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REVIEW 5 major objections 6 minor 1 cited by

Kolmogorov-Arnold Networks for Time Series Granger Causality Inference

T0 review · 5 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A Kolmogorov-Arnold Network can infer Granger causality from its first-layer base weights, matching or beating neural baselines on nonlinear, high-dimensional, and short time series.

desk verdict KANGCI is a legitimate KAN-based Granger causality extension with broad benchmarks, but its causal score uses only the unregularized base-weight path, and the paper never shows that base-weight sparsity equals Granger absence—so the central claim needs more evidence before it convinces me. read the letter →

arxiv 2501.08958 v2 pith:H2GHYHQQ submitted 2025-01-15 cs.LG cs.AI

Meiliang Liu , Yunfang Xu , Zijin Li , Zhengye Si , Xiaoxiao Yang , Xinyue Yang , Zhiwen Zhao This is my paper · ORCID
classification cs.LGcs.AI
keywords GrangercausalityKolmogorov-ArnoldNetworkstimeseriessparsity-inducingpenaltytime-reversedneuralcausaldiscoverynonlineardynamicalsystemsEEG
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 proposes KANGCI, a Granger causality inference method built on Kolmogorov-Arnold Networks (KANs). Instead of reading the whole network, it extracts the first-layer base weights of each component-wise KAN, applies group lasso to those columns and ridge regularization to deeper layers, and uses the resulting column norms as the Granger causality matrix. On Lorenz-96, gene regulatory networks, fMRI BOLD signals, VAR models, and real rat EEG recordings, the method matches or beats established neural Granger causality baselines, especially when the time series are nonlinear, high-dimensional, and short. A secondary contribution is a time-reversed fusion rule that chooses the better of the original and reversed inferences, or averages them element-wise, to suppress spurious connections.

What carries the argument

The load-bearing object is the first-layer base-weight matrix $W_b^{0}$ of a component-wise KAN. In KAN, each edge is a learnable univariate function combining a SiLU base term weighted by W_b and a B-spline term weighted by W_s; the paper reads causal influence from W_b alone, assuming that after group-lasso training the zero pattern of $W_b^{0}$ columns marks absent Granger causes. Group lasso on columns enforces whole-series sparsity, ridge regularization stabilizes deeper layers, and a time-reversed Granger fusion step (Algorithm 1) selects or averages the original and reversed matrices based on prediction and sparsity losses.

What would settle it

Train KANGCI on a synthetic bivariate system where x causes y only through the spline component of the KAN edge, with the base weight for that edge fixed at zero; if the inferred Granger matrix then shows no edge from x to y, the base-weight reading is the sole carrier and the method fails on spline-only influences.

Watch

Extended reading notes

Core claim

The paper's central claim is that the base-weight matrices of a Kolmogorov-Arnold Network carry the cross-variable influence needed for Granger causality, so a sparsity penalty on the first-layer base-weight columns turns a KAN regressor into a causal discovery method. Each component gi is a separate KAN, each edge function is φ(x) = W_b b(x) + W_s spline(x), and the group-lasso penalty is applied to columns of $W_b^{0}$; the Granger matrix row is G(i,:) = ||$W_b^{0}$(:,j)||_F. The authors report AUROC values on Lorenz-96, DREAM-3/4, all 28 fMRI BOLD simulations, VAR, and real EEG that are competitive with or higher than cMLP, cLSTM, TCDF, eSRU, NAVAR, GVAR, JGC, CUTS+, and JRNGC, with the largest gains on noisy high-dimensional and limited-sample settings.

Load-bearing premise

The method assumes that the causal influence of each input series is captured by the first-layer base weights of the KAN, so that a group-lasso penalty on those weights reveals true Granger absence and presence.

Editorial extensions

If this is right

  • If KANGCI is correct, causal discovery from nonlinear time series no longer requires recurrent or convolutional architectures; a KAN regressor with grouped sparsity suffices.
  • The method should stay practical for high-dimensional systems because the component-wise KAN's computational graph is smaller than an MLP's, and only the first-layer base weights are needed for the causal matrix.
  • The time-reversed fusion rule should reduce spurious connections in EEG and other volume-conduction-prone data, since it explicitly compares or blends original and reversed inferences.
  • Across the reported benchmarks, the largest AUROC advantages appear in the hard regimes: p=40 Lorenz-96 with T=500, DREAM-4 with T=210, and noisy fMRI simulations, suggesting base-weight sparsity is most useful when data are scarce.
  • The real EEG results map inferred drivers to known somatosensory responses (contralateral S1 at 10-20 ms, ipsilateral S1 at 20-30 ms), so the method recovers physiologically plausible causal structure.

Reading between the lines

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

  • Beyond the paper, a natural test is whether the spline weights W_s^0 also carry causal information; if they do, the base-weight-only reading could underestimate influences that KAN represents through splines.
  • The time-reversed fusion rule is heuristic (threshold 0.05, max-or-average), so a principled calibration of that threshold across noise levels and sample sizes would be a direct follow-up.
  • Since the paper's own review notes that perfect causal reversal holds only under specific conditions for nonlinear systems, the fusion rule's success on Lorenz-96 and EEG suggests it does not rely on exact reversal but on a robustness comparison; this distinction could be tested on systems where reversal provably fails.
  • For fMRI and gene regulatory benchmarks, KANGCI's advantage is largest in low-sample, high-noise settings, implying the model may be especially suitable for clinical or experimental data where long clean recordings are unavailable.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper proposes KANGCI, a Kolmogorov-Arnold Network architecture for inferring Granger causality from multivariate time series. For each target series, a component-wise KAN is trained with a group-lasso penalty on the columns of the first-layer base weight matrix, a ridge penalty on the remaining base weights, and a prediction loss. The Granger causality matrix is then read out from the norms of the first-layer base-weight columns. A second contribution is an algorithm that fuses or selects between the Granger matrices estimated from the original and the time-reversed series, using prediction and sparsity losses plus an element-wise threshold rule. The method is evaluated on Lorenz-96, DREAM-3/DREAM-4 gene regulatory networks, simulated fMRI BOLD signals, VAR processes, and a real rat EEG dataset, reporting AUROC against cMLP, cLSTM, TCDF, eSRU, GVAR, NAVAR, JGC, CUTS+, and JRNGC. The authors conclude that KANGCI is competitive and, in the final section, that it outperforms existing baselines.

Significance. If the base-weight causal measure is sound, KANGCI would be a useful addition to the neural Granger causality toolbox: it is conceptually simple, leverages the compact learnable-edge representation of KANs, and the authors evaluate it on a broader fMRI benchmark than most prior work, including real EEG data with physiological validation. The empirical coverage is a strength, as is the use of synthetic benchmarks with known ground truth, which gives the central claim external grounding. However, the paper's central methodological assumption—that the zero/nonzero pattern of the first-layer base weights after group-lasso training equals Granger absence/presence—is neither proved nor ablated, and the fusion algorithm's threshold is introduced after observing experimental divergences. These issues, together with missing hyperparameter reporting, mean that the current evidence supports 'competitive performance' but not the stronger 'outperforming' conclusion.

major comments (5)
  1. [§3.1–3.2, Eq. (5), Eq. (12), Eq. (17)] The Granger causality score in Eq. (17) is computed from the first-layer base weights W_b^0 alone, but each KAN edge is phi(x) = W_b b(x) + W_s spline(x) (Eq. 5), and the group-lasso penalty in Eq. (12) is applied only to columns of W_b^0. The spline weights W_s are unregularized, so the optimizer can encode a true causal dependency entirely in the spline pathway while driving W_b^0 to zero, causing Eq. (17) to miss the edge; conversely, a non-causal input can have nonzero W_b^0 if the spline pathway is trained to cancel the base contribution. The paper gives no approximation-theoretic or empirical argument that the zero/nonzero structure of W_b^0 coincides with Granger absence/presence under this parameterization. I recommend either proving this property under the training objective, or adding an ablation that directly tests it—for example, constructing synthetic mechanisms where the causal signal is carried only by the spline terms and reporting whether Eq. (17) still recovers the ground truth.
  2. [§3.3, Algorithm 1] The fusion rule and the threshold theta=0.05 are justified by the observation that 'in certain simulation trials, the causal relationship inferred from the original and time-reversed time series exhibit considerable divergence,' which is an explicitly post hoc motivation. The paper does not report a sensitivity analysis for theta, and the loss-based selection between original and reversed matrices (lines 7–10) is a heuristic whose connection to Granger causality is unclear, especially because the cited literature (Winkler et al., 2016; Korenek & Hlinka, 2021) shows that time reversal reverses linear Granger causality but can conserve or alter nonlinear causality depending on conditions. This is load-bearing because the fusion step is part of the proposed method and can affect all reported AUROC values; please provide a sensitivity analysis over theta and an independent validation of the loss-based selection rule, or remove the claim that the algorithm 'automatically selects' the better result.
  3. [§6 vs. Tables 3 and 5] The conclusion states that KANGCI 'outperform[s] the existing baselines,' but the tables do not support this uniformly. In Table 3 (DREAM-4), KANGCI is below JRNGC on Gene-2 (0.591 vs. 0.613) and is not the best on all rows; in Table 5 (VAR), CUTS+ achieves AUROC 1.000 in all three settings while KANGCI drops to 0.993 in the sparsity=0.3 setting. The abstract's phrasing 'competitive performance' is consistent with the evidence; I request that the conclusion be aligned with the tables, or that the claim be restricted to the specific settings where KANGCI is best.
  4. [§4, experimental setup] The manuscript does not report the values of the key hyperparameters—lambda (group lasso penalty), gamma (ridge regularization), theta (fusion threshold), number of KAN layers, hidden widths, spline grid size, spline order, learning rate, number of epochs, or parameter initialization—for any of the five datasets. Without these details the experiments cannot be reproduced, and it is unclear whether the reported AUROC numbers are robust to hyperparameter choices or were selected on the test benchmarks. Please include a full hyperparameter table and, where possible, a sensitivity analysis for the most influential parameters.
  5. [§3.1, Eq. (4) and Eq. (10)] The input notation is ambiguous regarding time lags. Equation (4) writes the input as xt, but Granger causality requires predicting the current value from past values (as in Eq. (1)–(2), which use x_<t). The paper never specifies how the lagged input matrix is constructed—e.g., whether xt in Eq. (4) denotes a vector of p variables at a single time point or a window of lagged observations. This is a central modeling detail, since without past values the fitted function is not a Granger-causal autoregressive model. Please define the lag structure explicitly and state how many lags are used in each experiment.
minor comments (6)
  1. [§3.2, Eq. (17)] Equation (17) defines G(i,:) = ||W_b^0(:,j)||_F without specifying the range of j; it should read G(i,j) = ||W_b^0(:,j)||_F for j=1,...,p to make the matrix construction unambiguous.
  2. [§2.2.3, Theorem 2.1] The statement of Theorem 2.1 uses inconsistent subscripts: the text introduces Phi_q and phi_{q,p}, but the displayed formula uses Phi_i and phi_{q,p} with i indexing the outer sum. Please align the notation.
  3. [Figure 1] Figure 1 is difficult to read at the resolution provided; the layer labels, activation labels, and the distinction between W_b and W_s are not clearly legible, and the arrows connecting the regularization losses to the weight matrices could be made more explicit.
  4. [Table 4 header] The header reads 'Dateset' (typo for 'Dataset') and 'Subject=50'; also the caption lists T=50/100/200/2000/5000 but the main text says T=200 in most cases, so please clarify which T values correspond to which simulations.
  5. [§5.3, EEG analysis] The one-way ANOVA results are reported only as p<0.0001; please also report effect sizes or the F-statistic and state whether multiple comparisons across channels or time epochs were corrected.
  6. [General] The paper states that it builds on efficientKAN but does not release the code for KANGCI; providing an implementation would substantially help reproducibility and is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central Granger inference claim is tested against external ground-truth benchmarks.

full rationale

KANGCI's derivation chain is self-contained in the relevant sense. The model takes only time series as input; the Granger matrix G(i,:) = ||W_b^0(:,j)||_F is computed from learned first-layer base weights, and the claimed ability to recover causal structure is assessed by AUROC against known synthetic graphs (Lorenz-96, DREAM, fMRI BOLD, VAR) and physiological EEG criteria. Those ground truths are not used in training or in the construction of Eq. 17, so the output is not equivalent to an input by construction. The paper's use of KAN (Liu et al., 2024; efficientKAN code) is an external architectural foundation, not a self-citation chain; no uniqueness theorem is imported from the authors' own prior work. The group-lasso penalty in Eq. 12 is the standard Tank et al. NAR approach applied to KAN base weights. One caveat is that Algorithm 1's fusion threshold (0.05) and the loss-based selection rule are described as motivated by observations made 'during the experiment' (Section 3.3), which suggests post-hoc tuning on the evaluation benchmarks; this is a reproducibility/overfitting concern, not definitional circularity. A further correctness risk, not circularity, is that spline weights W_s are unregularized while Eq. 17 reads only W_b^0, so the identification of Granger causality with base-weight sparsity is an unproven assumption. These concerns do not make the reported AUROC results forced by the paper's definitions.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The main load-bearing assumption is that KAN base weights alone are a faithful causal measure even though each edge also contains spline weights. The fusion algorithm further assumes that time-reversed causality behaves consistently across the tested nonlinear systems. Hyperparameters are free parameters that are not reported.

free parameters (4)
  • Group lasso penalty lambda = not reported
    Controls group sparsity on first-layer base weights and directly shapes the inferred causality matrix.
  • Ridge regularization gamma = not reported
    Regularizes later KAN layers; no values or sensitivity analysis are reported.
  • Fusion threshold theta = 0.05
    Hand-set threshold in Algorithm 1 for combining original and time-reversed matrices; no sensitivity analysis is provided.
  • KAN architecture and training hyperparameters = not reported
    Number of layers, hidden widths, spline grid size, spline order, learning rate, epochs, and optimizer are not specified.
assumptions (4)
  • standard math Kolmogorov-Arnold representation theorem
    Used in Section 2.2.3 to justify KAN as a function approximator.
  • domain assumption Component-wise nonlinear autoregressive model with Gaussian noise
    Section 2.2.1 assumes each time series component depends on past values of all series plus Gaussian noise.
  • ad hoc to paper First-layer base weights represent Granger causal influence
    Eq. 17 equates causal strength to the norm of W_b columns, but the spline pathway in Eq. 5 is not shown to be ignorable.
  • domain assumption Time-reversed Granger causality can be fused to reduce spurious connections
    Algorithm 1 relies on behavior cited from Korenek and Hlinka (2021), which holds only under specific conditions in nonlinear systems.

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Cite this review

Pith. "Pith review of Kolmogorov-Arnold Networks for Time Series Granger Causality Inference." pith.science (2026). https://pith.science/paper/H2GHYHQQ

@misc{pith2026250108958,
  author       = {Pith},
  title        = {Pith review of: Kolmogorov-Arnold Networks for Time Series Granger Causality Inference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H2GHYHQQ}},
  note         = {Machine review of arXiv:2501.08958}
}
read the original abstract

We propose the Granger causality inference Kolmogorov-Arnold Networks (KANGCI), a novel architecture that extends the recently proposed Kolmogorov-Arnold Networks (KAN) to the domain of causal inference. By extracting base weights from KAN layers and incorporating the sparsity-inducing penalty and ridge regularization, KANGCI effectively infers the Granger causality from time series. Additionally, we propose an algorithm based on time-reversed Granger causality that automatically selects causal relationships with better inference performance from the original or time-reversed time series or integrates the results to mitigate spurious connectivities. Comprehensive experiments conducted on Lorenz-96, Gene regulatory networks, fMRI BOLD signals, VAR, and real-world EEG datasets demonstrate that the proposed model achieves competitive performance to state-of-the-art methods in inferring Granger causality from nonlinear, high-dimensional, and limited-sample time series.

Figures

Figures reproduced from arXiv: 2501.08958 by the authors.

Figure 1
Figure 1. The architecture of KANGCI. γ > 0 is the ridge regularization hyperparameter that con￾trols the regularization strength. Finally, the predicted loss is defined as: Lp = Xp i=1 (xti − gi(xt))2 (15) Therefore, the loss function is defined as: L = Lp + Ls + Lr (16) Since the proposed model is a component-wise architecture, a total of p models are needed to construct the complete Granger causality matrix. We extract the… view at source ↗
Figure 2
Figure 2. The analysis pipeline of real-world whisker stimulation rat EEG signals. Step 1: EEG electrode positions. A solenoid is used to stimulate the unilateral whisker of the rat, with nodes 1-7 representing the ipsilateral electrodes to stimulation and nodes 9-15 representing the contralateral electrodes to stimulation. Step 2: Extracting the time period to be analyzed. Step 3: Inferring the Granger causality from time se… view at source ↗
Figure 3
Figure 3. (a) The inferred Granger causality in epoch -100-0 ms. (b) The Granger causality driving of each channel. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.0 0.2 0.4 0.6 Channel Total Driving (GC) ✱✱✱✱ [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: (a) The inferred Granger causality in epoch 10-20 ms. (b) The causal driving of each channel. Furthermore, KANGCI effectively identifies the causal re￾lationship from cS1 to the contralateral frontal and parietal regions during the 10-20 ms epoch, as illustrated in Fig…

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