-based detector.
The authors' suggestion that KANs can fit symbolic functions implies a concrete next step: converting each surviving first-layer spline into an explicit symbolic expression, which would turn GC-KAN into a dynamical-law extractor rather than only a causal detector.
The reported advantages appear mainly in compact architectures, so an equal-parameter comparison, rather than equal-neuron comparison, would clarify whether the gain comes from spline flexibility or from model size.

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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

I read the GC-KAN paper. Short version: the method is reasonable, the evaluation is not.

What's actually new: replacing the MLP in Tank et al.'s component-wise framework with a KAN, and adding a proximal soft-thresholding step on the first-layer weights (both base and spline) so that irrelevant lags get exactly zero contribution. That's a clean, simple extension and the appendix figure shows it works as advertised. The paper is clearly written, the synthetic setups (VAR and Lorenz-96) are standard, and using AUROC over a range of proximal strengths is a sensible way to measure detection.

The soft spots are mostly about whether the main claim holds. The headline comparison against cMLP uses numbers taken directly from Tank et al., with 100 hidden neurons and hierarchical group lasso, while GC-KAN uses 1 hidden neuron for VAR and 10 for Lorenz-96. Those are different architectures, different data draws, different preprocessing—so the reported margins (99.3 vs 98.4, 95.9 vs 95.1) aren't evidence of superiority. The one equal-capacity comparison they mention (10 hidden neurons, Lorenz-96, T=1000, F=40) is a single point from another source and still not matched on data. The abstract says 'outperform', but the actual numbers are mixed: GC-KAN is worse at low sample sizes for VAR and only marginally better elsewhere. The conclusion is more measured, but the framing overstates.

There's also a genuine technical ambiguity: Eq. 18 defines the Granger contribution C_{j,k} as a sum over hidden neurons of |phi_{j,k,i}|, where phi is a spline function, not a scalar. The paper defines an L1 norm for activation functions in Eq. 11, so a charitable reading is that Eq. 18 uses that norm, but the paper never says so. That needs to be explicit. And there's no code release, which makes it hard to check the ROC construction.

None of these are fatal to the idea. The method is coherent, the proximal sparsity step is a real contribution, and the synthetic experiments are a reasonable starting point. But the central comparative claim needs an equal-footing rerun: same data, same tuning budget, same hidden size for at least one configuration, and ideally code. I'd send this to a referee—it's the kind of paper where a careful revision with a controlled baseline would make it solid—but it isn't ready as is.

For your reading group, it's worth a slot if you want a concrete example of KANs applied to causal discovery with a methodological caveat to discuss. I wouldn't cite it in its current form; I'd wait for the revised version.

Referee Report

3 major / 4 minor

Summary. The paper introduces GC-KAN, a Kolmogorov-Arnold network based framework for nonlinear Granger causality detection. Each target time series is modeled by a KAN whose first-layer spline activations map lagged inputs to hidden neurons; sparsity is encouraged by L1 and entropy regularization plus a proximal soft-thresholding step applied to the first layer. The method is evaluated on synthetic VAR(1), VAR(2), and Lorenz-96 data with known ground truth, and AUROC scores are compared with cMLP results taken from Tank et al. [15]. The authors report comparable or slightly better accuracy for GC-KAN in several settings and argue that KANs offer a more parameter-efficient and interpretable alternative to MLPs for sparse high-dimensional Granger causality detection.

Significance. If the central claim were fully supported, the paper would make a useful contribution by adapting KANs to nonlinear Granger causality with an explicit sparsification mechanism and a compact architecture. The proximal-operator idea is a sensible adaptation of the cMLP pipeline to spline-based networks, and evaluating on synthetic systems with known ground truth is appropriate. The manuscript is also honest in reporting that many differences are small. However, the evidence for the headline claim that GC-KAN can outperform MLPs is weakened by an unequal benchmark: cMLP results are borrowed from [15] with a 100-hidden-neuron architecture and a different data pipeline, while GC-KAN results are generated in-house with 1 or 10 hidden neurons. In addition, the causal contribution metric in Eq. (18) is not fully specified because the norm of a spline function is not defined. A matched-condition rerun and a precise definition of the contribution score would be needed before the comparative claim can be accepted.

major comments (3)

[Section III, Tables I and II] The headline comparison is not made on equal footing. The cMLP AUROC values are "taken directly from [15]" and correspond to a 100-hidden-neuron cMLP optimized with Hierarchical Group Lasso on data from a different pipeline, whereas the GC-KAN results are produced by the authors with 1 (VAR) or 10 (Lorenz-96) hidden neurons on their own simulated data. The reported margins (e.g., 99.3 vs 98.4 at T=1000 in VAR(1), and 95.7 vs 95.5 at T=1000, F=40 in Lorenz-96) are small, and no matched-condition standard deviations are provided for the borrowed baseline. Consequently, the central claim that GC-KAN has the potential to outperform MLPs in Granger causality detection is not supported by controlled evidence; the authors should rerun cMLP under identical data, preprocessing, and capacity settings.
[Eq. (18) in Section II.C.2] The Granger contribution C_{j,k} is defined as a sum of |φ_{j,k,i}|, but φ_{j,k,i} is a learnable spline function, not a scalar. The paper never states whether the absolute value refers to the L1 norm defined in Eq. (11), the absolute value evaluated at a particular input, or some other functional norm. Since this quantity is used to construct the binary Granger causality matrix that underlies the ROC curves, the causal scoring rule is not mathematically specified and the experiments are not reproducible as written. Please define the norm explicitly.
[Section IV and Table II] The conclusion that GC-KAN shows "stronger performances in low-sample-size settings" for Lorenz-96 is overstated. At F=10, T=250 the AUROC values are 86.9±0.5 for GC-KAN versus 86.6±0.2 for cMLP, which is within the reported standard deviations; the only non-overlapping low-sample advantage appears at F=40, T=250 (86.3±0.2 vs 84.0±0.5). The claim should be qualified to refer to that specific setting.

minor comments (4)

[Section III.B] The phrase "10 add hidden neurons" should read "10 additional hidden neurons."
[Fig. 1 caption] The caption states that the data are generated from an n=10 VAR(3) model, while Section III.A describes the VAR experiments as n=20 VAR(1) and VAR(2). This inconsistency should be resolved.
[Eq. (14)] The GC-KAN loss omits the summation over time steps t=p,...,T that appears in the cMLP loss of Eq. (5). Please make the notation consistent.
[Eqs. (16)-(17)] The proximal operator is written as Prox_{λprox}, but the soft-thresholding rule uses ηλprox as the threshold. The role of the learning rate η in the definition of the proximal map should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the method is validated against known synthetic ground truth, and the cited baselines and KAN machinery are external prior work rather than self-citations that smuggle in the conclusion.

full rationale

The paper's central claim is that GC-KAN can detect Granger causality and can be competitive with or outperform cMLP. The evaluation is conducted on synthetic VAR and Lorenz-96 data with known ground-truth causal structure, and the reported AUROC numbers are computed by sweeping the proximal/sparsity strength and comparing against that ground truth. No parameter is fitted to the target causal matrix, and no prediction reduces to an input by construction. The KAN formulation (Eq. 7-14), the proximal operator (Eq. 16-17), and the causal contribution metric (Eq. 18) are all stated as algorithmic definitions rather than results derived from an assumed conclusion. The only notable weakness is that cMLP results are 'taken directly from [15]' (Tables I and II), which raises a benchmarking-fairness concern because the cMLP baseline was trained under different architecture choices and data draws than the GC-KAN runs. That is an experimental validity issue, not a circularity issue: the cMLP numbers are external evidence, not a fitted constant from this paper. Similarly, Eq. 18 is notationally underspecified because phi is a spline function and the paper does not explicitly say it is using the |phi|_1 norm of Eq. 11, but ambiguity is a reproducibility defect, not a self-referential reduction. There are no load-bearing self-citations by the present authors, no uniqueness theorem imported from their own prior work, and no ansatz smuggled in via their own citations. The honest finding is therefore no significant circularity, score 0.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the KAN architecture's ability to represent nonlinear functions, on the choice of hyperparameters (proximal strength, regularization weights, architecture size), and on the assumption that the simulated data satisfy Granger causality conditions. The contribution metric in Eq. 18 is an undefined function magnitude, which is a gap in the method. No new physical entities are introduced.

free parameters (5)

proximal strength λ_prox = swept over a range
The ROC curve is generated by retraining with different λ_prox values; the AUROC depends on the chosen range, which is not specified.
regularization weight λ = not specified (μ1=μ2=1)
Balances MSE and sparsity terms in Eq. 14; its value affects sparsity and detection performance.
number of hidden neurons = 1 for VAR, 10 for Lorenz-96
Chosen by hand for interpretability or motivated by a multiplication approximation identity, but not systematically justified.
max lag p = 5
Initialized to 5 for all experiments; affects input dimensionality and the lag structure of detected causality.
spline grid size and order = not specified
B-spline parameters are part of the KAN architecture but are not reported; they determine the flexibility of the learned functions.

assumptions (3)

standard math Kolmogorov-Arnold representation theorem ensures KANs can represent any continuous multivariate function
This is the theoretical basis for using KANs to model nonlinear Granger causality.
domain assumption The simulated VAR and Lorenz-96 data satisfy the Granger causality assumptions, including stationarity, no unobserved confounders, and a complete system
The experiments assume the data generation is consistent with the causal inference setup, as acknowledged in Section II.A.
ad hoc to paper Ten hidden neurons are sufficient to capture Lorenz-96 nonlinearities via the identity 2xy = (x+y)^2 - (x^2 + y^2)
The paper motivates 10 hidden neurons by citing KAN's ability to approximate multiplication, but does not justify the exact count.

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Cite this review

Pith. "Pith review of Granger Causality Detection with Kolmogorov-Arnold Networks." pith.science (2026). https://pith.science/paper/V5C2GW6A

@misc{pith2026241215373,
  author       = {Pith},
  title        = {Pith review of: Granger Causality Detection with Kolmogorov-Arnold Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V5C2GW6A}},
  note         = {Machine review of arXiv:2412.15373}
}

read the original abstract

Discovering causal relationships in time series data is central in many scientific areas, ranging from economics to climate science. Granger causality is a powerful tool for causality detection. However, its original formulation is limited by its linear form and only recently nonlinear machine-learning generalizations have been introduced. This study contributes to the definition of neural Granger causality models by investigating the application of Kolmogorov-Arnold networks (KANs) in Granger causality detection and comparing their capabilities against multilayer perceptrons (MLP). In this work, we develop a framework called Granger Causality KAN (GC-KAN) along with a tailored training approach designed specifically for Granger causality detection. We test this framework on both Vector Autoregressive (VAR) models and chaotic Lorenz-96 systems, analysing the ability of KANs to sparsify input features by identifying Granger causal relationships, providing a concise yet accurate model for Granger causality detection. Our findings show the potential of KANs to outperform MLPs in discerning interpretable Granger causal relationships, particularly for the ability of identifying sparse Granger causality patterns in high-dimensional settings, and more generally, the potential of AI in causality discovery for the dynamical laws in physical systems.

Figures

Figures reproduced from arXiv: 2412.15373 by the authors.

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Reference graph

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Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [15]

In these approaches, each target time series is independently modeled using a dedicated MLP or LSTM, with the lagged values of all time series serving as inputs

introduced the component-wise Multilayer Percep- tron (cMLP) and the component-wise Long Short-Term ∗ Corresponding author Email address: hongyu.lin.18@ucl.ac.uk Memory network (cLSTM) for multivariate nonlinear Granger causality. In these approaches, each target time series is independently modeled using a dedicated MLP or LSTM, with the lagged values of...

arXiv 2024

[2] [1]

Each KAN is tasked with predicting a single target series xt,i based on the lagged observations of all series

Mathematical Formulation for GC-KAN The GC-KAN model consists of one KAN for each time series in the multivariate system xt. Each KAN is tasked with predicting a single target series xt,i based on the lagged observations of all series. The input vector for each GC-KAN is defined as: z = [xt−1, xt−2, . . . ,xt−p] ∈ Rnp×1. The output of each KAN is the pred...

work page

[3] [2]

Granger causality is extracted from the functional mappings in the first layer, which en- code the contribution of each lagged input series to the prediction of the target series

GC-KAN Optimization for Granger Causality The first layer of GC-KAN maps time-lagged inputs to hidden representations. Granger causality is extracted from the functional mappings in the first layer, which en- code the contribution of each lagged input series to the prediction of the target series. By analyzing these map- pings, significant contributions c...

work page

[4] [3]

Net- work granger causality with inherent grouping structure

Sumanta Basu, Ali Shojaie, and George Michailidis. Net- work granger causality with inherent grouping structure. The Journal of Machine Learning Research , 16(1):417– 453, 2015

work page 2015

[5] [4]

Each time series generated has self dependencies and three ran- domly selected parents among the other n−1 series

Granger causality detection For linear Granger causality where the underlying dy- namics can be represented using a V AR model, we simu- late data from n = 20 V AR(1) and V AR(2) models. Each time series generated has self dependencies and three ran- domly selected parents among the other n−1 series. The influence of a parent series on a target series is ...

work page

[6] [5]

Lag selection comparison For cMLP with the Hierarchical Group Lasso penalty, as expressed in Eq.6, the regularization simultaneously selects Granger causal variables and determines the max- imum lag order of the interaction. This penalty ensures that for each variable, there exists a maximum lag ˜k such that all weights associated with lags greater than ˜...

work page

[7] [6]

Survey and evaluation of causal discovery methods for time series

Charles K Assaad, Emilie Devijver, and Eric Gaussier. Survey and evaluation of causal discovery methods for time series. Journal of Artificial Intelligence Research , 73:767–819, 2022

work page 2022

[8] [7]

Multi- variate granger causality and generalized variance

Adam B Barrett, Lionel Barnett, and Anil K Seth. Multi- variate granger causality and generalized variance. Phys- ical Review E—Statistical, Nonlinear, and Soft Matter Physics, 81(4):041907, 2010

work page 2010