Exploring Neural Granger Causality with xLSTMs: Unveiling Temporal Dependencies in Complex Data
Authors: Harsh Poonia
Felix Divo
Kristian Kersting
Devendra Singh Dhami
Venue: NeurIPS 2025 poster ()
Keywords: Granger causality
xLSTM
time series
Primary area: probabilistic_methods
TL;DR: GC-xLSTM jointly optimizes a sparse feature selector and predictive xLSTM to uncover a graph of Granger Causal dependencies

Abstract: Causality in time series can be challenging to determine, especially in the presence of non-linear dependencies. Granger causality helps analyze potential relationships between variables, thereby offering a method to determine whether one time series can predict—Granger cause—future values of another. Although successful, Granger causal methods still struggle with capturing long-range relations between variables. To this end, we leverage the recently successful Extended Long Short-Term Memory (xLSTM) architecture and propose Granger causal xLSTMs (GC-xLSTM). It first enforces sparsity between the time series components by using a novel dynamic loss penalty on the initial projection. Specifically, we adaptively improve the model and identify sparsity candidates. Our joint optimization procedure then ensures that the Granger causal relations are recovered robustly. Our experimental evaluation on six diverse datasets demonstrates the overall efficacy of GC-xLSTM.

Exploring Neural Granger Causality with xLSTMs: Unveiling Temporal Dependencies in Complex Data
Harsh Poonia 1.+
Felix 2
Kristian Kersting 2.3,5
Devendra Singh Dhami
'Caneggie Mellon University AA & ML Group, TU Darmstadt Phessian.Al
*DFKI
CCentre for Cognitive Science, TU Darmstadt
"TU Eindhoven
hpoonia cmu, edu d. dhami@tue. (felix. divo, kersting) 0cs. tu-darmstadt.c
Abstract
Causality in time series can be challenging to determine, especially in the presence of non-linear dependencies. Granger causality helps analyze potential relation- ships between variables, thereby offering method determine whether one time series can grediet-Granger cessful, Granger causal methods still struggle with capturing long-range relations between variables. To this end, we leverage the recently successful Extended Long Short-Term Memory (XLSTM) architecture and propose Granger causal xL- STMs (GC-xLSTM). It first enforces sparsity between the time series components by using a novel dynamic loss penalty on the initial projection. Specifically, we adaptively improve the model and identify sparsity candidates. Our joint opti- mization procedure then ensures that the Granger causal relations are recovered robustly. Our experimental evaluation six diverse datasets demonstrates the overall efficacy of GC-ML.TM..
cutuse-future values of another. Although
1 Introduction
Finding cause and effect among and within group of multivariate time series can lead better un- derstanding of the dynamics of the involved time series. For instance, in computational neuroscience and medicine, discovering brain connectivity assists better understanding natural cognition [Smith) et al. 2011]. Discovering inter-dependencies between time series also has critical impact on many other research areas, such as finance [Masini al.. 2023], climate science [Mudelsee] 2019), and industrial applications [Strem et al. 2025]. Although efforts have recently been made to improve the interpretability of time series models Ismail et al. 2020 Turbé et al. 2023), most methods are restricted to finding post-hoc interpretations and only focus on short-term dependencies.
The framework of Granger causality (GC) [Granee 1969 was introduced address the challenge of determining whether one variable's past values can help forecast another's future values, without implying direct causality. GC can be established using statistical hypothesis tests, determining whether one time series can predict another. The test traditionally involves estimating vector autoregressive model and examining whether lagged values of time series improve or degrade the prediction of the other, while controlling the past behavior of both series. Although GC does not imply direct cause-and-effect relationship between the involved time series [Heckman) 2008). recognizing these interdependencies can lead to better understanding of the dynamic relationships between variables over time [Marcinkevičs and Vogtl 2021 Shojaic and Fox 2022
Authorr contributed equally.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Many families of deep learning architectures have been explored for time series analysis over
the years, such as multilayer perceptrons [Zeng et al. 2023 Das et al., 2023). recurrent neural
networks (RNNs) [Hochreiter and Schmidhuber 1997 Cho et al.] 2014]. convolutional neural
networks [Wu et al. 2022 Wang et al.] 2022]. Transformers [Vaswani et al., 2017, Nie et al.
2023], state-space models (SSMs) [Wang et al., 2025]. or mixing architectures [Wang et al. [2024] Throughout this, recurrent models remained a natural choice for time series data since their direction of computation aligns well with the forward flow of time. This aligns particularly well with the goals of neural Granger causality. Although SSMs are comparable in that regard, RNNs tend to offer more powerful forecasting capabilities since they deteriorate less when modeling long-term dependencies. Furthermore, their inference runtime is typically linear in the sequence length at constant memory cost, making them much more efficient than, for instance, Transformers with quadratic runtimes and memory requirements, while remaining highly expressive. Recently, [Bek et al. (024) revisited recurrent models by borrowing insights gained from Transformers in many domains, specifically natural language processing. Their proposed Extended Long Short-Term Memory (xLSTM) model sparked resurgence of interest in recurrent architectures for sequence modeling and has already proven highly suitable for time series forecasting [Kraus et al. 2025 Alharthi and Mahmood] (2024)
Although most Granger causal machine learning methods assume linearity in time series as fundamental prerequisite (Siggiridou and Kugiumtzis, 2015 Zhang et al.] 2020], recent efforts capture non-linear dynamics in time series by using neural networks as the modeling choice instead of VARs Tank et al.] [022] Löwe et al. [002]] Cheng et al.) 2024]. Although successful, these non-linear methods require careful feature engineering to include time- based patterns. Thus. they may not capture interactions between time series and external factors as effectively as XLSTMs, which can learn non-linear patterns and adapt to the non-stationary nature of time series data.
We introduce GC-xLSTM, a novel method that leverages xLSTMs to uncover the GC relations in the presence of complex data, which inherently can have long-range dependencies CrraddenciesGGx....... enforces sparsity between the time series components by using a novel lasso penalty on the initial projection layer of the XLSTM. We learn a weight per time series and then adapt them to find the relevant variates for that step. Then, each time series component is modeled using a separate xLSTM model, enabling to discover interpretable GC relationships between the time series variables. After the forecast results by the individual xLSTM models, the important features are made more prominent, whereas the less important ones are diminished by joint optimization technique, which includes using a novel reduction coefficient. Thus, the overall GC-XLSTM model can be trained end-to-end uncover long-range Granger causal relations.
I GC-XLSTM first
Our main research contributions can be summarized as follows:
(i) We propose GC-xLSTM, novel model that can uncover Granger causal relations in non- linear time series.
(ii) Our novel algorithm jointly improves the forecasting model while adaptively enforcing strict sparsity.
(iii) Our empirical evaluations demonstrate that GC-xLSTM can robustly discover Granger causal relations in the presence of complex simulated and real-world data.
Outline. We start by recalling preliminaries and reviewing related research to contextualize this work in the broader body of research on neural Granger causality in Section 2 This allows us to introduce GC-XLSTM in Section D and empirically evaluate relation to other methods in Section Finally, we conclude with an outlook to future work in Section
2 Preliminaries and Related Work
We are interested in datasets of strictly stationary time series E VXT of V variates with length T. Let S, IR. denote the value of S at time f. A variate v (sometimes called a channel) can be any scalar measurement, such as the chlorophyll content of a plant or the spatial location of some object being tracked. Its value at time is S,, E HR. The measurements jointly at T' regularly spaced time steps. In forecasting, a model is presented with a time series of C context steps Se, before f, from which it shall predict the next value S, E RV
assumed to be carried out
'Code available ggthub.cco//rrpooii/CCC.XLS.

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2.1 Granger Causality
If the observed time series were generated by some underlying process g. which we can formalize as a structural equation model for all time steps t as Se., 9: (0,(S,,t)...... Sv,4))) + Ev. for all v E V. where E is some additive zero mean noise independent from all variates Sv and := (I...., V] is the set of all variates. In Granger causality [Granger) 1969). we aim to determine whether past values Sv.et of variate are predictive for future values S of another variate w. Following the notation of SShoaai and Fox [002], we formally define:
Definition (Granger Causality). Variate v is Granger non-causal for w if and only if Jv is invariant to Sw, <t for all & {1,...: : 7) i.e., and only if
9u (S = g.. Sv,ca Sw,
Else, we call Granger causal for w.
The set of all such relationships are the directed edges & C V V of the Granger causal graph (V,E) of the variates, which we eventually aim uncover.
2.2 Neural Granger Causality
Unfortunately, however, we cannot explicitly access g in most realistic settings. Using machine learn- ing methods, time series forecasting model Ma, RS go(S = Su.e Eu,t. We can do so by estimating the parameters O of the model based on the dataset of time series, minimizing the predictive mean squared error (MSE) loss
nonetheless estimate each of the V process components I. by an autoregressive
V
T
ppeed (0)
(S., - (S.i(
0=1 1=1
In the case of using neural networks for Mo... o. this approach is called Neural Granger Causality Tank et al. 2022
It would be very costly to train total of 2 models test if each variate Granger causes any other variate and thus construct the entire Granger causal graph. To avoid this, we can train merely a single component-wee
model Ma.,. for each variate v and inspect what inputs it is sensitive this does not ablate models as in classical GC, it reduces the number of models to be trained from quadratic to linear in V. This can, for instance, be achieved by optimizing the predictive loss Eposd (0) based on all model parameters with regularizer ((OO)) enforcing sparsity in the input features:
While
Σ 2 (0.).
min Epod (0) 0.0
+ A
(2)
1=1
where θ, are tunable parameters of the regularizer for variate and X E E, a hyperparameter to adjust the degree of sparsity. One such approach are cMI.Ps (Tank et al. [022]; multilayer perceptrons where the first weight matrix W E DxV projecting from V features at each time step to D hidden dimensions is regularized to encode sparse selection of input features. Ω is instantiated 2 norm of its columns: E |/ Note that O and θ can overlap.
Sparsity can then be extracted by binarizing the entries of W using user-defined threshold r. This is necessary the 1.2 penalty tends only to shrink parameters to small values near zero, yet not clamp them sharply to it. This, however, allows subsequent layers to amplify the dampened signal again and still use it for forecasting. We avoid this disad in GC-xLSTM by explicitly optimizing the feature extractor for strict sparsity. This more principled approach works without determining a sparsity threshold
Previous work has explored both more regularizers (Tank et al., 2022 and different means to extract Granger causal relationships. including using feature attribution via explainability (Atashgahi et al. 2024] and interpretability [Marcinkeviés and Vogt 202T]. Furthermore, several works have gone towards learning relevant representations that respect the underlying Granger causality (Xu et al. 2016 Varando et al., 202T [mochowskj] 2023). Zoroddu et al.] (2024) present another approach where prior knowledge is encoded in the form of noisy undirected graph, which aids the learning of
3

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after model estimation
Lookback
Sparse Feature Selector W
Mg
Forecast
Granger Causal Graph
XLSTM
XLSTM
Epo 0)
3
xLSTM
Reduction Coefficientsa
(a W)
L(
Figure I: GC-XLSTM performs three key steps to determine the Granger causal links: Firstly, for each time series component, all variates are embedded with sparse feature encoder W that is regularized through a novel sparsity loss with learned reduction coefficients a. xLSTM models then learn to autoregressively predict future steps from that embedding. Finally, once model estimation is complete, Granger causal dependencies can be extracted from W.
Granger causality graphs A more recent approach [Lin al. 2024 employs Kolmogorov-Armold networks [Liu et al., (2024) to learn the Granger causal relations between time series. For a discussion of when Granger causality implies (Pearlian) causality, we refer to Pettenuzzo and White [20TT] and 2023 allowing to connect to works such as the one Rubenstein et al. 2017]
Das and Babadi
2.3 Extended Long Short-Term Memory (XLSTM)
Beck et al. (2024) propose two building blocks to build up xLSTM architectures: the sLSTM and mL.STM modules vector-valued (i.e., multivariate) sequences. sLSTM cells improve upon classic LSTMs by exponential gating. For parallelizable training, mLSTM cells replace memory mixing between hidden states with an associative matrix memory. We will continue by recalling how sLSTM cells function since we found their memory mixing more effective in time series forecasting.
The standard LSTM architecture of Hochreiter and Schmidhuber 1997 updates the cell state e, through a combination of input, forget, and output gates, which regulate the flow of information across tokens. sLSTM blocks, owing to the contained sLSTM cells, enhance this by incorporating exponential gating and memory mixing [Greff et al.) [017]] to better handle complex temporal and cross-variate dependencies. Additional normalization states are introduced to stabilize training under the new exponential activation function. As Beck et al. have shown, it sufficient and computationally beneficial to constrain the memory mixing performed by the recurrent weight matrices R.,, R. R,, and R. to individual heads. This is inspired by the multi-head setup of Transformers (Vaswani al. 2017], yet more restricted and efficient. In particular, each token gets broken up into groups of features, where the input weights W f act across all of them, but the recurrence matrices Rz are implemented as block-diagonal. This permits specialization the individual heads to patterns specific to the respective section of the tokens and empirically does not sacrifice expressivity.
3 GC-xLSTM
We will now introduce the GC-XLSTM architecture and detail the optimization for strict sparsity jointly with the model parameters. At the end, will additionally discuss theoretical properties of the proposed system.
3.1 Overall Architecture
As Figure I shows, we estimate pipeline of sparse feature selectors and xLSTM models to predict the multivariate time series. Eventually, this allows to derive Granger causal dependencies from the selected features. Specifically, we learn for each variate v separate sparse projection and compute ar, W.S + b, shared across time. The matrix W, E DxV is shared across all lags of the time series for simplicity. We will lift this restriction in Section 4.2 Note that b does not affect the sparse use of inputs. We write of for the set of parameters W, and b, for all E V. addition to selecting dependencies, the sparse projection embeds the data into D-dimensional hidden space.

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reduction cocfficient update and model improvement
gradient descent
Retain high values
W
V
C C(..0.0.))
W <-
Ancc. [Wolis/
W
w
Quickly shrinks zero
proximal
grudient descent
compress W
(a) The optimization procedure for GC-XLSTM alternates between GD and proximal GD. A high value of a, entails strong compression of column We, i.c. more sparsity in depending variate w.
Visual representation the compression step in Algorithm L line The matrix preserves columns with high norm that are important for Granger causal- ity detection (dark) while suppressing others (light).
Figure 2: Optimization procedure and compression intuition for GC-xLSTM.
For Neural Granger Causality to successfully and faithfully extract the proper underlying depen- dencies, it is essential to employ models that can capture the complete set of dependencies. We, therefore, employ powerful deep-learning models with significantly higher capacity than the cMLPs and cLSTMs in prior work [Tank et al. 2022]. In particular, we instantiate the individual time series forecasters Ma... with sLSTM blocks introduced in Section 2.3] They can capture long-range dependencies in time series data, substantially enhancing the capabilities of traditional LSTMs in handling extended contexts. GC-XLSTM consists V sLSTM models, each modeling a different time series component. They are trained using established forecasting losses, such as the MSE loss Epodd (Ø, 0) from Eq.
3.2 Optimizing for Striet Input Sparsity
The purpose of the feature selector W is to only "pay attention" to as many variates necessary for successful forecasting. relationships is via the lasso regularization explained in Section 2.2 We use variation of the group lasso penalty [Yuan and Lin 2006 Simon and Tibshirani 2012 on initial projection layer of GC-XLSTM as structured indusit penalty that encourages the selection of entire groups of variables, encoded as the columns of W.. Note that our penalty differs from adaptive lasso (Yuan and Lin 2006]. where the weights are not learned but are treated as fixed heuristics. Note that standard gradient descent methods cannot optimize such a penalty directly due to its non-differemtiability We thus adaptively compress by learning reduction coefficient O₂ E RV that selects which of the V columns of W. are redundant. We perform this compression of W, in a joint procedure with the general optimization of the forecasting model, as provided in Algorithm Specifically, two updates per optimization step for each of the variates, as Figure 2a depicts. Firstly, we optimize the projection weights Φ, the reduction coefficients a, and the xL.TM parameters using mini-batch Algorithm It optimizes the following loss
common approach to achieve this sparsity on the Granger causal
perform
gradient descent. This corresponds to lines to 15 expected over the time series data S:
min Crea (S; Φe, 0.) + -Alog a. 0g
Σ a |sg (W
Eme(au,, W.)
Note that we, crucially, only descend on the reduction coefficient a in Eed and not on W., as the stop-gradient sg(-) denotes. This sparsity optimization is instead performed by the second step in the procedure, shown in I6 where a proximal gradient descent step dynamically shrinks W, proportional to C.. The compression update takes descent step towards the gradient of
Σ
W
followed by a soft thresholding. Intuitively, the Lied component of Eq. 3 keeps W P fixed while learning Ce,, and Eq. (4) keeps Or fixed the proximal step to compress Figure 2h depicts the intuition of the proximal gradient step and soft- soft-thresholdigg in line
5

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Details on learning the reduction loss Ledd is worth briefly discussing the use of the logarithm in Eq. 3 mainly gives more equal weight to the decreases in W, column norms and encourages learning of better sparse Granger causal relations. It furthermore normalizes the gradient updates to Ogp- Empirically, this loss engineering allowed training models that were significantly more robust to noise and changes to the sparsity hyperparameter X. This was reflected by more stable variable usage and predictive loss pped
Ensuring non-negativity of the reduction coefficients For the proximal update step to be well-behaved, we need to ensure that a, results in convex combination column weights, i.e., that a > 0 for all w V and ai 1 = 1. We achieve this by re-parameterizing it as a, softmax (Bo), and learning G, instead of at..
Intuitive dynamics of the gradient update step. The weights W, in the reduction loss Lxed of Eq. (3) only serve to learn good reduction coefficients O,, and are not optimized themselves. Deriving the gradient of the penalty term Legg with respect to the underlying ,3 provides helpful intuition of the training dynamics:
V
V
Opp
W
12
Σ
softmax
[ = a. 20
Σ
W
355
w=1
==
w=1
0 033
0
feg
log
Σ o [W.'ll
0.310
W
We can see that if the norm of a column |W'll2 is large, that corresponding σø Crog will be large. Gradient descent will thus decrease a and effectively allocate less weight to its removal in the compression step. Furthermore, a σअ. Legg also scales with a."" resulting in self-reinforcing loop that aids learning sparse representations.
Practical considerations. Furthermore, we perform staged optimization of o, which is initialized to uniform distribution by setting all O. We only start training after exploring the prediction loss and having obtained a reasonably compressed forecaster, which is controlled by the hyperparameter K in Algorithm| (see line [3). While we present the method with mini-batch gradient descent for conciseness, modern optimizers, such as Adam KKngmma and Ba 2017]. can further improve convergence.
3.3 Theoretical Analysis
Ultimately, we want to ensure that, provided real-world data, we can find hyperparameters such that Algorithm T discovers all and only those edges of the unique underlying GC graph (V, E') as per Definition Providing convergence guarantees in full generality is notoriously hard for such practical architectures and optimization schemes, and thus rarely attempted. However, we can at least investigate whether the chosen model class containing Me., can approximate to arbitrary precision. that is the case, we can be reasonably sure that gradient- based optimization schemes will yield satisfactory approximations, even without formal guarantees.
The forecasting component of GC-xL consists of two main steps: the sparse initial projection W. and the subsequent STM blocks. One might think that the sparsity of W, hinders learning the correct f.- However, the true underlying So is independent of all variates without ingoing edges into variate v. Thus, depending on an appropriate choice of the sparsity hyperparameter X, the projection W, can encode exactly those as zero entries. It remains investigate the approximation capabilities of the sLSTM blocks, which we present in Appendix B in more detail. In summary, we can assume that sLSTM blocks are at least as powerful RNNs, which are, in turn, universal function approximators. Thus, the overall GC-XLSTM architecture is sufficiently rich to model to adequate precision. We continue by confirming this empirically in the next section.

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Table 1: GC-XLSTM is highly accurate at discovering GC relations in the chaotic and linear Lorentz-96 system. For each setting and baseline, we provide the accuracy (Acc.), balanced accuracy (BA), and AUROC. The best models are highlighted as bold.
F 10
F = 40
Model
Year
Acc. (1)
BA (T)
AUROC (1)
Acc. (1)
BA (1)
AUROC (1)
VAR
2021
83.8 91.8+112
94.0+1.6
86.4+0.8
58.5
74.5+4.7
cLSTM
2022
97.0=1.0
95.0+2.8
95.8+2.6
84.4+1.
65.623.7
66.
cMLP
2022
97.2.0.5
95.6+1.6
96.3+1.8
68.3+2.7
80.5=1.7
97.9#166
GC-KAN
2024
92.1=0.3
87.1
TCDF
2019
87.1+1.2
70,9
85.7+2.7
77.5+2.3
62.2=3.0
67.943.1
eSR
2020
96.6+1.1
95.1+2.0
96.3+2.0
86.7=0.9
88.6=1.4
93.442.1
GVAR
2021
98.2+0.3
98.2+0.6
99.7=0.1
94.5+1.0
88.524.6
97.0=0.9
GC-XLSTM
ours
99.1+0.2
98.5+1.0
99.3=0.3
96.3+0.3
96.6-0.3
88.0+0.2
4
Experimental Evaluation
We conduct extensive experiments on six datasets to assess the practical effectiveness of GC-XLSTM. We will now explain the chosen architecture and parameters used train GC-xLSTM and then discuss the datasets used before presenting our obtained results in detail. The results are split into investigating the general GC modeling capabilities on diverse range of applications (Section and a subsequent model analysis, including an ablation study and using different numbers of variates (Section
Architecture Details. For our component-wise networks, we use a single xLSTM block comprised of one sLSTM layer, followed by a linear layer to predict the next time step from the preceding C' = 10 time steps (see Table 5 in Appendix C for deviations). We then directly optimize the architecture on each time step. The hidden dimension of the sLSTM block is set to 32 for all datasets. We find that the presence of a gated MLP to up- and down- -oojeet the hidden states of the sLSTM block does not significantly improve performance, so we omit it in all our experiments for simplicity. We deliberately do not use any mLSTM blocks, as we also find that the sLSTM blocks are superior at capturing long-range dependencies the data (Kraus et al. 2025
Training and Evaluation Details. We use the Adam optimizer [Kingma and Ba 2017 for full gradient descent training with a weight decay of 0.1. We schedule the learning rate to follow linear warmup 2,000 iterations to 7 = 10-4 followed by cosine annealing until the end of training for a total of 13,000 steps. We start learning the reduction coefficients after a warmup period of K = 1, 500 iterations, during which the uniform compression across all columns combined with the prediction loss gives reasonable priors for the gradient directions of the reduction coefficients. Due to the moderate size of the datasets, we performed full-batch gradient descent. Only the sparsity hyperparameter À was tuned specifically for each setting. This allows obtaining degrees of sparsity specifically tailored to the characteristics and requirements of each dataset and task. Note that, except for the customization of X, set of datasets, underpinning the robustness of GC-xLSTM. Following (Tank et al. [022] Sec. 6.1], we compute all AUROC scores by sweeping over À (5, ..... 15) in steps of one. This effectively integrates this hyperparameter as it covers the entire empirically viable range. All training runs were carried out single NVIDIA RTX A6000 GPU and concluded in at most 1.5 hours. Mini-batching could further decrease this modest training time.
essentially the same hyperparameter configuration for extensive
Datasets. We evaluate GC detection with GC-XLSTM on six diverse datasets. Obtaining objective truth about the underlying graph for real-world scenarios a constant challenge in Granger causality research. We thus employ the Lorenz-96 system of differential equations (Karimi and Paul 2010]. realistic fMRI brain activity simulations (Smith et al. 20 and simulated linear VAR data following Tank et al. [2022]. For further qualitative insights on real-world data, we additionally analyze
the Moléne weather dataset [Girault 2015), human motion capture recordings [CMU 2009], and company fundamentals (Divo al. 2025 An overview and more details are provided in Appendix
7

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(a)
(b) À = 10.
Figure 3: GC-xLSTM uncovers dynamic GC weather patterns in the Moléne dataset. We observe that the sparsity of the learned Granger causal relations increases with higher X.
(a) Salsa with À Salsa with
(c) Running with
(d) Running with 6.
Figure 4: GC-XLSTM captures complex human motions. GC-xLSTM is able uncover complex real-world dependencies in the Human Motion Capture dataset, giving us an intuitive understanding of the learned interactions.
4.1 Main Results
Lorenz-96. Table 1 evaluates GC detection with GC-XLSTM on three metrics. As quali- tative comparison, we provide results for seven well-known baselines: VAR as classic ftttsts for Granger causality taken from Marcinkevvis
Ground Truth GC
Estimated GC
Affected series
and Vogt [2021). cMLP and cLSTM [Tak al. 2022], GC-KAN [in et al. 2024].
TCDF Nauta et al.) 2019). eSRU Khanna and Tan 2020), and GVAR [Mareinkeviés and Vogt,
2021]. Their scores are taken as reported in original papers. First, GC-XLSTM outper- forms all baseline methods for both F 10 and F= 40 both accuracy and balanced accuracy. This shows that GC-XLSTM reliably captures the underlying Granger causal relationships in the presence of limited and noisy data. Second, demonstrates that the sparsity hyperparameter is rather well-behaved, as the AUROC score resulting from sweep over range of À values still provides very competitive results. Third, qualitative validation, Figure visually confirms the strong prediction accuracies of GC-XLSTM.
Causal series
Causal series
Figure 5: GC-XLSTM uncovers the vast major- ity of GC edges. In the highly chaotic F 40 setting of the Lorenz-96 system GC-xL is ac- curate in predicting the GC edges, shown dark blue Errors are marked red
Moléne. Unlike Zoroddu et al. 2024 who incorporate graph prior knowledge based sensor locations, our approach learns the GC structure solely from the temperature observations. This ensures that GC-XLSTM does not inherently favor regional connections over long range dependencies,

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allowing it to discover dominant weather patterns operating both locally and across broader spatial scales. Adjusting allows balancing granularity and interpretability for insights into both local and regional dependencies. The dense structure of the resulting Figure 3a exhibits richer set of GC interactions, while the more sparse Figure 3b highlights only the most pronounced edges.
IMRI. Next, we evaluated the efficacy of GC-XLSTM in noisy settings by considering the rich and realistic BOLD deconvolved time series provided by the fMRI data. As the balanced accuracies (BA) in Table 2 show, GC-XLSTM significantly outperforms the baseline models.
Table 2: GC-xLSTM dis- covers brain connectivity highly accurately.
Model
BA (1)
Human Motion Capture. To analyze GC relations between body joints, focus on two specific activities: Salsa dancing and running, which provide interpretable motion patterns. Figure (eggure4showe shows the results
TCDF
72.
GVAR
65.2+4.5
VAR
51.3+1.5
of the learned graphs for those activities. A closer look offers an intuitive understanding of the learned interactions. For example, in the Salsa dance, we observe edges from the feet to the knees and the arms, sup- porting the characteristic movements of the lower driving the upper body. We can also see the cross-limbe correlation, with movements initiating on one side of the body and propagating across. Similarly, the results for running strongly establish the lower limbs as primary motion drivers, with edges from the feet and knees to the arms. The eyclic dependencies between the knees, ankles, and feet capture the repetitive, alternating nature of the gait.
cMLP
61.426.8
cLSTM
65.5
GC-xLSTM
73.3+3.0
Company Fundamentals. Lastly, we evaluated how well GC-XLSTM can uncover relationships be- tween financial indicators of large companies. The dataset consists of short time series of quarterly per- formance indicators of 2527 large publicly traded companies. As the excerpt in Figure[]]shove and as verified by a financial expert, those extracted edges are mostly economically sensible. Results on the full set of 19 features are provided Appendix
Other Operating Expenses
Cost of Revenues
Income Tax Expense
Operating Income
Net Income
Figure 6: GC-XLSTM extracts relations be- tween company fundamentals.
4.2 Model Analysis
Ablation Study. GC-XLSTM comprises two key innovations: Employing the xLSTM ar- chitecture (cf. Section 3.1 and a novel joint optimization strategy (cf. Section 3.2 To disentangle the empirical impact of each, we performed an ablation study. Table to the right shows balanced accuracy results on the Lorentz (F 40) and IMRI datasets. First, to assess the architectural contribution, we replaced the XLSTM block with a standard LSTM. The resulting substantial performance drop (GC-xI
Table 3: Both the xLSTM forecaster and the novel joint opti- mization of GC-xLSTM drive its improved performance.
Ablation Forecaster
Optimization
Lorentz
fMRI
++ (I)) validates the importance of the xLSTM's advanced model- ing capacity. Second, to evaluate the optimization strategy, we sub- stituted it with standard Group Lasso [Simon and Tibshirani 2012]. This also led to marked decline in performance (GC-xLSTM + (II), confirming its ability to enforce strict input sparsity.
GC-xLSTM (I)
XLSTM
Joint
96.6=0.3
73.3=3.0
LSTM
Joint
93,020.3
62.8=2.0
(II)
XLSTM
Group Lasso
73.0+4.6
65.4+2.0
Variable Number of Lags. GC-XLSTM can naturally be extended to learn separate projec- tions ( per time lag effectively inflating W to rank three. We evaluate it on the simulated VAR dataset across different time series lengths T and number of variates V in Table proving its ability to learn multiple lag-specific relations without training additional xLSTM models.
Table 4: GC-xLSTM discovers specific relations per lag. Balanced Accuracies for simulated VAR at different lengths T and number of variates V.
BA (1)
T = 250
T 500
T 1000
V 10
93.1
92.5+1.0
95.5+1.0
= 20
83.9+1.0
89.1
88.5+2.0

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Complexity and Scaling Behaviour: To discover the entire GC graph, we need to fit V models Mo... obtain the respective sparse projection matrices W, containing all edges arriving each v. Assuming for brevity that the latent dimension D x V, i.e., is a fixed multiple of the number of variates V. each fitting runs in O(TV2 time and requires ((TV2/N,, space, with T being the time steps and N₁ the number of sLSTM heads. Their block-wise structure permits that, despite the squared effort, forward and backward passes are efficiently computable even for thousands of dimensions. Depending on whether all V models are estimated in sequence or parallel, either the time or memory complexity multiplies by V to arrive at the total cost. In practice, however, GC-XLSTM is extremely efficient contemporary computing platforms due the availability of highly optimized implementations for xLSTM layers. Figure 8 in Appendix E shows that it effectively scales linearly in the number of variates for the ranges relevant to standard GC detection settings.
Inspecting Training Dynamics. Here, we elaborate on the utility of the logarithm in the reduction loss fied of Eq. (3). The logarithm term incentivizes the model to explore sparser solutions to GC discovery by allowing the training to move forward over any local minima that use the complete set of input variates. As the projection matrix becomes sparser and the input variates vanish from consideration by the model, the prediction loss Cpred will slightly increase, As seen in Figure 9 in Appendix F an increase in the sparsity of the feature selectors W drives down the loss and enables learning more meaningful GC relations. It also shows how the variable usage quickly stabilizes.
Conclusion
We presented GC-xLSTM, novel <LSTM-based model uncover Granger causal relations from the underlying time series data. GC-xLSTM first enforces sparsity between the time series components and then learns weight per time series decide the importance of each time series for the underlying task. Each time series component is then modeled using a separate XLSTM model, which enables it better discover Granger causal relationships between the time series variables. We validated GC-xLSTM in six scenarios, showing its effectiveness and adaptability in uncovering Granger causal relationships even in the presence of complex and noisy data.
Limitations. While Section 3.3 provides a theoretical analysis of GC-XLSTM, it does not give guarantees in the form of mathematical proofs. The rigor is limited by lack of formal analysis of XLSTM blocks (see also Appendix B which are yet to be formally analyzed to the same degree as more established architectures like LSTMs. While we discuss and measure the scaling behaviour of GC-XLSTM in Section 4.2 we only focused on dozens of variates as common in the literature.
Future work. This includes using more sophisticated architectures such as TimeMixer (Wang] et al. 2024) or XLSTM-Mixer [Kraus et al. 2025). Furthermore, discovering causal links specific to certain lags could be refined, where per-lag projections are learned for the near past and a remainder projection for the more distant lags. Finally, extending our evaluations to more real-world datasets encompassing domains such as climate change or ecology is essential next step.
Acknowledgments and Disclosure of Funding
This work received funding from the ACATIS Investment KVG mbH project "Temporal Machine Learning for Long- Term Value Investing" and the KompAKI project of the German Federal Min- istry of Research, Technology and Space within the ''The Future of Value Creation Research on Production, Services and Work" program (funding number 02L.191555 Management Agency Karlsruhe (PTKA). The TU Bindhoven author received support from their Dep. of Mathematics and Computer Science and the Findhoven AI Systems Institute. Furthermore, this work benefited from the DYNAMIC Centre funded by the LOEWE program of the Hessian Ministry of Science and Research, Arts and Culture (HMWK) as LOEWE / 11//////////..// ,001 (0009)98 and the HMWK project "The Third Wave of Artificial Intelligence - 3AI". It also benefited from the early stage of the German Federal Ministry for Economic Affairs and Energy project "Souveräne KI für Europa" (13IPC040G) as part of the EU funding program IPCEI-CIS: funding has not started yet, and from early stages of the Cluster of Excellence "Reasonable Al" (EXC-3057) funded by the German Research Foundation (DFG) under Germany's Excellence Strategy; funding will begin in 2026. Funding for H. Poonia to attend NeurlPS was provided by the CMU G/A/rrovost Conference Funding. The authors are responsible for the content of this publication. Map data OpenStreetMap contributors, licensed under the ODbL and available from poonoppenttretmmap.org
managed by the Project
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A Full Algorithm
Algorithm| T describes the entire procedure for extracting a Granger causal graph using GC-XLSTM. (+)+ denotes truncation max(-,0 0).
Algorithm Granger causality detection with GC-xLSTM
I: Input: Training data D { )eE N)) sparsity hyperparameter IR,,, learning rate
E I,,, and compression schedule start K E N
2: Output: Granger causal graph
3: for all v NEW VEVan do
D Training decomposes over
4:
Φo W., by (-VVV,
Kaiming/He initialization
5:
a, 1/v1
Uniform reduction coefficient
6:
0, (
Standard xLSTM initialization
7:
k +
8:
repeat
9:
Sample random mini-bated B ~
8 EsEB (Sp Φu. 0.) + X log (Ew= a ||sg (W.") 1l2
10:
9 f- 0.a00
11:
Φe d, 7
GD step > GD step
12:
O, + 0, ngee
if k K then
b Optimize after K steps
14:
On &- On 119a
b GD step
end if
W,
Ana [. +
W
Compression step with proximal GD
4-
18:
until convergence
19: end for
20:
E E (v) e | | l2 1>>00000 0}
21: return extracted graph (V,8)
B On the Approximation Capabilities of sLSTM blocks
Classic RNNs have long been known to be extremely powerful models of computation. Specifically, they are Turing complete Siegeelman and Sontag 1995] and, by extension, universal function approximators [Song al. 2023 Traditional I. LSTMS proposed by Hochreiter and Schmidhuber [1997] are universal function approximators being least as powerful the RNNs [Song et al. 2023 Corollary 16). Due their novelty, the sLSTM cells as presented in Section 2.3 have not yet received the same degree of theoretical analysis Yet, it appears natural to extend the same reduction to RNNs as has been shown for LSTMs, since the main technicality is the differing normalization of the hidden state [Beck et al. 2024 Eq. 36) and the alternative exponential activation function, which [Son et al.
2023 abstract away with. sLSTM blocks then wrap these cells with additional operations (cf. Beck ct al. 2024]. App. A.4). They, however, can be carefully configured to reduce to only rescaling Layer Normalizations and residual connections, which are known to, again, yield universal function approximators for many architectures [Tabuada and Gharesifard] 2020 Marchi et al. 2021]. This can be shown by first omitting the optional upfront convolution and Swish activation. We, furthermore, set the number of heads to N₂ = 1, causing the per-head Group Normalization after the cell to degenerate yet another Layer Normalization. Lastly, the final post up- up-projection of zin Wour is defined as
Wa ((W + b.) GeLU (Waæm b2)) + b3.
Cot
A discussion of expressivity hierarchies be found in Auer
2025
App. A].
21

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With the right instantiation, where a is chosen such that (a) = 1, we obtain
π
0
Cout
0
1
Zin +
c GeLU (Oxin + al)
+0
0
0
0/
0
0
I
0
11
0
0
1
0
0
I
0
0
0
11
2in
0
0
0/
Din
This, again, shows that sLSTM blocks are least as general LSTMs. While rigorous proof is far beyond the scope of this work, this investigation still underpins the strong capabilities of this architecture. These theoretical considerations align well with empirical findings, showing that xLSTM blocks are at least as effective as LSTMs [Beck et al. 2024
C Dataset Details
This section details all six datasets used in our empirical evaluation of GC-xLSTM. An overview is provided in Table 5
Table 5: Overview of the diverse collection of datasets. It also shows the number of variates V. time steps T, samples N, and look-back context steps of GC-xLSTM C.
Name
Origin
Type
Lorenz-96
Karimi and Paul
2010
Simulated
20
500
1
10
fMRI
Smith al.
2011
Real-world
15
200
1
10
Moléne
Girau
2015
Real-world
32
744
1
10
Human MoCap (Run)
CMU
2009
Real-world
54
1232
61
10
Human MoCap (Salsa)
CMU
2009
Real-world
54
4136
30
10
Company Fundamentals
DIvo
2025
Real-world
56
2527
40
VAR
Karım and Paul
[2010]
Simulated
10/20
mult.
1
5
Lorenz-96. The V-dimensional Lorenz-96 model Karimi and Paul 2010 is chaotic multivariate dynamical system governed by the differential equations
d (5r. Tt,i 2) L1,1 F.
dt
with the external forcing coefficient F regulating the non-linearity of the system. Low values of F correspond to linerl dynamics, while higher values induce chaotic behavior. There are two sources randomness in the system. Firstly, we sample i.i.d. starting conditions from (0 0.01). Secondly, in each step of the simulation, we add ii.d. noise sampled from 0.1) to Zt, Following the setting of Tank et al. [0022 for best comparability, we simulate V 20 variates with a sampling rate of At 0.05 for a total of T 500 time steps after a brief burn-in time. We use two forcing constants E (10, 40) to test our model under different levels non-linearity.
fMRI. Discovering connectivity networks within (human) brains is a key application of GC de- tection methods. To this end, brain activity is measured non-invasively using functional magnetic resonance imaging (fMRI) over time and grouped into regions between which connections looked for. Specifically, we used the realistic simulations of deconvolved data of Smith et al. 2011).
(BOLD)
The Moléne dataset Girault 2015 contains hourly temperatures recorded by sensors at V 32 locations in Brittany, France, during T 744 hours. The objective is to understand the spatio-temporal dynamics of the temperature and assess the extent to which the model can uncover complex relationships in weather by considering only local observations.
Moléne.
22

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Human Motion Capture. We also apply our methodology to detect complex. nonlinear dependen- cies human motion capture (MoCap) recordings. In contrast the Lorenz-96 simulated dataset results, this analysis allows us to visualize and interpret the learned network more easily. We consider a data set from the CMU MoCap database [CMU] [009]. The data comprises V 54 joint angle and body position recordings across multiple subjects. Since some regions, like the neck, have multiple degrees of freedom in both translation and rotation, we consider the GC relations between two joints based on edges between all movement directions. The motion ranges from locomotion (e.g., walking) over physical activities such as gymnastics and dancing to day-to-day social interactions.
Company Fundamentals. Another common field of application of GC detection methods is discovering links in economic data, where it is otherwise hard to maintain an overview. We, therefore, benchmark GC-xLSTM on company fundamentals data [Divo al.) 2025]. The dataset contains V = 19 economic variables, such as the Net Income and the Total Liabilities of 2527 companies.
The data was collected quarterly from 2009 Q1 2023 Q3, resulting only T 56 time steps.
VAR. Following the well-known setup of Tank et al. [022], we generate two-step linear auto- regressive process in E {10, 20} variates. All variables depend on themselves, and an additional three random dependencies are added targets to be discovered. The number of time steps varies in E (250, 500, 1000}, and we start recording the data after a brief burn-in time.
D Extended Experimental Results
This section supplements the experimental findings of Section 4.1
Specifically, Figure provides the complete set of relations extracted from the Company Fundamentals dataset. On this challenging full feature set, only some of the discovered GC relations are economically plausible.
Capital Expenditure
Cash from Financing
Cash from Investing
Cash from Operations
Cost of Revenues
Income Tax Expense
Net Interest Expense
Levered Free Cash Flow
Net Income
Operating Income
Other Operating Expenses
Total Assets
Total Cash and Investments
Total Current Assets
Total Current Liabilities
Total Debt
Total Equity
Total Liabilities
Total Revenues
Capital Expenditure
Cash from Financing Cash from Investing
Cash from Operations
Cast of Revenues
Income Tax Expense Net Interest Expense
Levered Free Cash Flow
Other Operating Expenses
operating Income
Net Income
Total Cash and Investments
Total Assets
Total Current Assets
Total Current Liabilitie
Total Debt
Total Equity
Total Liabilities
Total Revenues
Figure 7: GC-xLSTM uncovers relationships between economic variables within individual companies. Established GC links are highlighted in dark blue.
23

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E Scaling Behaviour
Figure shows the training time and peak GPU memory reserved during the training of GC-xLSTM on the Lorenz-96 dataset with T 1000 for various numbers of variates V.
600
40
550
Training Time (minutes)
Memory Reserved (MB)
35
500
30
450
25
400
20
350
15
300
10
250
15 Number of Variates (V)
10
20
25
Figure 8: GC-XLSTM effectively scales linearly in the relevant range of variate counts.
F
Additional Insights into Training
Figure 9a shows how the different loss components change during training. The robustness of training reflected the stability of variable usage once an optimal is found, Figure 9b shows.
Training Loss
Prediction
Reduction Loss
100
180
LO05
50
Vemaalle Usage
0
-50
33-
2000
10000
12000
8020
Training steps
Training steps
(a) Visualization of the different loss components.
(b) Variable drops once the threshold 2 K is reached and then quickly stabilizes.
Figure Algorithm Djointly optimizes the prediction loss while adaptively establishing sparsity. Results show training on Lorenz-96 with F 40 and T 1000.
24


Reviews:
[{"review_id": "PmYWvZJxgV", "reviewer": "Reviewer_jaiu", "summary": "This paper investigates the problem of learning Granger causality from time series with long-range dependencies. The authors propose an Extended Long Short-Term Memory (xLSTM) architecture, designed to enhance the model's ability to capture temporal dependencies relevant for Granger causal inference. Experimental results on both synthetic and real-world datasets, such as the Moléne dataset, demonstrate improved performance over existing methods.", "questions": "See Weaknesses.", "limitations": "See Weaknesses.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1. The paper is well-organized and clearly written.\n\n2. Experiments on real-world data validate the practical utility and efficiency of the proposed approach.\n\nWeaknesses:\n\n1. A central concern is the identifiability of Granger causality using GC-xLSTM. Although the authors acknowledge this issue in the conclusion, it remains a critical point. In the causal discovery community, theoretical guarantees for identifiability are often essential to justify empirical findings.\n\n2. The set of baseline methods appears outdated. I recommend organizing the baseline algorithms in a table with details such as publication year and key differences, to clarify the novelty and time-effectiveness of the proposed method.\n\n3. The claim that existing methods fail to capture interactions between time series requires more justification. Many neural network-based methods, especially with sufficient depth and capacity, can model complex temporal dependencies. Furthermore, it remains unclear how modeling interactions enhances the identifiability of Granger causality. A more detailed explanation or illustrative example would help clarify this point.", "was_revised": true, "final_justification": "Thank you for the authors’ response. Most of my concerns have been addressed, and I will raise my score accordingly. However, due to the relatively weak theoretical contribution of this work, I am not inclined to give it a high score.", "rebuttal": "Dear Reviewer jaiu,\n\nThank you for recognizing the clear writing as well as the wide set of experiments on synthetic and real-world data.\n\nWe gladly clarify the raised comments and questions:\n\n**W1:** In the causal discovery community, guarantees on identifiability are indeed cornerstones of many important works. However, we want to point out that (classic) causality and our focus, Granger causality, are different fields closely related in purpose, not methodology (see also ll. 26-29). Granger causality is a statistical test that does not yield Pearlian causal graphs and is centered on the prediction task. Thus, Granger causality can indeed be helpful in identifying potential causal relationships, but it doesn't necessarily imply true causal mechanisms in the classical sense. There has been some work on equating Granger causality with classical structural causality, such as by Bodik and Paluš (2024), where, however, the focus was on extreme events. Thus, we hope that the reviewer will appreciate that proving “identification” in case of Granger causality is an open, non-trivial problem and warrants a completely separate study.\n\nIn GC, identifiability is *defined* (see Def. 1) to be given when we can correctly learn the underlying time-evolving process $g$. This is the case whenever we use models with sufficient capacity and flexibility (universal approximators), which we discuss being the case for GC-xLSTM in Sec. 3.3 and extend on in Appendix B. We further note that even the seminal work of Tank et al. (2022), having introduced Neural Granger Causality, recognizes this analysis as the limit of current theoretical guarantees in their conclusion section.\n\nWe do agree that this is indeed important future work. However, it goes beyond this work and overarches the entire family of Neural Granger Causality models, such as cMLP, cLSTM, GC-KAN, and our GC-xLSTM. We thus mention it in ll. 338-341.\n\n**W2:** Adding the publication year to Tab. 1 is a very good suggestion. We thus added the following column and will reorder the rows to follow an ascending timeline.\n\n| Name | Year |\n|------|------|\n| VAR | 2007 |\n| cLSTM | 2022 |\n| cMLP | 2022 |\n| GC-KAN | 2024 |\n| TCDF | 2019 |\n| eSRU | 2020 |\n| GVAR | 2021 |\n| GC-xLSTM (ours) | (2025) |\n\nAs can be seen, we compare against recent works, such as cLSTM, cMLP, and GC-KAN. We will extend the manuscript with a more detailed description of the methods and how they differ from xLSTM-Mixer.\n\n**W3:** This appears to be a misunderstanding. We never claim that models such as cMLP “fail to capture interactions”. Instead, we state that they “may not capture interactions between time series and external factors *as effectively as xLSTMs* [emphasis added]” in ll. 48f. This improved modeling capability was also shown by Beck et al. (2024).\nThis modeling capability is important for uncovering the data-generating process with sufficient fidelity, as discussed in W1 and Sec. 3.3. The improvement is most clearly shown in the results of Tab. 2, where the benchmark is sufficiently challenging to differentiate the models based on that. There, methods with simpler models (specifically, GVAR, VAR, cMLP, and cLSTM) discover far fewer correct edges than more advanced models such as the attention-based TCDF and our GC-xLSTM (overall best).\n\nThank you again for your thoughtful review. We hope it answered all outstanding questions. Specifically, having clarified the questions regarding identifiability, choice of baselines, and modeling complex temporal dependencies, we would greatly appreciate it if you considered raising your score to reflect this.\n\nBest\n\nThe Authors\n\n**References:**\n- Tank, A., Covert, I., Foti, N., Shojaie, A., & Fox, E. B. (2022). Neural Granger causality. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(8), [https://doi.org/10.1109/TPAMI.2021.3065601](https://doi.org/10.1109/TPAMI.2021.3065601).\n- Beck, M., Pöppel, K., Spanring, M., Auer, A., Prudnikova, O., Kopp, M., Klambauer, G., Brandstetter, J., & Hochreiter, S. (2024). xLSTM: Extended long short-term memory. Advances in Neural Information Processing Systems, 37, 107547-107603.\n-Bodik, J., Paluš, M. & Pawlas, Z. Causality in extremes of time series. Extremes 27, 67–121 (2024). https://doi.org/10.1007/s10687-023-00479-5"}, {"review_id": "vXt4qqEXVH", "reviewer": "Reviewer_w7eG", "summary": "The paper proposes Granger causal xLSTMs (GC-xLSTM), which leverage the recent xLSTM architecture to capture long-range relations between variables. GC-xLSTM enforces sparsity between the time series components. The authors test the method on six diverse simulated and real-world datasets.", "questions": "I wonder how recent deep state space models perform. Some examples include [1-2], which has excelled in long-sequence modeling. I would like to get a justification for using xLSTM over other deep state space models.\n\n[1] Smith, Jimmy TH, Andrew Warrington, and Scott W. Linderman. \"Simplified state space layers for sequence modeling.\" arXiv preprint arXiv:2208.04933 (2022).\n\n[2] Gu, Albert, and Tri Dao. \"Mamba: Linear-time sequence modeling with selective state spaces.\" arXiv preprint arXiv:2312.00752 (2023).\n\nIn the architecture details paragraph, it says the linear layer predicts the next 10 steps from the preceding 10 timesteps. This number seems quite small compared to some of the long-range sequencing tasks that are used to evaluate modern long-range sequence models.\n\nIn the Main Results section, I am not fully convinced that GC-xLSTM outperforms other baseline models or provides an intuitive understanding.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Quality: The submission is technically sound, with appropriate explanations in the background and methods sections.\n\nClarity: The paper is clearly written and well-organized, with informative background and methods sections. I especially liked the paragraph on explaining the intuition behind the gradient update step.\n\nSignificance: Although the paper is well-structured, I do not think the paper goes beyond the current neural Granger causality methods to be impactful for the community. I think the contribution of the paper is lacking.\n\nOriginality: Although the paper explores how the recently developed xLSTM architecture performs in recovering the Granger causal relations,  I do not think that the work provides sufficiently new insights, deepens understanding, or highlights important properties of existing methods.", "was_revised": true, "final_justification": "Through rebuttal, the authors answered my concerns on significance and originality. I thus decided to increase my score.", "rebuttal": "Dear Reviewer w7eG,\n\nThank you for recognizing the work as well-written and technically sound, as well as the diverse evaluation of the method and baselines.\n\nWe gladly clarify the raised comments and questions:\n\n**Regarding Significance and Originality:** We want to point out that the contribution of this work is twofold: While the exploration of the xLSTM architecture of GC detection is one part, as identified by the reviewer, the other part is the novel optimization scheme introduced and analyzed in detail in Sec. 3.2. It is summarized in Algorithm 1 and implemented in the provided code. The latter is a contribution beyond just xLSTMs.\nTo more clearly isolate the empirical contribution of these two components, we perform ablation studies of (a) the use of xLSTMs and (b) the joint optimisation strategy. The results are provided for two datasets and shown below.\n\nFirst, we replace the xLSTM block in the architecture with a standard LSTM layer, keeping the rest of the architecture unchanged. We can clearly see from (base) -> (a), that the performance drops substantially without the modeling capabilities of the xLSTM blocks.\n\nSecond, we employ the baseline of Group Lasso (Simon and Tibshirani, 2012) instead of our novel procedure for optimizing for strict input sparsity. Again, we can see from (base) -> (b) that our strong results indeed stem from the contribution of the new optimization procedure.\n\nResults on Lorenz-96 ($F=40$):\n\n| Model | Balanced Accuracy |\n|------------------------------|-----------------------------------|\n| (base) xLSTM + Joint Optimization   | 96.6% (±0.3) |\n| (a) LSTM + Joint Optimization    | 93.0% (±0.3) |\n| (b) xLSTM + Group Lasso | 73.0% (±4.6) |\n\nResults on fMRI:\n\n| Model | Balanced Accuracy |\n|------------------------------|-------------------------|\n| (base) xLSTM + Joint Optimization   | 73.3% (±3.0) |\n| (a) LSTM + Joint Optimization    | 62.8% (±2.0) |\n| (b) xLSTM + Group Lasso | 65.4% (±2.0) |\n\n**Q1:** When choosing model families, there often is a trade-off between efficiency and fidelity. Indeed, state space models (SSMs) make for highly efficient time series models. However, xLSTM-based models tend to generally provide better modeling accuracy while still limiting the compute necessary (see, e.g., Beck et al., 2024). This is also confirmed empirically by the effectiveness of GC-xLSTM. We will ensure to make this discussion more explicit and balanced in the final version of the paper, where § ll. 30ff currently only discusses why no Transformers have been employed.\n\n**Q2:** We first note that the xLSTM is a recurrent model, where forecasting 10 steps into the future auto-regressively is quite typical. While a look-back of 10 steps is indeed little compared to many long-term time-series forecasting methods, it is a lot in the regime of (Granger) causal discovery we contribute to. For instance, the seminal work of Tank et al. (2022) introducing Neural Granger Causality mostly uses a lookback of 2 or 5. Marcinkevičs and Vogt (2021) use values of 1 or 5. We did not scale beyond that, as higher numbers were not necessary to sufficiently forecast the datasets we investigated. Also, not that this regime is beneficial in many real-world datasets, where limits in length do not allow for substantially larger look-backs or forecast horizons.\n\n**Q3:** Firstly, regarding outperforming other baselines: We quantitatively compare the results of GC-xLSTM to those of baselines on the Lorentz-96 dataset in Tab. 1 and on the fMRI dataset in Tab. 2. For Lorentz-96, GC-xLSTM always wins in both Accuracy and Balanced Accuracy while staying very competitive for AUROC. We further reiterate that this AUROC score is computed by sweeping over different sparsity configurations $\\lambda$ ll. 244ff, essentially removing that hyperparameter. For fMRI, GC-xLSTM outperforms all five baselines.\n\nThank you again for your thoughtful review. We hope it answered all outstanding questions. Specifically, having clarified the questions regarding significance and originality, as well as providing the suggested ablation study, we would greatly appreciate it if you considered raising your score to reflect this.\n\nBest\n\nThe Authors\n\n**References:**\n- Tank, A., Covert, I., Foti, N., Shojaie, A., & Fox, E. B. (2022). Neural Granger causality. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(8), [https://doi.org/10.1109/TPAMI.2021.3065601](https://doi.org/10.1109/TPAMI.2021.3065601).\n- Marcinkevičs, R., & Vogt, J. E. (2021, January). Interpretable models for Granger causality using self-explaining neural networks. In International Conference on Learning Representations.\n- Simon, N., & Tibshirani, R. (2012). Standardization and the group lasso penalty. Statistica Sinica, 22(3), 983–1001. [https://doi.org/10.5705/ss.2011.075](https://doi.org/10.5705/ss.2011.075)\n- Beck, M., Pöppel, K., Spanring, M., Auer, A., Prudnikova, O., Kopp, M., Klambauer, G., Brandstetter, J., & Hochreiter, S. (2024). xLSTM: Extended long short-term memory. Advances in Neural Information Processing Systems, 37, 107547-107603."}, {"review_id": "YXaWo17lOh", "reviewer": "Reviewer_dBaa", "summary": "This paper introduces GC-xLSTM, a model to discover Granger causal relations from complex time series data, with a focus on long-range dependencies.\nThe method is based on a three step procedure. First, a sparse feature selector projects the input variables onto an embedding space using a sparse projection, that enables understanding what variables are relevant for predicting the future for each variable. Second, this time series of embeddings is fed to a xLSTM model trained to predict the next token. Finally, the coefficients of the sparse projection can be analyzed to reveal Granger causality. The whole method is trained end-to-end using a joint optimization technique that simulateously learns the parameters of the model and enforces sparsity. \nThe authors evaluate their method of diverse time series causality tasks such as Lorenz96 and fMRI datasets, showing it outperforms previous baselines.", "questions": "- I would like the experiments to demonstrate the added value of the optimization strategy proposed by the authors. Could the authors have an ablation of their model where they use common regularization on W and $\\alpha$ and binarize entries using a threshold $\\tau$ ? In my sense, this would validate the added value of the optimization strategy.\n- Could the authors demonstrate more directly the impact of modeling long term dependencies in practice ? \n- Similarly, the same method could be applied to any auto-regressive model. Could the authors show that using xLSTM rather than another autoregressive approach is beneficial ?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "- The paper tackles an important problem of inferring causality from time series. It is well written and proposes an elegant solution to the problem.\n- The authors propose an effective optimization scheme to jointly train their model while enforcing strict sparsity in the coefficients.\n- Despite the focus on long range  dependencies that may not be captured by other models, I could not find much experimental evidence for this advantage in the paper.\n- The theoretical section 3.3 is not really a theoretical section but rather a high-level motivation for their architecture. Although obvious from the architecture, the authors could consider formalizing the Granger causality inferred by their model from the sparse coefficients.\n- The long term dependency feature is directly inherited by using xLSTM and does not represent a major contribution.", "was_revised": true, "final_justification": "The reviewers have addressed my concerns", "rebuttal": "Dear Reviewer dBaa,\n\nThank you for recognizing the clear description of the novel optimization scheme and comprehensive evaluation on a diverse set of benchmarks.\n\nWe gladly clarify the raised comments and questions:\n\n**Q1:** We agree that this is an important ablation and since added it to the paper for the datasets Lorenz-96 and fMRI. Specifically, we employ the baseline of Group Lasso (Simon and Tibshirani, 2012) instead of our novel procedure for optimizing for strict input sparsity. We can see from (base) -> (ablated) that our strong results indeed stem from the contribution of the novel optimization procedure. It provides substantial and consistent improvements over the prior Group Lasso procedure.\n\nResults on Lorenz-96 ($F=40$):\n\n| Model | Balanced Accuracy |\n|------------------------------|-----------------------------------|\n| (base) xLSTM + Joint Optimization   | 96.6% (±0.3) |\n| (ablated) xLSTM + Group Lasso | 73.0% (±4.6) |\n\nResults on fMRI:\n\n| Model | Balanced Accuracy |\n|------------------------------|-------------------------|\n| (base) xLSTM + Joint Optimization   | 73.3% (±3.0) |\n| (ablated) xLSTM + Group Lasso | 65.4% (±2.0) |\n\n**Q2:** The motivation for employing xLSTM-based models is more than just capturing long-term dependencies, and also their generally improved modeling accuracy while still limiting the compute necessary (see, e.g., Beck et al., 2024). This is also confirmed empirically by the effectiveness of GC-xLSTM. Specifically, for the Company Fundamentals dataset, the model uses 40 time steps (10 years) instead of 10 steps (2.5 years), which we deem as already a long-term span in (Granger) causal discovery. See also the next paragraph for empirical proof of the benefit of using xLSTM modules over the classic LSTMs.\n\n**Q3:** To specifically show the benefit of this improved modeling capability, we ablated GC-xLSTM by replacing the xLSTM with a classic LSTM. The following table shows the results of replacing the xLSTM block in the architecture with a standard LSTM layer, keeping the rest of the architecture unchanged. We can clearly see from (base) -> (ablated), that the performance drops substantially without the modeling capabilities of the xLSTM blocks, especially in the case of the real-world data set of fMRI.\n\nResults on Lorenz-96 ($F=40$):\n\n| Model | Balanced Accuracy |\n|------------------------------|-----------------------------------|\n| (base) xLSTM + Joint Optimization   | 96.6% (±0.3) |\n| (ablated) LSTM + Joint Optimization    | 93.0% (±0.3) |\n\nResults on fMRI:\n\n| Model | Balanced Accuracy |\n|------------------------------|-------------------------|\n| (base) xLSTM + Joint Optimization   | 73.3% (±3.0) |\n| (ablated) LSTM + Joint Optimization    | 62.8% (±2.0) |\n\nThank you again for your thoughtful review. We hope it answered all outstanding questions. Specifically, having clarified the questions regarding the long-term dependencies, as well as providing the suggested ablation study, we would greatly appreciate it if you considered raising your score to reflect this.\n\n\nBest\n\nThe Authors\n\n**References:**\n- Simon, N., & Tibshirani, R. (2012). Standardization and the group lasso penalty. Statistica Sinica, 22(3), 983–1001. [https://doi.org/10.5705/ss.2011.075](https://doi.org/10.5705/ss.2011.075)"}, {"review_id": "6ces3nfKts", "reviewer": "Reviewer_TkdF", "summary": "This paper improves the detection of Granger causality using recent advances in deep learning for time series. In particular, it uses xLSTMs, which have demonstrated the ability to track longer-horizon relationships than traditional LSTMs. The paper introduces a loss function consisting of a sparsity-promoting feature selection and the xLSTM reconstruction. The new architecture is tested against a variety of GC methods on real and simulated data, where the method demonstrates good performance.", "questions": "1. What is the effect of the xLSTM vs. the new sparsity penalty?\n\n2. Do you have any practical guidance on how to choose $\\lambda$? This would greatly improve the practicality of the method, as it seems quite sensitive.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\n\n**Clarity** The paper is very well-written and easy to understand.\n\n**Performance** The proposed method performs better than baselines, which are largely state-of-the-art. This is occasionally by somewhat large margins in the case of cLSTMs and cMLPs.\n\n## Weaknesses\n\n**Theoretical Analysis** I don't find the theoretical analysis particularly convincing. In particular, the point of the paper is that cLSTMs/cMLPs/etc. are not sufficient for Granger causality in complex settings, but unless I'm mistaken, the theoretical analysis applies equally well to those other architectures.\n\n**AUROC Metric** GC-xLSTM is never the best model in terms of the reported AUROC, but I'm not super sure this is a meaningful metric, anyways. If I understand correctly, it is computed by sweeping over $\\lambda$ -- but because of the group LASSO procedure done here, it's not obvious that $\\lambda = 5$ for xLSTM is similar to $\\lambda = 5$ in some other model.\n\n**Lack of Ablation** This paper contributes both xLSTM-based GC, and a novel group LASSO-based sparsity penalty -- but these are not properly ablated. Thus, we cannot properly understand to what extent performance gains are due to xLSTMs, due to the new sparsity penalty, or both.\n\n**Novelty** The use of xLSTMs rather than LSTMs is practically relevant and useful for the community, but is not significantly novel.\n\n## Minor Typos\n- Figure 6 Caption: \"extracs\" should be \"extracts\".\n- Line 313, Left Column: \"effectively inflating W it to rank\" shouldn't have \"it\".", "was_revised": true, "final_justification": "I still believe the use of xLSTMs is not particularly novel, and don't think the theoretical results are very strong. However, the authors have effectively addressed my concerns about ablation studies, which have further clarified that the optimization method proposed is actually a rather large component of the empirical success. I therefore raise my score, with the suggestion that the authors further emphasize the optimization algorithm over the use of xLSTMs in a camera ready version.", "rebuttal": "Dear Reviewer TkdF,\n\nThank you for recognizing the work as well-written, as well as the diverse evaluation and superior performance.\n\nWe gladly clarify the raised comments and questions:\n\n**W1:** One *strength* of the theoretical analysis is that parts of it, indeed, apply to other works as well. You are correct in that models such as cLSTM would also, in theory, be able to learn GC relationships, which we never question. We merely posit that due to the increased modeling capabilities of xLSTMs, more complex relationships can be learned (Beck et al., 2024). We validate this hypothesis with empirical results. We also note that a major contribution besides the xLSTM is the novel optimization scheme, going beyond cLSTM etc.\n\n**W2:** Comparing very different algorithms by “integrating out” their hyperparameter choices, such as the sparsity penalty $\\lambda$, is, indeed, challenging. They are comparable if the range of HPs each is complete, i.e., that the range covers both extremes where the behaviour saturates. This is the case for our AUROC computation. Also, note that the scores for cLSTM and cMLP were obtained analogously to our work. However, particularly when omitting this metric, GC-xLSTM clearly outperforms all baselines (cf. Tab. 1 & 2).\n\n**W3/Q1:** We agree that an ablation study to more clearly isolate the empirical contribution of these two components would be very valuable. We thus perform an ablation study of (a) the use of xLSTMs and (b) the joint optimisation strategy. The results are provided for two datasets and shown below.\n\nFirst, we replace the xLSTM block in the architecture with a standard LSTM layer, keeping the rest of the architecture unchanged. We can clearly see from (base) -> (a), that the performance drops substantially without the modeling capabilities of the xLSTM blocks.\n\nSecond, we employ the baseline of Group Lasso (Simon and Tibshirani, 2012) instead of our novel procedure for optimizing for strict input sparsity. Again, we can see from (base) -> (b) that our strong results indeed stem from the contribution of the new optimization procedure.\n\nResults on Lorenz-96 ($F=40$):\n\n| Model | Balanced Accuracy |\n|------------------------------|-----------------------------------|\n| (base) xLSTM + Joint Optimization   | 96.6% (±0.3) |\n| (a) LSTM + Joint Optimization    | 93.0% (±0.3) |\n| (b) xLSTM + Group Lasso | 75.0% (±4.0) |\n\nResults on fMRI:\n\n| Model | Balanced Accuracy |\n|------------------------------|-------------------------|\n| (base) xLSTM + Joint Optimization   | 73.3% (±3.0) |\n| (a) LSTM + Joint Optimization    | 62.8% (±2.0) |\n| (b) xLSTM + Group Lasso | 65.0% (±2.0) |\n\n**W4:** Beyond the innovative and effective use of xLSTMs, we also propose a novel optimization procedure (Sec. 3.2, Alg. 1, Code), which we discuss in detail and validate empirically.\n\n**Minor typos:** Thank you, they are now fixed.\n\n**Q2:** Choosing hyperparameters is a common challenge in machine learning. However, $\\lambda$ is rather well-behaved in two regards. Firstly, due to the adaptive joint optimization of both the sparsity projection and the forecast xLSTM, in many cases, results are rather stable across choices of $\\lambda$. This really shows with the rather high AUROC scores of 88.0 and 99.3, which are computed over sweeps of $\\lambda$ (cf. ll. 244ff). Secondly, when $\\lambda$ does have a large effect, we can often visually inspect the resulting graph and coordinate with domain experts to determine the desired level of sparsity. As Fig. 4 shows, choosing $\\lambda$ in this setting is not about model performance, but instead about the desired connectedness of the resulting graph, and therefore, an inherent user preference much like the threshold in other Granger causal detection methods.\n\nThank you again for your thoughtful review. We hope it answered all outstanding questions. Specifically, having clarified the questions regarding the theory, the AUROC metric, and the hyperparameter choice, as well as providing the suggested ablation study, we would greatly appreciate it if you considered raising your score to reflect this.\n\nBest\n\nThe Authors\n\n**References:**\n- Maximilian Beck, Korbinian Pöppel, Markus Spanring, Andreas Auer, Oleksandra Prudnikova, Michael K. Kopp, Günter Klambauer, Johannes Brandstetter, and Sepp Hochreiter. xLSTM: Extended Long Short-Term Memory. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, November 2024.\n- Simon, N., & Tibshirani, R. (2012). Standardization and the group lasso penalty. Statistica Sinica, 22(3), 983–1001. [https://doi.org/10.5705/ss.2011.075](https://doi.org/10.5705/ss.2011.075)"}, {"review_id": "IgHbXFJnVl", "reviewer": "Reviewer_K18p", "summary": "This paper develops Granger causal xLSTMs that use xLSTMS to learn granger causal relations from complex time series data with long time dependencies. The paper introduces a new model (GC-LSTM), formulates an optimization process for strict sparsity and provides evaluation of the model compared to benchmark models using six datasets.", "questions": "(Please see additional discussion related to these questions 1a and 1b in the Strengths & Weaknesses section).\n1a. Why are other methods not sufficient for this problem?\n1b. What is the significance of the GC-xLSTM model? Put another way, what is the particular advantage of this model? Does it better find long temporal dependencies? Can the authors provide evidence of this, and show that it has some major improvement over prior methods (as, with the current results it seems other methods actually work well.)\n\n2. Appendix C – how many samples are in each dataset? The number of variables and timesteps are listed, but not the total number of samples.\n\n3. I have additional concerns about scale, since this is a common issue with neural granger causality. Appendix F shows the scale with the variates V but how does the model scale over the number of timesteps T or the number of samples in the dataset?", "limitations": "Yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "This paper has good quality and clarity: it is organized logically, explains the background extremely well, and describes the framework in a straightforward manner. The formulations seem technically sound and the model architecture and training seem logical. In terms of originality, the paper does propose a new model type of granger causal xLSTMs and includes an additional optimization and loss function focused on sparsity. The results show that the model is able to robustly find temporal dependencies from diverse data sources.\n\nHowever, the point that is unclear for me is the significance: Why do we need this new model type? The introduction does a good job at defining granger causality, framing recent renewed interest in recurrent models and xLSTMs, and describing the combination of granger causality and xLSTMs, but what is missing here is a discussion about *why* these methods specifically should be combined, and why previous methods are not sufficient for learning long range dependencies in time series (and thereby justifying the need for this new model type).\n\nThis problem is further reiterated in the results section. I appreciate that the results section includes a wide range of results, including an ablation study, compares to many benchmarks, uses multiple datasets and provides some intuitive figures (e.g., Figure 4 in particular is very nice) showing the ability of the model to find underlying dependencies. However, these results do not convince me that this *specific* GC-xLSTM model is needed to solve the problem. For example, in Table 1 there is not a huge improvement compared to GVAR (and in some cases GVAR actually does better). Moreover, the figures (e.g., figures 3, 4, 5) do not compare with other models, making it unclear if this model learns dependencies any better than other methods.\n\nOverall, I am willing to reevaluate if the authors can provide more compelling evidence about (a) why other methods are not sufficient and (b) show a major performance improvement with this model that justifies its need. For instance, if the authors are trying to make the claim that this method is better at finding long range dependencies between time series, from the current results it is not clear that the other methods do poorly for such dependencies – addition of other figures that might show much better found dependencies for the GC-xLSTM model compared to others would be helpful (e.g., like figures 3, 4 or 5), or additional performance tables (like Table 1 or 2) on the other datasets that show a large performance improvement across a range of tasks.", "was_revised": true, "final_justification": "The authors have addressed my core concerns in their rebuttal relating to significance and scalability. Therefore, I have raised my score from a 3 to a 4.", "rebuttal": "Dear Reviewer K18p,\n\nThank you for recognizing the work as clearly written and original.\n\nWe gladly clarify the raised comments and questions:\n\n**Q1a:** Granger causality is not yet solved. For instance, looking at Tab. 2, we see that few methods even surpass just 70%, and all fall far short of the desired detection accuracies of 90+%. GC-xLSTM contributes toward reaching this by integrating xLSTM modules into a novel optimization procedure specifically designed to discover sparse inputs.\n\n**Q1b:** GC-xLSTM improves the detection accuracy of Granger causal graphs, as the varied empirical results shown specifically in Tab. 1 and 2 confirm. However, this does not yet show which of the two major contributions, (a) integrating xLSTMs and (b) the joint optimization procedure for sparse feature selection, contributes how much. We thus perform a respective ablation study on two datasets. Results are shown below.\n\nFirst, we replace the xLSTM block in the architecture with a standard LSTM layer, keeping the rest of the architecture unchanged. We can clearly see from (base) -> (a), that the performance drops substantially without the modeling capabilities of the xLSTM blocks.\n\nSecond, we employ the baseline of Group Lasso (Simon and Tibshirani, 2012) instead of our novel procedure for optimizing for strict input sparsity. Again, we can see from (base) -> (b) that our strong results indeed stem from the contribution of the new optimization procedure. This new procedure is a contribution beyond GC-xLSTM and could, in the future, be used in other models too.\n\nResults on Lorenz-96 ($F=40$):\n\n| Model | Balanced Accuracy |\n|------------------------------|-----------------------------------|\n| (base) xLSTM + Joint Optimization   | 96.6% (±0.3) |\n| (a) LSTM + Joint Optimization    | 93.0% (±0.3) |\n| (b) xLSTM + Group Lasso | 75.0% (±4.0) |\n\nResults on fMRI:\n\n| Model | Balanced Accuracy |\n|------------------------------|-------------------------|\n| (base) xLSTM + Joint Optimization   | 73.3% (±3.0) |\n| (a) LSTM + Joint Optimization    | 62.8% (±2.0) |\n| (b) xLSTM + Group Lasso | 65.0% (±2.0) |\n\n**Q2:** An overview would help maintain a more complete image. We thus added the following table to Appendix C, amending the missing numbers:\n\n| Name | Origin | Type | Variates $V$ | Time Steps $T$ | Samples $N$ |\n|------|--------|------|---|---|---|\n| Lorenz-96 | Karimi and Paul [2010] | Simulated | 20 | 500 | 1 |\n| fMRI | Smith et al. [2011] | Real-world | 15 | 200 | 1 |\n| Moléne | Girault [2015] | Real-world | 32 | 744 | 1 |\n| Human MoCap (Run) | CMU [2009] | Real-world | 54 | 1232 | 61 |\n| Human MoCap (Salsa) | CMU [2009] | Real-world | 54 | 4136 | 30 |\n| Company Fundamentals | Divo et al. [2025] | Real-world | 20 | 56 | 2527 |\n| VAR | Karimi and Paul [2010] | Simulated | 10/20 | 250/500/1000 | 1 |\n\n**Q3:** We agree that scaling behavior is one of the key questions for any Granger causality method. Usually, though, the main challenge lies in scaling the number of variates $V$ (cf. ll. 97ff). We, therefore, specifically analyze it in Appendix F and already successfully apply the flexible GC-xLSTM framework to time series lengths between $T=56$ and $T=4136$ (see table directly above). Lastly, we also consider a wide range of samples from $N=1$ to $N=2527$. Note that here larger numbers of variates $N$ can actually be beneficial, since it reduces the likelihood of overfitting on smaller datasets.\n\nThank you again for your thoughtful review. We hope it answered all outstanding questions. Specifically, having provided additional results on the main concern (significance), we would greatly appreciate it if you considered raising your score to reflect this.\n\nBest\n\nThe Authors\n\n**References:**\n- Simon, N., & Tibshirani, R. (2012). Standardization and the group lasso penalty. Statistica Sinica, 22(3), 983–1001. [https://doi.org/10.5705/ss.2011.075](https://doi.org/10.5705/ss.2011.075)"}]

Author rebuttal:

Meta-review: This work introduces a model to discover Granger causal relations from complex time series data, with a focus on long-range dependencies. The method is based on a three-step procedure. First, a sparse feature selector projects the input variables onto an embedding space using a sparse projection, which enables understanding what variables are relevant for predicting the future for each variable. Second, this time series of embeddings is fed to an xLSTM model trained to predict the next token. Finally, the coefficients of the sparse projection can be analyzed to reveal Granger causality. The whole method is trained end-to-end using a joint optimization technique that simultaneously learns the parameters of the model and enforces sparsity. The authors evaluate their method of diverse time series causality tasks such as Lorenz96 and fMRI datasets, showing it outperforms previous baselines.

All reviewers agree that this work is worth being accepted at NeurIPS, and I recommend accepting this paper, under the condition that the authors highlight better in camera camera-ready version what piece of their framework is driving the improvements they see e.g. sparsity, neural network architecture etc., to make the contribution clear.
Decision: Accept (poster)