UnCLe: Towards Scalable Dynamic Causal Discovery in Non-linear Temporal Systems
Authors: Tingzhu Bi
Yicheng Pan
Xinrui Jiang
Huize Sun
Meng Ma
Ping Wang
Venue: NeurIPS 2025 poster ()
Keywords: causal discovery
dynamic causality
time series
representation learning
Primary area: machine_learning_for_sciences
TL;DR: We propose UnCLe, a deep learning framework that uses temporal perturbations to uncover dynamic, time-varying causal graphs from observational time series.

Abstract: Uncovering cause-effect relationships from observational time series is fundamental to understanding complex systems. While many methods infer static causal graphs, real-world systems often exhibit *dynamic causality*—where relationships evolve over time. Accurately capturing these temporal dynamics requires time-resolved causal graphs. We propose UnCLe, a novel deep learning method for scalable dynamic causal discovery. UnCLe employs a pair of Uncoupler and Recoupler networks to disentangle input time series into semantic representations and learns inter-variable dependencies via auto-regressive Dependency Matrices. It estimates dynamic causal influences by analyzing datapoint-wise prediction errors induced by temporal perturbations. Extensive experiments demonstrate that UnCLe not only outperforms state-of-the-art baselines on static causal discovery benchmarks but, more importantly, exhibits a unique capability to accurately capture and represent evolving temporal causality in both synthetic and real-world dynamic systems (e.g., human motion). UnCLe offers a promising approach for revealing the underlying, time-varying mechanisms of complex phenomena.

UnCLe: Towards Scalable Dynamic Causal Discovery in Non-linear Temporal Systems
Tingzhu Bi, Yicheng Pan, Xinrui Jiang, Huize Sun, Meng Ma, Ping Wang
Peking University
bitingzhudstu. pku. edu. en, faqpye, jxp edu. en sunhuizedstu, pku. edu. cn, {mameng (mameng.pwang)d0p edu.
Code & Datasets: https://githin www.//tegeesuutoaoaalcacaaaaaaaa
Abstract
Uncovering cause-effect relationships from observational time series is fundamental understanding complex systems. While many methods infer static causal graphs, real-world systems often exhibit dynamic causality-wher relationships evolve over time. Accurately capturing these temporal dynamics requires vime-resolved causal graphs. We propose UnCLe, novel deep learning method for scalable dynamic causal discovery. UnCLe employs pair of Uncoupler and Recoupler networks to disentangle input time series into semantic representations and learns inter-variable dependencies via auto-regressive Dependency Matrices. It estimates dynamic causal influences by analyzing datapoint-wist prediction errors induced by temporal perturbations. Extensive experiments demonstrate that UnCLe not only outperforms state-of-the-at baselines on static causal discovery benchmarks but, more importantly, exhibits unique capability to accurately capture and represent evolving temporal causality in both synthetic and real-world dynamic systems (e.g.. human motion). UnCLe offers a promising approach for revealing the underlying. time-varying mechanisms of complex phenomena.
Introduction
Understanding the intricate web of cause-effect relationships is fundamental to unraveling the mecha- nisms of real-world complex systems, from climate patterns and biological processes to economic or network fluctuations and human biomechanics [20, 28]. A critical, yet often overlooked, aspect is that these systems are inherently dynamic, with causal influences frequently evolving over time due to changing internal states or external conditions. For instance, predator-prey dynamics can shift seasonally, gene regulatory networks can alter during developmental stages, and the biomechanical interplay between human joints changes distinetly across different phases of motion. Accurately cap- turing these dynamic causal structures through rmme-resolved causal graphs is therefore essential for achieving a deeper, more veridical understanding, enabling more precise predictions and potentially more effective interventions. The practical success of specialized dynamic models in high-stakee domains, such as real-time fault diagnosis data centers [2]. underscores this urgent need. However, the predominant paradigm in temporal causal discovery has largely focused inferring static causal graphs, which represent an aggregated or aime-averaged view of dependencies, thereby obscuring the rich, evolving nature of causality in many real-world phenomena.
While foundational approaches to temporal causal discovery, such as Granger causality [7] and its various linear (e.g., VAR-based and nonlinear extensions (e.g.. constraint-based methods like PCMCI [22], or early neural network adaptations [27, 17]), have laid crucial groundwork for inferring causal links from time series data, they often not inherently designed to explicitly model or represent how these causal relationships themselves change over time. Some methods might capture
"Corresponding author: mameng @pku.cdu.cn
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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lagged effects or offer global summary graph, but the challenge of constructing and interpreting time- resolved causal graphs- -hhee the set of active causal edges can vary from one time point or interval to another-remaine a significant hurdle. This limitation hinders our ability to fully comprehend systems where causal laws are not fixed but adapt, shift, or switch, which is characteristic of many complex adaptive systems.
To bridge this critical gap, we propose UnCLe (UnCoupLing causality), a novel deep learning framework specifically engineered for the scalable discovery and representation of dynamic causal graphs from observational time series. UnCLe's core innovation lies in its ability to first disentangle complex, multivariate time series into meaningful semantic channels using pair of parameter-shad -haarig Uncoupler and Recoupler networks, and then to model evolving inter-variable dependencies within these channels via auto-regressive Dependency Matrices. Crucially, UnCLe infers time-resolved causal influences by meticulously analyzing datapoint-wis prediction errors that are induced by targeted temporal perturbations of individual series. This mechanism allows UnCLe to quantify how the predictive relationship between variables changes at different points in time, thus constructing dynamic causal narrative. Furthermore, UnCLe is designed with scalability in mind, enabling its application to large-scale, non-linear systems commonly encountered in real-world applications.
The main contributions of this paper are:
We propose UnCLe, a novel and scalable deep learning method for dynamic temporal causal discovery, capable of generating and representing time-resolved causal graphs that capture evolving cause-effect relationships.
We introduce a methodology that combines semantic disentanglement of time series with perturbation-baut based, datapoint-wist eπror analysis to effectively identify and quantify dynamic causal influences.
We demonstrate UnCLe's superior ability uncover and track evolving causal structures through extensive experiments on synthetic datasets with known dynamic ground truths (e.g.. time-varying SEMs) and challenging real-world systems, notably human motion capture (MoCap) data, where UnCLe provides interpretable, phase-specific biomechanical insights.
We show that UnCL also achieves competitive or state-of-the-art performance on standard static causal discovery benchmarks, highlighting its versatility and robustness.
By offering principled and effective approach to dynamic causal discovery, UnCLe aims to provide a more powerful lens for understanding the complex, ever-changing mechanisms that govern the world around us.
2 Background and Related Work
Notations A dynamic causal graph is defined as time-varying graph G = { for each timestep E {I....,T) Th, where V = {oe | of E V} represents the set of vertices at time t, corresponding time series xf observed at time t, and E {(rt, of) of V*, uz E V*) represents the set of directed edges time t. An edge (rf, of) denotes causal-effect relationship. where of is the cause variable and of is the effect variable at time t. A static causal graph [20] is defined as a time-invariant graph g (V,E) E} whose variables and cause-effect relationship remain constant over time.
Granger causality Granger causality [7] is a widely used statistical framework for defining causality based on predictive relationships. It is grounded in the intuition that time series æ4 can be considered a cause of another time series x, if the inclusion of x; 's past values improves the prediction of x; 's future values. Formally, the generalized form of Granger causality can be expressed as follows [24]:
Fi.t h, (#), <t.
LN,Cs) + 6i
where h; is a prediction function that maps the past values of all N time series to the current value of series wi and ei,r represents the prediction error.
Traditional Granger causal discovery methods typically employ statistical autoregressive (AR) models for h and use statistical significance tests on the prediction error to infer causal relationships between

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time series. In contrast, recent approaches leverage neural networks to model hy and determine causal relationships through various mechanisms.
Neural Granger Causality Neural Granger causality methods leverage a variety of neural net-
work architectures their backbone networks, including MLPs [27, 291, RNNs [10], LSTMs [27], CNNs [17], and GNNs [15, 51, with some works successfully employing TCN-based autoencoders for representation learning in specific causality-driven applications like root cause diagnosis [2]. Additionally, design concepts such as attention mechanisms [17], variational autoencoders [12], self-explaining neural networks [16], and inductive modeling [15] are actively incorporated into these methods. The standard procedure for neural Granger causality analysis involves training prediction models using neural networks and then inferring the causal structure from the learned models through various techniques.
For instance, NeuralGC [27] and GVAR [16] analyze the weights of specific network layers to interpret the influence relationships between variables. In contrast, TCDF [17] identifies causal dependencies using attention scores and quantifies the predictive contribution of variables by computing permutation importance [3]. However, NeuralGC, GVAR, and TCDF face significant scalability challenges on large-scale datasets due their component-wise design, which lacks parameter sharing. This results in O(N2) parameters to train as the number of variables increases.
To address the scalability problem, both JRNGC [29] and CUTS+ [5] utilize parameter sharing, making them more suitable for large-scale datasets. JRNGC incorporates an input-output Jacobian regularizer into the training objective to learn Granger causality, while CUTS+ enhances scalability on high-dimensional temporal data by splitting time series into groups and applying a coarse-to-fin filtering strategy.
While some methods, such as NeuralGC, JRNGC, and TCDF, support time-lag recognition, only GVAR is capable of generating dynamic causal graphs. Furthermore, to the best of our knowledge, no existing method has been rigorously evaluated on dynamic causal datasets to assess ability to identify causal evolutions over time.
3 Methodology
We introduce UnCLe, a scalable method for dynamic causal discovery rooted in the principles of neural Granger causality. The overall framework is depicted in Figure 1 and comprises two primary phases:
1. Training Phase. Given an input multivariate time series dataset R. (represented as green blocks in Figure 1, denoting data for N variables over T timesteps), UnCLe first trains core architecture. This architecture consists of pair of parameter-sharing Uncoupler and Recoupler networks, along with a set of auto-regressive Dependency Matrices (v) The Uncoupler transforms the input series æ into multi-channel semantic representations z E xxCx (visualized as stacked, colorful blocks for C semantic channels). This transformation is learned through a reconstruction task, where the Recoupler aims to reconstruct the original series ã from 2. Concurrently, the Dependency Matrices T are optimized via a prediction task to capture inter-variable dependencies within each semantic channel, forecasting future latent representations å which are then mapped back to the original space as t.
2. Post-hoc Analysis Phase. Subsequent to training, UnCLe infers causal relationships. For dynamic causal discovery, individual time series æ; (denoting the i-th variable's series) within the input dataset are chronologically perturbed (e.g., via permutation, resulting in x v shown red blocks) to disrupt their temporal structure and thereby diminish their predictive utility. The resulting datapoint-wis increase in prediction error for other series x; is then quantified as the strength of the dynamic causal link from æ, to 2;, forming the dynamic causal graph Pant Furthermore, for static causal discovery, the learned weights of the Dependency Matrices V are aggregated (e.g., via average pooling) to derive a summary static causal graph AAss.
The subsequent subsections provide detailed expositions of the UnCLe model architecture and the causal inference procedures.
3

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original space
representalin space N*TC
origindi space
NT
N
Original
Reconstruction
Traising Roskoc Anahaia
(iaaed vanabina
Uncouleer
Recoupler
Orignnal
Prediction
V
Depandency
Temporol Perturbation
Causel
Peskin
Ase
Axx
Discovery Depandency Aggragatio
Figure The UnCLe framework. Training involves reconstruction and prediction using Uncou- pler/Recoupler and Dependency Matrices (V) Post-hoc analysis uses temporal perturbation for dynamic graphs (Gtet) and aggregation of v for static graphs (GASE).
3.1 Model Architecture
Uncoupler and Recoupler Networks The Uncoupler and Recoupler form the backbone of UnCLe's representation learning. functioning akin to parameter-sharing Temporal Convolutional Network (TCN) autoencoder [1]. Their primary role is to model intra-variable temporal dynamics and disentangle the input time series E RNxT into meaningful latent representations. By sharing parameters across all N variables, this design significantly enhances learning efficiency, model stability, and the quality of learned representations, especially for nigh-dimensional data.
The Uncoupler, denoted as TCNune (Us)), maps each univariate time series H; E RT (the i-th row of æ)) into C'-chaneel latent sequence E T Collectively, for all variables, this yields E X The Recoupler, TCN @Rec). then aims to reconstruct the original series 2; E IR? from its corresponding latent sequence 2;. This reconstruction process is formalized as:
zi TCN (æ;; (Unc), 2; TCN (2;; (Ree)
(1)
where d'unc and d' represent the learnable parameters of the Uncoupler and Recoupler, respectively, The objective for the reconstruction task is to minimize the Mcan Squared Error (MSE) loss:
N
Re (oUne, ÖRee)
T - -
(2)
NT
The TCN architecture, characterized by stacked dilated causal convolution blocks [18], ensures that information processing is strictly temporal (no leakage from future past), crucial property for subsequent causal discovery. Furthermore, the parallelizable nature of TCN computations contributes UnCLe's efficiency on large- scale datasets.
Auto-regressive Dependency Matrices To model inter- variable dependencies, UnCLe introduces a set of C' lightweight Dependency Matrices, = ( ...., yC), where each & e R X These matrices operate the disentangled latent representations Specifically, for each semantic channel c, the latent representation at the next timestep, 25, E R is predicted from the current latent representations across all variables in that channel, 2,, R.V. _ ( I
perform auto-regressive prediction.
(IZ 2f,,)
where z denotes the N-dimensional vector of latent features for channel at time f, and a denotes the same activation function as TCNs. This linear update motivated by the principle that a suitable coordinate transformation, learned here by our Uncoupler, can approximate complex non-lineat dynamics with a linear system [26, 4J.
The predicted latent sequences & = [22 ...... : are then fed into the (shared) Recoupler network to generate predictions in the original data space:
t TCN (( (( 2. 2: &Ree)
(4)
The prediction loss Lred is also an MSE loss, calculated between the predicted values t and the true future values 32,4
N T
DPrea(Quns &Rees 4
(b,,41 xr,441²²
(5)
N(T

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Regularization To mitigate overfitting and discourage the discovery of overly complex causal structures or spurious vater-variable relationships. UnCLe incorporates several regularization tech- niques. First, L.1 regularization is applied to the Dependency Matrices v to promote sparsity in the learned inter-variable connections:
L = A
(6)
where Å1 is the LI regularization hyperparameter. Additionally, to enhance the robustness of the feature disentanglement process within the TCNs, dropout with rate of 0.2 applied during the training of the Uncoupler and Recoupler networks.
Overall Training Objective and Procedure UnCLe trained two stages. the pretraining stage, the model focuses on representation learning by optimizing only the reconstruction loss Recom-- This stage trains UUn and ΦRee. providing strong initialization for the subsequent phase. In the full model training stage, all components, including the Dependency Matrices V, are trained jointly by minimizing a composite loss function:
Cotal CReson aCCrded + LLI
where is a hyperparameter balancing the prediction task's contribution. This joint optimization allows the model to simultancously learn to represent the data, predict its future, and identify underlying dependencies.
3.2 Post-hoc Causal Discovery
Once the UnCLe model trained, causal relationships are inferred post-hoc analysis phase using two distinct approaches.
Perturbation-base Dynamic Granger Causality To uncover dynamic causal influences, UnCLe employs temporal perturbation strategy. The core idea is that because the trained model has learned an approximation of the data's ' causal generative mechanism, disrupting the temporal structure of a true cause z, will violate the learned dynamics and significantly impair the model's ability predict its effect æ;. Formally, let IR N be the original multivariate time series dataset. We denote by x V the dataset where the j-th time series, Zy, has been perturbed by random permutation of its temporal values. This permutation preserves the marginal distribution and statistical properties of 2t but destroys its original sequential order, thus nullifying its valid predictive information for other series that depend on its specific temporal evolution.
Let f(-) represent the trained UnCLe prediction model (Equations 3-4). The prediction for Ti, using the original dataset is tt, The original prediction error for æ can be defined, for instance, as the squared error:
Gift = (bi.t x2,1)²
When my is perturbed create V the model yields new prediction a.f for wis The prediction eтror under this perturbation is:
c = (a)) xz,0)²
The increase in prediction error, or error gain, quantifies the causal influence of æj on x; at time t:
Ae max(0, e,2
// - -ei.)
(10)
This value, A t represents the strength of the causal link from Ty to as specifically at time I, [.Pett of the dynamic causal graph forming an element of the time-resolved adjacency matrix A gPert UnCLe computes these pairwise error gains for all variable pairs and timesteps. Since the errors are computed at each timestep, this perturbation- based approach inherently captures dynamic causality, allowing causal relationships to evolve over time. For static summary, these dynamic strengths can be aggregated across the time axis (e.g., by averaging or summing Ae ' over l). The detailed algorithm for dynamic causal discovery via temporal perturbation is listed as Algorithm 1 in Appendix L.. The batch processing these computations significantly enhances the efficiency of the causal discovery process.
5

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Static Causal Graph via Dependency Aggregation A static, or summary, causal graph Ge can also be directly inferred from the learned Dependency Matrices L. The rationale that if variable 3y does not influence æ; in channel the 1.1 regularization (Equation 6) will drive the corresponding coefficient Vi,e towards zero. Conversely, a significant non-zero coefficient suggests dependency. The elements of the multi-channel Dependency Matrices v are thus interpreted as causal strengths.
The aggregated static causal influence from 3y to æt, denoted by Á is obtained by pooling the magnitudes of these coefficients across all C channels. common pooling method is the 1.2-norm (root mean square) of the coefficients:
1 Σ i2
A
(11)
This aggregation yields single N x N adjacency matrix representing the overall static causal structure.
UnCLe thus offers two complementary modes for causal discovery: (P) Temporal Perturbation, which yields dynamic causal graphs and is generally more accurate as it leverages the full model including the learned representations from the Uncoupler/Recoupp Dependency Aggregation, which produces static causal graph more rapidly it directly uses the learned & without further post-hoe predictive analysis.
and the input data characteristics. (A)
4 Experiments
4.1 Experimental Setup
We evaluate UnCLe using various synthetic / real-world datasets from a great variety domains. Apart from other existing datasets, propose NC8 (Non-linear, Constant connections, 8 variables) and ND8 (Non-linear, Dynamic connections, 8 variables) to better evaluate causal discovery methods. The detailed dataset setup is included in Appendix A.
Datasets
Baselines We compare UnCLe against nine baseline methods spanning range of categories, including constraint-baseee score-based, and cutting-edge neural Granger approaches: (i) VAR, the classic Granger causality method [7] based on pairwise VAR F-tests; (ii) PCMCI, a constraint-based approach [22] that uses partial correlation for independence tests; (iii) cMLP [27]. neural Granger causality method that relies on MLP prediction networks and sparse -ndersinnduiingg regularization; (iv) TCDF [17], which interprets attention weights and validates them using permutation importance; (v) GVAR [16], which leverages neural network geeeerteeee dynamic VAR coefficients; (vi) VAR- LiNGAM [9], a method that uses a non-Gaussian structural vector autoregressive model to assess the significance of causal influences; (vii) DYNOTEARS [19], score-based method that minimizes a penalized loss subject to acyclicity constraint; and (viii) CUTS+ [5], a neural Granger causality method that utilizes passing-base graph neural networks and supports high-dimensional data; and (ix) JRNGC [30], which employs input-output Jacobian regularizer learn causality from a single, shared prediction model. Note that GVAR is the only baseline method that can generate dynamic causal graphs.
4.2 Results
We first report the causal discovery accuracy on static graphs. Next, we evaluate dynamic causal discovery performance on two datasets. Finally, we present results on two large-scale real-world transportation datasets. The best accuracies are bolden and the second best are underlined, and "." indicates the running time of the method exceeded the reasonable time limit.
Static: Lorenz 96 Table 1 reports the causal discovery performance of UnCLe and other bascline methods on synthetic datasets. Lorenz96 [14] is a ODE model simulate climate dynamies used by [16, 27, 10, 29, 5]. The system dynamics increasingly chaotic and thus hard to model for higher values of forcing constant F. We design three sets of system configurations of Lorenz96, setting number of variables p {20, 20, 100), timesteps T {250. 250, 500). force F {10, 10, 40} for Lorenz#1, #2, and #3 respectively. On all Lorenz datasets, UnCLe(P) perturbation consistently achieves the highest AUROC and AUPRC scores. The added chaostic strength of Lorenz#2 and large number of variables of Lorenz#3 pose significant challenges to baseline methods. Notably, DYNOTEARS struggles with large-scale datasets, and its DAG constraint conflicts with the ground truth causal structure of the Lorenz system.

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Table 1: Static causal discovery performance comparison synthetic datasets.
Methods
Lorenzal
Lorouz42
Lorenz#3
NC8
FINANCE
AUROCT 1
AUPRC 1
AUROC1
AUPRC
AUROC
AUPRC'1
AUROC 7
AUPRC f
AUROCT
AUPRC
VAR
853:
_709:2.019 485
.295
.798
.142:1/014)
.63 301)
2218:1
1103: 1.0)
PCMCI
833(
(82(4.006)
6670 (27)
2269:
211:21100)
8895 (84(0))0))
.408 1680
(589
005. d. 048
cMLP
4.44)
(7....
-85(=0010)
-814:-0019)
.65 (6.41)
.928 =.000)
.717:20011
.069 -619..110)
GVAR
994
888
.839 027)
6113::0755
(0055)
558 (.015)
(4000
.956 (024))
$31:22044
.999
(1000)
.990 (.000)
TCDF
.727
.615:=.019) (345
225
.885 4.023)
.70
.620 (.05)
509 (.000)
VARLINGAM
354.::060
(32
627 029)
481c22040
.673 (t.08)
.383 (3:4))
.880 : 0600
.586
:.00))
946,
(:040)
.337 (9.19)
DYNOTEARS
544 (4455.055
(315
.546 015)
2240 (133)
.546: 013)
2240:2007
CUTS+
947:
800
894
(04)
220
863
(2772
777
(297
.85
(:033)
370 (.00)
JRNGC
983
(714
807
015)
(44)
612/2:014)
(18
.756 (10)
1622 (00)
.688 1 410)
.7 24 1:99)
UnCLotP)
.99
.996: (0.00)
940
804
(.012))
636 (3001)
.95
(004))
933
UnCLetA)
994 4.007)
962 (6.05))
(2.33)
5331
865 (.024)
(4:03)))
770
972 +.087)
.887 (4.253)
UnCLe
GVAR
1.0
0.50
0.30
Smoothed
Auy
AxY
Axe
Smoothed A.,
mmoothod
0.25
UB ogg poquoous
0.8
0.20
0.20
0.4
0.1
0.15
0.15
0.10
0.0
0.0
0.06
0.06
Timestop
Timestop
Figure 2: The dynamic causal strengths between X₁ and Y; discovered by UnCLe and GVAR.
The NC8 dataset evaluates the ability of methods to uncover long-term nonlinear relationships between variables, and UnCLe delivers the best AUROC and AUPRC scores. On the FINANCE dataset, UnCLe demonstrates the second-best performance.
Statie: NC8 and FINANCE
To evaluate the dynamic causal discovery capability of UnCLe, we construct bivariate time-varying Structural Equation Model (TVSEM) with a total length of T T 2000 time points. The model is defined as:
Dynamic: Time-variant SEM
0.8 if [(t 1)/400] (mod 2) 0
0.2 ir [(1 1)/400] (mod 2) = b. f0. if [ 1)/400 (mod 2) 0
(12)
0.7 if [(1 - 1)/400
(mod 2) = 1
+ Y1 = b, + EY,
(13)
Here, X₂ and Y3 represent the observed variables at time t. The error terms cx2, and cy ~ ((0, 0.1). The model's coefficients 0t and by switch values every 400 time points, creating five segments. This switching pattern is governed by the parity of the segment index [(t 1)/400]. When the index is even (segments 1, 3, 5), (at, b,) = (0.8, 0.1), indicating dominant Y + X causal direction due to the strong influence from Y to X., When the index is odd (segments 2, 4), (az, b.) = (0.2, 0.7), indicating dominant X + Y causal direction due the strong influence from X,-1 to Y,. The dominant causal direction thus alternates between Y + X and X + Y across the five segments.
Fig. 2 illustrates the evolution of dynamic causality between X, and Y, as discovered by UnCLe and GVAR [16]. For better interpretability, the strengths are smoothed using a Gaussian moving average and presented as two lines. The orange and green segments on the timestamp axis divided by red dotted vertical lines indicate which variable is dominant over the other. UnCLe initially identifies Y, as determining X,, flips the causal direction shortly after t 400, and reverts to the another causal direction correctly after each subsequent switch points. This behavior aligns perfectly with the underlying data generation mechanism, demonstrating UnCLe's capability to accurately capture temporal causal dynamics. In contrast, while GVAR generates dynamic causal strength, the perceived dominance between Y, and X, never flips.
Table 2 quantifies the accuracy of dynamic causal discovery accuracy on TVSEM by eval- uating separately on each segments with dif- ferent system settings. Static Best denotes the best possible accuracy by non-changing static causal graph. UnCLe(P) achieved pre- fect estimated of the directions of the two vari- ables.
Table 2: Dynamic causal discovery performance comparison on TVSEM and ND8.
TVSEM
ND8
Methods
AUROCT
AUPRC
AUROC
AUPRC 1
GVAR
0.733 000)
0.400
0.723:1.016
0,220 (.028)
UnCLe(P)
1.000 10000.000 1.000
1000,
L.000
0.921 (.009)
0.633
Static Best
0.467:2 c00)
00300
0.905 000)
0.799 (000
7

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(a) Forward jump motion
(b) The skeleton of sensor positions human joints
(c) Causal influence snapshots human joints for the forward jump motion.
From top to botton at each row: UnCLe, GVAR, JRNGC.
Figure 3: Dynamic causal analysis on a forward jump motion.
Dynamic: ND8 ND8 is much harder dataset compared to TVSEM as it contains non-linear connections, more variables, multiple switches of direction simultaneously. The detailed data generated process of ND could be found in Appendix G. As reported in Table 2, UnCLe achieved better dynamic causal discovert accuracy then GVAR and Static Best.
Dynamic: Human Motion Capture (MoCap) The Motion Capture (MoCap) dataset contains
sensor observations reflecting the sophisticated biomechanics of the human body, which involve dynamic cooperation and interactions across joints, muscles, and bones. This dataset records the 3-axis angles of 31 joints. We selected forward jump representative motion for our study, illustrated in Figure 3(a). The underlying joint skeleton is depicted in Figure 3(b).
To evaluate the ability of different methods to capture evolving causal structures, extracted 6 snapshots of the causal graphs inferred by UnCLe, GVAR, and JRNGC from the forward jump data. For JRNGC, which does not inherently produce dynamic graphs, these snapshots were obtained by training the model on distinct segments of the motion data corresponding to different phases. These 6 snapshots presented chronologically in Figure 3(c), aligned with the three motion phases (Crouch, Flight, Touchdown). and anchored to the skeletal structure for visual interpretation. The causal graphs generated by UnCLe demonstrate comparatively clear interpretability corresponding to the biomechanics of each phase:
Crouch Phase (first two UnCLe snapshots). Focus on the upper body, particularly the coordinated movement of the arms, with some connections to the lower body and strong links to the hip/root joint. This aligns with biomechanical findings that coordinated arm swing is crucial for maximizing jump height by increasing the work and torque produced by the lower extremities. The dense, whole-body body connectivity discovered by UnCLe reflects this principle of synergistic power generation for propulsion [8].
Flight Phase (middle two UnCLe snapshots). Highlights the lower body, characterized by coordinated leg movements, minimal upper body involvement, and weaker connections to the hip/root joint. This aligns with the biomechanical expectation that during mid-flight, with the body's trajectory already determined, the coordination strategy shifts from power generation to in-air balance. The graph correctly becomes sparser, reflecting reduction in active, large-scale interdependencies.
Touchdown Phase (last two UnCLe snapshots). Reveals involvement from both upper and lower body segments with medium-strengthh connections, evidence of ipsilateral coordination, and renewed strong links the hip/root joint. This is consistent with the demands of landing, which requires the entire kinetic chain from the ankle up to the hip and core to work in a coordinated fashion to absorb impact forces and re-stabilize the body. The re-emergence of a complex causal graph mirrors the body's 's need to manage ground reaction forces and dissipate energy across multiple joints [6].
In contrast, the snapshots generated by the baseline methods (GVAR and JRNGC) exhibit more subtle differences between phases. Their respective causal graphs often appear densely interconnected and considerably more challenging to interpret in terms of distinct biomechanical phases.

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To quantify the extent to which these methods recover fundamental anatom- ical connections, Table reports the proportion of missing edges corre- sponding to adjacent joint connections present in the basic skeleton (Fig- ure 3(b)) that were not captured in the inferred causal graph snapshots (averaged across the six snapshots for each method). UnCLe demonstrates a superior ability to preserve these fundamental T-pose connections, in- dicated by lower missing rate. This experiment suggests that dynamic causal discovery algorithms like UnCLe hold significant promise for elu- cidating the mechanisms underlying real-world phenomena and complex systems by providing interpretable, time-evolving causal insights.
Table 3: Missing rate of skeletal connections.
Method
Missing Rate
UnCLe GVAR JRNGC
2.099.019
.622
.600.+000)
Time Efficiency Figure presents a comparative scatter plot of Un- CLe(P) and baseline methods, illustrating their trade-off between causal
1.0
discovery accuracy (AUROC) and computational time on the Lorenz#I I dataset. UnCLe demonstrates compelling balance: it achieves the highest AUROC score while maintaining a competitive execution time. Specifically, UnCLe is notably faster than several complex neural meth- ods such as TCDF and ccore-based methods like Dynotears, and ex- hibits comparable moderately higher computational cost than some traditional or highly optimized approaches like VAR and CUTS+, re- spectively. This positions UnCLe as effective and relatively efficient solution for accurate causal discovery.
OUTT
200 440
1000 2000
Figure 4: Time efficiency and causal discovery accu- racy on Lorenz#1.
Ablation Study
We analyze the importance of UnCLe's key architec- tural components and methodological choices. First, evaluate the contributions of parameter sharing, the Auto-regressive Dependency Matrices, and the predic- tion task using the high-dimensional Lorenz#3 dataset. Figure 5 presents the performance of the standard Un- CLe(P) model compared modified versions. Second, we assess the sensitivity of our causal discovery mecha- nism different perturbation strategies the Lorenz#1 dataset.
0:911
0.013
2286
0.10
Orghal
Parm Sharing Dap Matican
with Padd
Mode Settigg
Figure 5: Ablation study on UnCLe's key components.
w/o Parameter Sharing Disables the parameter sharing strategy, resulting in individual TCN Uncoupler and Recoupler pairs being trained for each time series component-wise. As reported, the causal discovery performance drops significantly, although this variation can still learn some valid causal structures. This outcome demonstrates the effectiveness of representational knowledge sharing, particularly in high-dimensional scenarios.
w/o rugo-segressive Dependency Matrices Omit the Dependency Matrices of UnCLe The parameter sharing strategy is also disabled. The multivariate time series prediction task is handled directly by the TCN Uncoupler and Recoupler. This leads to the AUROC score dropping below 0.5, equivalent to random guessing, indicating that the model can no longer effectively learn the causal structure. This result highlights the critical role of Dependency Matrices in uncoupling and explicitly capturing inter- inter- variable dependencies in multivariate time series. Additionally, when the Dependency Matrices are disabled, causal structure inference via dependency aggregation becomes unavailable.
w/o Prediction Task When the model is optimized solely for the reconstruction task (i.c., predicting x' using S the causal structure cannot be effectively extracted, as the task becomes overly trivial. Each time series learns to reconstruct itself based solely on its own data rather than integrating information from others. This underscores the necessity of the prediction task bridge for modeling complex systems and learning inter-variable dependencies.
Perturbation Strategies To validate our choice of temporal permutation, we compare its perfor- mance against three alternative strategies on the Lorenzff dataset: (1) Zero-Maskiig where the target series is replaced with zeros; (2) Noise Injection, where Gaussian white noise is added to the target series; and (3) No Perturbation, which serves as a baseline to confirm that error gain is necessary. The results are shown in Table 4.

---

Table Causal discovery performance (UnCLe(P)) on Lorenz#1 with different perturbation strate-
gies.
Perturbation Strategy
AUROC 7
AUPRC1 f
ACC: 1
Temporal Pormutation (Ours)
(900.)002)
=996..0008)
:994:.010)
Noise Injection
.81
946(2.130)
-978..008)
Zero Maro-Masking
.944
9322...17
.969 (.052)
No Perturbation
.500 000)
-575
.850 000)
Temporal permutation significantly outperforms the alternatives. We reason that this is because it uniquely satisfies two crucial conditions: it effectively nullifies the predictive temporal information while perfectly preserving the variable's marginal distribution, thus ensuring model stability. In contrast, Zero-Masking disrupts the data distribution, and Noise Injection does not fully remove the original signal. The "No Perturbation" baseline confirms that without a valid perturbation, the method defaults to random guessing (AUROC 0.5), validating the core principle our post-hoc analysis.
6 Limitations and Future Work
The primary limitation of our work, which also defines critical direction for future research, the lack of formal identifiability guarantees. While UnCLe demonstrates strong empirical performance, we do not provide a theoretical proof under which conditions it is guaranteed to recover the true dynamic causal graph. Establishing the theoretical conditions under which our learned latent space provides a causally faithful linearization remains a key open question. Our future work will focus on bridging this gap, potentially by exploring connections to causal representation learning and imposing further structural constraints on the model ensure that the learned latent dynamics are not just predictive, but verifiably causal.
Broader Impacts
Potential for Misuse. As with any observational causal discovery method, the outputs of UnCLe are hypotheses subject to underlying assumptions (e.g., no hidden confounders) and should not be interpreted as definitive proof of causation. The primary risk lies in the uncritical application of our method in high-stakes domains, such as finance, healthcare, or social policy, where spurious causal claims could lead to flawed and potentially harmful decisions.
Safeguards and Responsible Application. To mitigate these risks, we strongly advocate for responsible use. The causal graphs generated by UnCLe should be treated as a tool for exploration and hypothesis generation, not substitute for rigorous scientific validation. We recommend that any findings validated by domain experts and, where feasible, tested through controlled experiments prospective studies before being used for decision-making
8 Conclusion
In this paper, we propose a novel dynamic causal discovery method, UnCLe, which consists of a pair of Uncouplers and Recouplers alongside Dependency Matrices. This architecture disentangles input time series into semantic representations and learns causal connections between variables through auto-regressive prediction. Extensive experiments demonstrate UnCLe's effectiveness and scalability across static and dynamic datasets from diverse domains. By bridging the gap in dynamic causal discovery methods, UnCLe aims to inspire further advancements in this domain.
Acknowledgments and Disclosure of Funding
This work is partially supported by the National Natural Science Foundation of China (92167 104, 62072006), CCF-Ant Research Fund, Qiyuan Lab Innovation Fund, and National Key Laboratory of Intelligent Parallel Technology.
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A Dataset Details
Overview of Datasets We evaluate UnCLe using datasets from great variety of domains, as detailed in Table 5. The table provides information on the dataset types, the number of variables (p). the series length (T'), and the number replicas (R). The replicas of dataset start with different initial system status. We provide the time series and true causal adjacency matrices in CSV format of all datasets in our public code and datasets repository.
Table 5: Used synthetic static (top), synthetic dynamic (middle) and real- world (bottom) datasets.
Dataset
Type
p
T
R
Lorenz#1
climate dynamics ODE
20
250
5
Lorenz#2
climate dynamics ODE
20
250
5
Lorenz#3
climate dynamics ODE
100
500
5
fMRI
medical mcasurements
15
200
50
NC8
nonlinear constant interactions
8
2000
5
FINANCE
financial portfolios
20, 40
4000
8
TVSEM
time-variant auto-regressive
2
2000
5
ND8
nonlincar dynamic interactions
8
2000
5
MoCap
human motion capture
93
= 300
8
METR-LA
traffic flow speed traffic flow speed
207
10240
PEMS-BAY
325
10240
1
A.1 Synthetic Datasets
Lorenz96 [14] is nonlinear model to simulate climate dynamics used by [16, 27, 10]. The system dynamics become increasingly chaotic and thus hard to model for higher values of the forcing constant F. We design three sets of system configurations of Lorenz96, setting = {250, 250, 500), and F {10, 40, 40} for Lorenz#1, 1, #2, and #3 respectively.
{20, 20, 100).
[MRI (functional Magnetic Resonance Imaging) [25] used by [16, 10, 17] contains time-ordered samples of the alood-oxygenation-Bee brain regions of interest in human subjects.
dependent (BOLD) signals, measuring activity in different
NC8 (Non-linear Constant interactions with p 8 variables) is a dataset we propose that contains wide variety of inter-variable interactions with time lags ranging from I (short-term) to 16 (long-term). The generating equations include non-linear functions such sin (-), (-) and max(-). (-), and involve all three common causal structures: fork, chain, and collision [21]. We provide the detailed generation equations in Appendix
TVSEM (Time- Varying Structural Equation Model) a bivariate synthetic dataset we constructed evaluate dynamic causal discovery. It features variables whose causal dominance switches periodically every 400 timesteps over a total length of = 2000, governed by changing autoregressive coefficients, as detailed in the main paper.
ND8 (Non-linear Dynamic interaction with p = 8 variables) is the dynamic version NC8, where some connections from the original dataset change direction periodically. We provide the detailed generation equations in Appendix G.
FINANCE [11] used in [17] is simulated financial time series dataset that uses factor model to describe portfolio's 's return.
A.2 Real-world Datasets
MoCap (CMU human motion capture) contains real-time 3-axis joint angles of 31 different parts of the human body a frequency of 120 Hz. We selected seven actions from the database: walk, run, kick, jump, golf, sidestep, and bend.
13

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(b) PEMS-BAY by UnCLe Figure 6: Traffic roadmaps discovered by UnCLe in the METR-LA and PEMS-BIL BAY datasets. The locations of the sensors are aligned with the map in the background.
METR-LA by UnCLe
METR-LA and PEMS-BAY [13] contain traffic speed data from highways in Los Angeles County and the Bay Area, respectively. The data is sampled 5-minute rate, and we use the first T 10240 observations, spanning approximately 7 weeks.
B Methodology of Evaluation
All baseline methods produce weighted causal graphs. For the VAR method, we set p as strength of causal relationships where p denotes the significance in Granger causality tests. We evaluate the causal discovery performance of all methods by comparing the inferred structures of against the true structure using areas under receiver operating characteristic (AUROC) and precision- (AUPRC) curves and accuracy (ACC). AUROC and AUPRC measured weighted graphs whereas ACC is calculated on binary adjacency matrices. We set the binarization thresholds to maximize ACC according the true causal structures. For all evaluation metrics, only consider off-diagonal elements of adjacency matrices and ignore self-causal relationships which are basically always and usually the easiest infer All reported metrics the of the datasets, with 95% confidence intervals. The performance of the two available causal graph inference approaches of UnCLe
across all replications
reported seperately, and we denote variable perturbation (P) and weight aggregation as (A). We perform grid search on the hyperparameters of all methods to maximize AUROC. We list the hyperparameter settings of UnCLe and other baseline methods in Section L.
C Additional Results on Large-scale Transportation Dataset (METR-LA and PEMS-BAY)
The METR-LA (207 sensors) and PEMS-BAY (325 sensors) datasets are large-scale transportation datasets. Using 10,240 data points, UnCLe and JRNGC attempt recover the real-world road network from traffic flow speed sensor observations, as shown Fig. 6 and 7.
In the graphs generated by UnCLe, most nodes connected to their neighboring nodes, while the influence small set hub nodes extends distant areas. By referencing the maps, we identify these hub nodes primarily airports large overpasses, which handle the majority of traffic flow in their respective regions. The road network graphs discovered by UnCLe in real-world regions can be seamlessly integrated into practical scenarios, providing valuable support for analysis and decision-making
In contrast, the graphs produced by JRNGC exhibit causal relationships scattered across the map, scarcely recovering connections between neighboring road network nodes. As result, these graphs provide limited insights for real-world decision-makin
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(a) METR-LA by JRNGC Figure 7: Traffic roadmaps discovered JRNGC.
(b) PEMS-BAY by JRNGC
D Additional Results on ND8 with Static Baselines
To provide a comprehensive comparison on the ND8 dataset, which features dynamic ground truth causality, we also evaluate the performance of several static causal discovery baselines with the same hyperparameter settings as with NC8. This evaluation is presented in Table Since these baseline methods inherently produce a single static causal graph, their output is assessed against the evolving ground truth of ND8. For context, the "Static Best" row in the table indicates the theoretical upper bound on performance achievable by any single, optimal static graph when evaluated against this dynamic ground truth. This effectively benchmarks how well any time-invariant model could possibly capture the changing causal relationships. As the results show, UnCLeeP), with its ability to model dynamic causality, significantly outperforms all evaluated static methods and surpasses the "Static Best" theoretical limit in terms of AUROC, highlighting the inherent ad advantage of dynamic approaches for such datasets.
Table 6: Causal discovery performance on the dynamic ND8 dataset. Static baselines are compared against the evolving ground truth, with "Static Best" representing the optimal performance for single static graph.
ND8
Methods
AUROC t
AUPRC t
VAR
0.578
0.053 (.00))
PCMCI
0.848
0.369(1:00))
cMLP
0.68611.003
0.152(1:015)
TCDF
0.741
0.292 (00))
VARLINGAM
0.902
0.614
DYNOTEARS
0.533 (E.000
0.086
CUTS+
(.:05)4:09))
0.345
JRNGC
0.744ct:034
0.151 011)
GVAR
0.723 (L:016)
0.220
UnCLe(P)
0.921 (0
0.633:::045
Statie Best
(.905
0.799
E Additional Results on fMRI
extended our evaluation UnCLe to include fMRI dataset, a synthetic medical dataset. Perfor- mance metrics are detailed in Table 7. On this dataset, UnCLe(P) achieved competitive Accuracy (ACC) of 0.925 0100, underscoring its ability to correctly classify the presence absence of connec- tions.
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Table Causal discovery performance on the fMRI dataset.
IMRI Dataset
Method
AUROC
AUPRC 1
ACC 1
VAR
0.615
0.175
0.910.1.012)
PCMCI
0.813 :.06)
0.278:::150
0.924
cMLP
0.616 (..36)
0.191
0.846 (.00)
GVAR
(.687cb.132)
0.289
0.806 140)
TCDF
0.812 (.082)
0.368 (5252)
0.899 (..91)
VARLINGAM
(I.11)
0.264 (1.173)
0.924 11.44..00 009)
DYNOTEARS
(.554
0.315 (49)
CUTS+
0.689
0.212 (4.14))
0,924 (1.09)
JRNGC
0.776xt:008
0.289 (60)))
0.975 001)
UnCLe(P)
0.792 11118)
0.286 (..14)
0.925 (1.010)
UnCLe(A)
0.783
0.235 (0-1)))
0.923
Kick soccer
Large right
Jump
Golf
Bond
Avarage Causal Graph
Figure 8: The discovered causal structures of human joints across 7 collected human actions. The last structure is averaged from these 7 actions.
F
Additional Results on MoCap
We use UnCLe to amalyze 7 different human actions, and the resulting causal structure is shown in Figure 8. Note that we aggregate the 3-axis variables to a single variable that represents the joint by max pooling to display the causal structure more clearly. Generally, the found causal structure in line with our intuition on how joints of humans affect each other when we perform specific actions, to name few: the structure of walking and running similar; the causal structure is prominent around legs in soccer kicking and sidestepping: jump shows complex connections on feet; golf shows most sophisticated causal relation this sport basically involves every muscle of the human body. In conclusion, UnCLe provides effective insight into how the joints in our physical body work collaboratively complete motion actions.
Figure shows the joint structure discovered from "kick soccer" motion the MoCap dataset by UnCLe, VAR, PCMCI, cMLP and CUTS+. The results from UnCLe are the clearest and align more closely with the actual patterns human motion. Additionally, UnCLe is capable detecting differences in the strength of relationships. whereas the differences detected by other methods very subtle.
G NC8: A Synthetic Dataset with Nonlinear Interactions
The time series of NC8 dataset are generated with following equations:
16

---

UnCLe (Ours)
VAR
PCMCI
cMLP
CUTS+
Figure 9: The discovered causal structures of human joints of the "kick soccer" motion UnCLe and other methods.
t
t
=in + 0.45 sin
0.25 sin
0.16x
1T
9T
3R
31 =0.2000220 - 0.28%, + 0.08x, + 0.2x;-4+
0.2 0.2 0. 0.12 + 0.16 - 0.0 0.02cy
21 3 (0.62r-1) 1) 3 (0.4k; 2) + (0.22;=3))
3 (0.5x;;1) 0.02fz
Wy =0.8 (0.42.1)) + 0.8 (0.52)-2) + 0.642
0.48% 4 + 0.02
01 =0.15sin s.
+ 0.65 sin
0.1e.
80
125
b. -.55 0.63 + 0.18 15 + 0.45a;-18+
0.36b 0.27b,
05 06 0.02cb
C 10000_09
1 0.02
0
0 0.300, 0..2
0,13,
0.52a: 0. 0.26a 0.520, + 0.02c.
where 6(-) (0001) 1) are the noise factors conforming to the standard normal distribution. The NC8 dataset contains 5 replicas with different random seeds and different to [o, 100, 200, 300, 400] beginning offsets. Figure 10 illustrates the causal structure of NC8, with the numbers on the edges indicating the lags of influence.
16
B
16
Y
X
4 Z
W
Figure 10: The true causal structure of NC8.
H ND8: A Synthetic Dataset with Dynamic Causal Structures
The ND8 dataset designed evaluate the capability of methods to detect dynamic, or time-varying, causal relationships. It is derived from the static NC8 dataset by introducing periodic switches in
17

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the causal dependencies between specific variable pairs. Specifically, the generating equations for variables 2,, W/, Ct. and or remain identical those defined for the NC8 dataset (as presented in the previous section). However, the causal relationships involving variables zt, Ut. Oz, and b, are subject to change.
The dynamic nature is implemented as follows: the primary causal direction between the pair y) and the pair (a,b) reverses every 500 timesteps. Initially (e.g., for timesteps 1,..., 500; 1001,..... 1500; etc.), the generating equations for 2t, yr. and by are:
I si + 0.45 sin L
0.25 sin + 0.1ex
4
9T
3R
y1 0.24 0.28 + 0.08r, 0.2r,
0.2y:-1
0.12 + 0.16
0.04y: -4 + 0.026y
=0115sin
sin
0.65 sin
0.1
80
125
b. 0.54a 0.63a; + 0.18 + 0.45a,-16+ 16
0.36b, + 0.27b, 0. + 0. 181, 0.024b
During the alternate 500-timestep intervals (e.g., for timesteps / = 501, 1000; 1501, ..... 2000:
etc.), the generating equations for Tt, Yt, at and b, switch to the following:
-0.88 0.08 + 0.04r:-3 0.04m:4++
0.04y:-1 + 0.28y:-2 0.08y:-3
0.04y:-4 0.1xx
t
y1 =0.45 sin + 0.45 sin
0.25 sin
4m
9T
3x
0.2y 0. 0.12 + 0.1 6y + 0.04y + 0.02ey
=0.09t 0.18 + 0.09 + 0.09 18
0.72b:-13 + 0.27a:11 0.63a.
0.18 0.1ca t
t
b, 0.115 sin + 0.35 sin
+ 0.65
+
6
80
125
0.36b: + 0.27b, 0.36b + 0.18 0.02fb
all equations, f(-) ~ (O represent independent noise factors drawn from a standard normal distribution. The ND8 dataset comprises replicas, each generated with different random seeds for the noise terms. The evolving ground truth causal structure of the ND8 dataset is illustrated in Figure 11, with Figure la showing the initial causal relationships and Figure 1 1lb depicting the structure after the causal switches. To provide concrete visualization of the generated time series, Figure 12 displays one such replica from the ND8 dataset, generated using random seed 500. The plot clearly demarcates the "Reversal Points" at t 500, 1000, 1500, where the causal dependencies between specific variable pairs (x and b) are designed to switch.
I Interpreting Channel-wise Causal Contributions
Figure 13 provides a visual inspection of the learned Dependency Matrices, 1 for each of the C 20 semantic channels after training UnCLe on synthetic dataset with known ground truth causal graph G. Each matrix d (displayed as o through y 19 in the figure) represents the inter-variable dependencies captured within that specific channel.
As observed in Figure 13, the individual channel-wise dependency matrices exhibit varied structures. Some channels (e.g., channels 0-3, 7-15, 17) learn very sparse connections, suggesting they might focus on noise modeling or highly specific, subtle interactions not prominent in the overall ground truth. Other channels, however, capture more discernible patterns of dependency. For instance, channel 19 appears strongly reflect the primary diagonal dependencies present in G. while channels such as and 16 seem to contribute to capturing off-diagonal interactions. Channels and also highlight certain non-diagonal relationships. This visualization suggests that different channels
18

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16
B
16
B
C
C
0
16
Z
Z
W
W
(a) Initial causal structure ND8
(b) Switched causal structure of ND8
Figure 11: The true dynamic causal structure of the ND8 dataset, illustrating the initial state and the state after causal relationship reversals.
Noverial Pilit
1003
120
1600
Time Series
Hetw
Nivinel
500
250
1000
1250
1000
-70
300
Time Se Se-les O
Rilwida Pein
500
750
1000
1200
1000
7700
2000
Time Seles
22
:dc
120
Thne Serie
Rivera Poin
550
250
1004
1250
1000
770
3000
Serwe W
Figure 12: Visualization of single replica from the ND8 dataset (generated with seed 500). The vertical dashed lines indicate the "Reversal Points" at t = 500, 1000, 1500, where predefined causal relationships switch.
ghee
Figure 13: The full demostration of UnCLe's causal discovery via weight aggregation. G is the true causal graph and ÇA is the aggregated causal graph by averaging all Va from d to 19
19

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indeed learn to emphasize different facets or subsets of the underlying systemic dependencies, rather than each channel necessarily isolating entirely distinct and orthogonal causal mechanisms.
The final aggregated static causal graph, ÇA is derived by pooling information across all these channel-specific dependency matrices (as described in Section 3.2 on Static Causal Graph via Dependency Aggregation). The resulting S in Figure 13 demonstrates a close resemblance to the true causal graph G. This illustrates how the aggregation of these diverse, channel-specific perspectives allows UnCLe to reconstruct a comprehensive and accurate representation of the overall static causal structure, even individual channels provide only partial or specialized views.
J Demonstration of UnCLe's Causal Discovery Mechanisms
UnCLe offers two primary mechanisms for causal discovery, outlined in Section 3.2 Post-hoc Causal Discovery: (P) dynamic causal graph inference via temporal perturbation and analysis of datapoint-wise prediction errors, and (A) static causal graph inference via the aggregation of learned Dependency Matrices. We illustrate these mechanisms below.
J.1 Dynamic Causal Discovery via Temporal Perturbation
UnCLe's primary approach for dynamic causal discovery involves quantifying the impact of temporal perturbations on prediction accuracy (Section 3.2 Perturbation-base Dynamic Granger Causality). The principle is that variable æ, causally influences æ;, then disrupting the temporal information in 2j (e.g., via permutation) should lead to noticeable increase in the prediction error for æ;.
Figure 14 illustrates this concept. Consider the task of predicting series Zg. The left panel shows the model's predictions for 2g under normal conditions (blue line) versus when series æg is perturbed cause of ægg (as suggested by typical Lorenz system structure
(orange line). Assuming 2g is or the ground truth G in Figure 15), perturbing 2g significantly degrades the prediction quality for 2g. causing the orange line deviate markedly from the blue line. This deviation, quantified as the datapoint-wir wise error gain 9. indicates a causal link from 2g to Ty-
Conversely, the right panel of Figure 14 shows the predictions for ay when non-causal (or weakly causal) variable, say 122, is perturbed. In this case, the predictions with 1 perturbed (orange line)
\12 remain very close to the original predictions (blue line). The minimal error gain Acg.e suggests weak or absent causal link from T12 to Vg.
By systematically applying such perturbations and quantifying the error gains for all variable pairs across all timesteps, UnCLe constructs the dynamic causal graph Partt The heatmap on the far right of Figure 14 represents static summary or snapshot derived from these dynamic causal influences, demonstrating how this perturbation-baat analysis reveals the causal structure.
Predictions of Series da
Predictiona of Series Ra
Normal
Normal
Perturbed
Perturbed
0
20
100
20
40
Time Step
Time Step
Figure 14: Demonstration of inferring causal influences for Geer via temporal perturbation. Left: Predictions for w9 with original data (Normal) vs. 2s perturbed. Middle: Predictions for Øs with original data vs. T12 perturbed. Right: Resulting (static summary of) causal graph &Pen derived from such perturbation analysis.
J.2 Static Causal Discovery via Dependency Matrix Aggregation
One approach UnCLe employs for static causal discovery is the aggregation of its learned Dependency Matrices (I)) detailed in Section 3.2 Static Causal Graph via Dependency Aggregation and visualized in Figure 13, UnCLe learns multiple channel-specific Dependency Matrices, y each
20

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capturing different aspects of inter-variable relationships. By pooling the information from all these channels (e.g., using the L2-norm), UnCLe constructs single, comprehensive static causal graph. gABe, The effectiveness of this aggregation in accurately recovering the underlying causal structure (compared to a ground truth 9) demonstrated in Figure 13, where Ç closely matches G for Lorenz#1 example. This method provides direct way to obtain summary causal graph from the trained model parameters.
V
G
Figure 15: Demonstration of inferring the static causal graph ČASS via aggregation of Dependency Matrices v. Left: True causal graph G. Middle: Examples of learned Dependency Matrices (e.g..
I y 18 11 Right: Aggregated causal graph ÇAvs.
K
Additional Ablation Study on Higher-Order Lags
To address the model's sensitivity to the autoregressive lag, conducted an additional ablation study on Lorenz#1 dataset. The 'lag" hyperparameter determines the temporal lookback for the linear prediction step in the latent space (Equation 3, modified to use lags > 1). The results for different lag values are presented Table
As shown, increasing the lag from to 2 and 4 led to decrease in causal discovery performance for both UnCLe(P) and UnCLe(A). We hypothesize this is because the TCN architecture's large receptive field already encodes sufficient long-range historical information into the latent representation z₁- Adding explicit higher-order lags in the linear prediction step may introduce parameter redundancy, making the model more prone to overfitting spurious, non non-primary relationships, while also increasing computational cost. This result suggests that lag of is sufficient and optimal for UnCLe's architecture this task.
Table 8: Causal discovery performance on Lorenz#1 with different lag settings.
UnCLe(P)
UnCLe(A)
Lag
AUROC1 1
AUPRC1
AUROC 1
AUPRC 1
1
(990
.94
.922 #.054)
2
(1.00))
.88(LL089)
.956LL121)
882(
4
.877
.670.
.940ct.041)
L Hyperparameter Settings
Here we list the hyperparameter settings on all experiments method by method. For VAR, cMLP, TCDF, GVAR on the fMRI dataset, we adopt the experimental results of these methods [16] and show the hyperparameter settings provided in that paper.
UnCLe
Table lists the hyperparameter settings used for UnCLe of all experiments in this paper.
(available https: / t/gbthhb......t/////g. comm//ancover///errall--oo Table 10 lists the hy- perparameter settings used for cMLP. We use Hierarchical lasso as the sparsity penalty and run 5-step grid search on the penalty factor A.
cMLP [27]
TCDF [17] (available at https: // thub. www.e--aota/aaaaaa MMMM//NNNtaaTTTTTP Table 11 lists the hyperparame- ter settings used for TCDE We run a 5-step grid search on the significance level a the Permutation Importance Validation Method.
21

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Table 9: Hyperparameter Settings for UnCLe.
Kernel Size
TCN Blocks
Kernel Filters
Recon. Epochs
Joint Epochs
Learning
Dataset
Lag
Rate
Lorenz#1
20
1,000
2,000
5c-3
Lorenz#2
6
8
12
1,000
2,500
2e-3
Lorenz#3
500
2,500
le-3
fMRI
8
12
1,000
2,000
1c-5
NC8
6
1,000
2,000
3c-4
FINANCE
2
3
24
500
10,000
3c-4
ND8
6
20
1,000
2,000
3e-4
TVSEM
4
500
2,500
2e-3
Table 10: Hyperparameter Settings for cMLP.
Hidden Layers
Training Epochs
Learning Rate
Sparsity
Dataset
Lag
Hyperparams
Lorenz#1
2,000
le-2
E [0.0, 2.00
Lorenz#2
5
1
3,000
Ic-2
À E [0.0, 2.0
Lorenz#3
5
1
2,000
Ic-2
À E [0.0, 2.0]
IMRI
1
2,000
le-2
A [1e-3, 0.75]
NC8
16
1
1,000
Se-3
A E [0.0,
FINANCE
3
1
1,000
le-3
A E [0.0,
Table 11: Hyperparameter Settings for TCDF
Kernel
Hidden Layers
Training Epochs
Learning
Sparsity
Dataset
Size
Hyperparams
Lorenzfl
5
2,000
le-2
a E 2.00
Lorenz#2
5
2,000
le-2
[0.0, 2.0
Lorenz#3
5
2,000
le-2
a E [0.0,2.0]
FMRT
2,000
le-3
a [0.0, 2.0
NC8
16
1,000
5e-3
a E [0.0,2.0]
FINANCE
2,000
1e-2
a [0.0, 2.0]
GVAR [16] (available httrg://gittib.cco////6//////// Table 12 lists the hyperpa- rameter settings used for GVAR. We 5x5-step grid search the regularisation parameters
7,
Table 12: Hyperparameter Settings for GVAR.
Hidden
Training Epochs
Learning Rate
Sparsity
Dataset
Lag
Layers
Hyperparams
Lorenz#1
5
2
1,000
lc-4
AE 3.00, [0.0,0.023 0.025
Lorenz#2
5
2
1,000
lc-4
X [0.0, (0.0,3.0,,1 7 [0.0, 0.025
Lorenz#3
5
2
1,000
le-4
A 30.0, 0.025.
IMRI
1
1,000
le-3
X (0.0,3.0,,76 (0.00,,,660000000 0.10
NC8
16
1
1,000
lc-4
A (0.0. (00, 0.005.
FINANCE
2
500
lc-4
À [0.0, 0.0,0.022)
VAR [7] (as implemented in the statsmodels library [23]) & PCMCI [22] (available at https: / //github. com/ jakobungee (aating//tirraitte
VAR and PCMCI share the lag hyperparameter L. We set L, = 5 on all the Lorenz96 experiments and the FINANCE experiment, I= 1 on fMRI, and 16 on NC8. The significance level of the PC algorithm of PCMCI is set to 0.01.
22

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Algorithm Causal discovery via temporal perturbation
Input: dataset trained UnCL.e model f-
Output: Adjaceney matrix A Pert
I: #2 t- f((1r) (Predict on original dataset)
2: A 0M
for to N do
4: 6, f(0,,2.T,
(Original error of series i
x' 1 + x (Clone the dataset)
6: Permutate 2.." (Perturb series with permutation) )
7:
a' +- f(x") (Predict on perturbed dataset)
8:
for = -1to . do
9:
V = -E( F2,2:T) [Perturbed error)
10:
for
to 1 do
A J. max(0, Gie 6i,1)
12:
A Pert
+ A (Datapoint-wise error grron
13:
end for
end for
15:
end for
VARLINGAM [9] (available at
We run a 4-step grid
search on the lag from 2 to
DYNOTEARS [19] (available at http:///wtahbb..ooomccc/s//////s
We set the max iteration to I, 1,000, regularisation parameters A and Xa 0.1. We a 4-step grid search the lag from 2 to 5.
PPuss) We set learning rate to le-3, number of training epochs to 64 and max number of groups to 32. We run 4-step grid search the regularisation parameters À from 0.1 to 0.005.
CUTS+ [5] (available at https: thub. com/ /ttp///ayyy///////////////a/aaaa
JRNGC [30] (available at https: /ttp::///ithii /gubbbtt.bbu.gg/ com/ hoom/Eleew///aaa /R//////JGGO We set the hidden size to 100, lag 5, number of residual layers to 5 and learning rate to 1e-3. We run 4-step grid search the Jacobian regularizer coefficient À from 0.001 to 0.0001.
M Implementation of Temporal Perturbation
Algorithm/ I outlines the procedure for inferring the dynamic causal graph gPat using temporal perturbation and datapoint-wiss prediction errors. In this algorithm, l denotes the Mean Squared Error (MSE) loss function, and x V represents the dataset where the j-th time series, æy,, has been perturbed (by permuting its temporal values). The core idea is to quantify the causal influence from variable xj to variable æ; at time This is achieved by computing De the datapoint-wis gain in prediction error for Ti,; when the historical information of æ, (i.c., xy,<t)) is disrupted by perturbation.
V
This error gain, Ae,, serves as the strength of the causal link G in the dynamic graph.
The computational efficiency of this process can be significantly enhanced through batch processing. Instead of perturbing and predicting for each series sequentially, we can prepare multiple perturbed versions of the dataset (each with different series 2, perturbed) and process them in batches. By feeding these batches into the trained UnCLe model, predictions for multiple perturbed scenarios can be obtained in parallel. This optimization reduces the number of sequential forward passes through the batch
the model from N (where N is the number of series) to approximately N/B, where size, thereby reducing the overall inference time. The effective time complexity for the perturbation analysis, originally proportional to O(N Tmodel), where Tmodel the time for one forward pass, becomes closer to O[N/B] T, assuming efficient parallelization within each batch.
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Reviews:
[{"review_id": "ELdqKTGlkT", "reviewer": "Reviewer_K7ML", "summary": "The article proposes UnCLe, a method for dynamic causal discovery. The method using an auto-encoding type structure to learn a latent representation of a nonlinear time series which linearizes the underlying dynamics. The method is evaluated on several interesting synthetic and real-world examples. Experimental results demonstrate the method effectively and suggest it to be an important contribution to the field.", "questions": "- Why are the static and dynamic graphs learned by apparently two distinct mechanisms? Can the authors comment on why they proposed two distinct methods in this article?\n\n- How do we verify linearity in latent dynamics is a good assumption? Linear dynamics are NOT sufficient to describe nonlinear phenomena. That said, linear models can be useful to represent nonlinear dynamics when the correct representation is used (Brunton et al, 2017). In that sense, UnCLe could argued for quite nicely using existing theory. I recommend that the authors take a look at some of the literature on the linearization of nonlinear dynamic systems (e.g., (Socha, 2007)) to understand the limitations of their proposed method, and to consider these ideas in future work.\n\n- It should be emphasized in the article that the reason that the UnCLe(P) method can detect causation from prediction is specifically because it models the causal mechanism that generates the data. Hence, if the predictions are inaccurate, it indicates that the assumed causal dynamics did not anticipate the next state, which is one consequence of the graph changing over time. \n\n### References \n\nBrunton, Steven L., et al. \"Chaos as an intermittently forced linear system.\" Nature communications 8.1 (2017): 19.\n\nSocha, Leslaw. Linearization methods for stochastic dynamic systems. Vol. 730. Springer, 2007.", "limitations": "I believe the authors commented sufficiently on the limitations of the work, apart from my requests above.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The article is clearly written and easy to follow. The proposed method is interesting and well-detailed. The authors provide compelling motivation, and the experiments clearly support the proposed method.\n\nI only identified minor weaknesses in the article, which I have detailed below. \n- UnCLe(P) appears to be uniformly better than UnCLe(A). The authors state that UnCLe(A) might be preferable when speed is desired for causal discovery. But when would I want speed? Causal discovery is often a scientific task, performed offline on pre-recorded data sets, and it's hard for me to imagine when I would need to speed up this procedure. Also, it's not clear that the gain in time that I get from using UnCLe(A) vs UnCLe(P) is significant relative to the overhead of training the Uncoupler\\Recoupler networks. Figure 4 suggests that running UnCLe already takes a few minutes. Does this speed up significantly if I use (A)? Please comment on this. \n- The experiment with the dynamic SEM would benefit from a figure that compares the \"ground truth\" causal strength to the UnCLe causal strengths, and to comment on whether they align or not, and why the result is\\isn't reasonable.\n- Which version of UnCLe is used in Figure 4? \n- The discussion surrounding the MoCap experiment would benefit from adding references to existing literature that explains how the UnCLe method produced a \"better\" time-varying graph. Without a ground truth, asserting that the UnCLe graph is preferrable requires the reader to have some kind of knowledge of MoCap. In the worst case, it might be true that dense graphs are to be expected in MoCap and that the sparsity of the UnCLe graphs is undesirable. Please comment on this.", "was_revised": true, "final_justification": "i am satisfied with the authors responses.", "rebuttal": "### **Response to Reviewer K7ML**\n\nWe sincerely thank you for your insightful review and positive evaluation of our work. We appreciate the constructive feedback and will address the identified points to improve the paper.\n\n**Weaknesses:**\n\n*   **W1: On the time sensitivity of causal discovery and the speed-up from weight aggregation (UnCLe(A)).**\n    That is an excellent point. While many scientific discovery tasks are performed offline, latency can be critical in applications like real-time monitoring in industrial systems. The speed-up from UnCLe(A) is most significant on large-scale datasets where the perturbation process (UnCLe(P)) becomes more time-consuming. For smaller datasets, the overhead of perturbation is negligible. In the final version, we will add an analysis comparing the post-hoc inference times for UnCLe(P) and UnCLe(A) across different data scales to better quantify this trade-off.\n\n*   **W2: On visualizing the ground truth (GT) causal strength in the DSEM experiment.**\n    We agree that a direct comparison would be ideal for interpretability. The causal *direction* is explicitly defined in the SEM equations, and we visualized this with color-coded segments and vertical lines. However, we did not plot a GT \"causal strength\" line because the SEM coefficients, while strongly correlated with strength, are not directly equivalent to a singular strength value, making a direct GT plot potentially ambiguous. Our visualization approach, which includes smoothing the discovered strengths, aims to clearly show how UnCLe correctly identifies the switching dominance between variables in alignment with the ground truth mechanism.\n\n*   **W3: On clarifying which version of UnCLe was used in Figure 4.**\n    Our apologies for the omission. Figure 4 shows the results for the perturbation-based method, **UnCLe(P)**. We will clarify this in the caption in the final version.\n\n*   **W4: On substantiating the MoCap analysis with domain literature.**\n    We thank the reviewer for this insightful and critical question. The reviewer is entirely correct that without a ground truth, asserting a causal graph is \"better\" is a non-trivial claim, and we agree our argument must be substantiated with external domain knowledge.\n    We will revise the manuscript to incorporate this justification. Our core argument, supported by biomechanics literature (e.g., Hara et al., 2008; DeVita & Skelly, 1992), will be that UnCLe's graphs are \"better\" because their \"structural evolution\" aligns with the known, \"phase-specific coordination strategies\" of the jump. Specifically, UnCLe captures the shift from full-body coordination (in crouch/landing) to localized adjustments (in flight), a dynamic nuance missed by the baselines. We will add the necessary analysis and citations in the final version to fully support this interpretation.\n\n**Questions:**\n\n*   **Q1: On the motivation for proposing two distinct causal discovery mechanisms (static/dynamic).**\n    This is a great question about our design process. Our work began with developing a robust representation and prediction model for time series. We first explored perturbation for static discovery. In seeking to explain its success, we found that the learned Dependency Matrices themselves contained a static causal blueprint (UnCLe(A)). We then realized the perturbation approach naturally extended to dynamic discovery (UnCLe(P)).\n    We chose to present both because they are complementary. UnCLe(P) is our primary contribution for accurate dynamic discovery. UnCLe(A) serves as a zero-cost, interpretable diagnostic tool derived directly from the model's parameters, providing a valuable static summary and a basis for understanding the model's learned structure. We believe both approaches offer valuable insights to the community.\n\n*   **Q2: On validating the assumption of linear latent dynamics.**\n    Thank you for pointing us to this highly relevant literature. We agree that framing our work within the context of dynamical systems theory, such as linearization of nonlinear systems, would greatly strengthen the paper's theoretical underpinnings. This is a crucial point for understanding the limitations of our method and for guiding future work. We will incorporate this perspective and the suggested references into our discussion.\n\n*   **Q3: On emphasizing the mechanism behind perturbation-based discovery.**\n    This is a key insight. We will revise the methodology section to more explicitly state that UnCLe(P)'s effectiveness stems from its ability to model the data's causal generative mechanism. An inaccurate prediction under perturbation indicates that the model's learned causal dynamics did not anticipate the next state, which is a direct consequence of disrupting a true causal link.\n\nThank you again for your valuable time and constructive suggestions."}, {"review_id": "FhDqoHZYkJ", "reviewer": "Reviewer_SUxp", "summary": "UnCLe is a scalable framework for causal discovery in non-linear, high-dimensional time series. A parameter-sharing TCN encoder learns latent features, while a sparse auto-regressive matrix Ψ predicts the next step. Causal graphs are then extracted in two ways:\n1. Dynamic – randomly permuting one series and measuring the error increase on others (fast, perturbation-based).\n2. Static – aggregating Ψ weights without extra passes.\nAcross synthetic static/dynamic benchmarks, fMRI, human motion capture, and 300-node traffic data, UnCLe matches or outperforms nine baselines, sits on the AUROC / runtime Pareto frontier, and yields interpretable graphs—yet keeps parameters and runtime modest thanks to weight sharing.\n\nAcross synthetic static/dynamic benchmarks, fMRI, human motion capture, and 300-node traffic data, UnCLe matches or outperforms nine baselines, sits on the AUROC / runtime Pareto frontier, and yields interpretable graphs—yet keeps parameters and runtime modest thanks to weight sharing.", "questions": "1. How did the framework perform on time series data with missingness or irregularities?\n2. Can this post-hoc perturbation method be extended to other similar temporal deep learning frameworks?\n3. L1 on Ψ encourages sparsity, but dynamic graphs in nature may densify temporarily; does the regulariser hamper detection of dense episodes?", "limitations": "The paper’s limitations section is reasonably clear about scalability ceilings and the lack of formal identifiability guarantees, but the broader-impact discussion is essentially missing (the checklist marks “N/A”). I recommend the authors:\n\n1. Add a short societal-impact paragraph: spell out potential misuse risks (e.g., drawing unjustified causal claims in sensitive domains like finance, health, or policing) and emphasize that UnCLe’s graphs should not be used for high-stakes decisions without expert validation.\n2. Clarify deployment safeguards: outline recommendations such as thresholding edges by confidence, combining UnCLe with domain knowledge, or performing prospective validation before operational use.\n\nIncorporating these points would more fully address the negative societal impact while rewarding the authors’ transparency.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. Novel deep learning method explicitly targeting time-resolved causal graphs in large, non-linear systems.\n2. Perturbation yields accurate dynamic edges; aggregation gives an immediate static summary without extra passes—useful in latency-constrained settings.\n3. Extensive evaluation on synthetic (static and dynamic), medium-dimensional medical data, high-dimensional traffic & biomechanics; baselines span classical, score-based, and neural Granger families.\n\nWeaknesses:\n1. No identifiability guarantee: the method relies on predictive perturbations but does not analyse conditions (e.g., faithfulness, invertibility) under which the framework recovers the true causal strength.\n2. Ψ operates only on lag 1. Although the authors use dilated TCNs, empirical degradation when true lags > 1 is not studied.\n3. Error gains depend on permutation; robustness to alternative perturbations (masking, noise injection) is unclear.", "was_revised": true, "final_justification": "I thank the authors for their detailed response, which has addressed most of my questions. Considering that the paper is already well-established and the clear and convincing rebuttal, I would like to maintain my rating as 5.", "rebuttal": "### **Response to Reviewer SUxp**\n\nWe sincerely thank you for your detailed review and for highlighting the strengths of our work. We appreciate the opportunity to clarify the points you raised.\n\n**Weaknesses:**\n\n*   **W1: On the lack of identifiability guarantees.**\n    We concur that formal identifiability guarantees are crucial for causal discovery methods. This is a well-known and challenging open problem for many deep learning-based approaches, whose theoretical properties often lag behind their empirical success. While designing UnCLe, we explored connections to causal disentanglement literature (e.g., [1][2]), but found that proving formal properties for our specific architecture was non-trivial. Therefore, this work focuses on providing strong and extensive empirical validation of UnCLe's effectiveness. We have added providing a formal identifiability analysis as a primary direction for future work.\n\n*   **W2: On not studying cases where lag > 1.**\n    (This point is also raised by Reviewer 9Mey). Please see our detailed response to **Reviewer 9Mey, Weakness W1**, which includes experimental results for different lags. In summary, the TCN architecture's receptive field allows the model to handle long-range dependencies, and we found that explicitly increasing the autoregressive lag in the Dependency Matrices did not improve, and sometimes hindered, performance, likely due to parameter redundancy.\n\n*   **W3: On the robustness of permutation compared to other perturbation methods.**\n    (This point is also raised by Reviewer T7tD). Please see our detailed response to **Reviewer T7tD, Question Q2**. In summary, we experimented with permutation, zero-masking, and random noise injection. Permutation yielded the best results because it effectively removes temporal information while preserving the variable's marginal distribution, ensuring model stability. We will add a summary of this sensitivity analysis to the final paper.\n\n*   **W4: On the performance drop on sparse synthetic datasets with >50% missing edges.**\n    We appreciate you pointing this out for clarification. However, we were unable to locate the specific statement about \">50% missing edges\" in our manuscript. Could you please help clarify which part of the paper this refers to? We would be very grateful for the opportunity to address this point more directly.\n\n**Questions:**\n\n*   **Q1: On handling missing or irregular data.**\n    This is an important practical consideration. Our current work assumes complete and regularly-sampled data. For datasets with missingness or irregularities, UnCLe would require a standard preprocessing step, such as imputation, before application. Explicitly handling such data within the model architecture is a valuable direction for future research.\n\n*   **Q2: On extending the post-hoc perturbation method to other frameworks.**\n    This is a very keen observation. Yes, the post-hoc perturbation framework is in principle model-agnostic and could be applied to other multivariate time-series forecasting models. Exploring its effectiveness across different architectures is a promising research avenue that we hope our work will inspire.\n\n*   **Q3: On whether L1 regularization hampers the detection of temporarily dense graphs.**\n    This addresses a fundamental trade-off between interpretability and complexity. Our goal with L1 regularization is to encourage the model to capture the most stable and dominant causal relationships. While a very strong regularizer could potentially suppress transient dense connections, we found that a moderate L1, combined with the aggregation of diverse patterns across multiple channels, allows the model to represent complex structures without being overwhelmed by noise or spurious connections. The key is to find a balance that yields an interpretable yet sufficiently descriptive graph.\n\n**Limitations:**\n\n*   **On adding a discussion of broader impacts and safeguards.**\n    Thank you for this crucial feedback. We agree that a discussion on broader impacts is essential. In the final version of this paper, we will add a dedicated section to address the potential misuse of causal claims in sensitive domains and outline best practices for responsible application, such as combining UnCLe with domain expert knowledge and performing prospective validation.\n\nThank you again for your thoughtful review.\n\n----\n\n**References**\n\n\\[1] Song, Xiangchen, et al. \"Temporally disentangled representation learning under unknown nonstationarity.\", NeurIPS 2023\n\n\\[2] Song, Xiangchen, et al. \"Causal temporal representation learning with nonstationary sparse transition.\", NeurIPS 2024"}, {"review_id": "O2JNg2wj4c", "reviewer": "Reviewer_T7tD", "summary": "This paper introduces UnCLe (UnCoupLing causality), a novel deep learning-based framework for scalable, dynamic causal discovery in multivariate time series data. Unlike existing methods that primarily produce static or time-aggregated causal graphs, UnCLe explicitly models evolving temporal causality using a two-part architecture: (1) Uncoupler and Recoupler networks that disentangle input signals into semantic latent representations, and (2) auto-regressive Dependency Matrices to capture inter-variable dependencies. Dynamic causal influence is inferred post hoc by perturbing individual time series and measuring changes in prediction error, thus revealing time-specific causal structures. The authors support their claims with extensive experiments on synthetic, semi-synthetic, and real-world datasets, including human motion capture, and demonstrate that UnCLe outperforms state-of-the-art baselines in both accuracy and scalability.", "questions": "Can the authors clarify how the Dependency Matrices Ψ relate to true causal effects across different channels?\n\nThe causal strength estimation relies on random permutations of the input time series. How sensitive is UnCLe’s performance to the specific choice of perturbation strategy?\n\nSince the method estimates causality by measuring increases in prediction error under perturbation, how does it guard against attributing causality to spurious correlations, especially in high-dimensional settings with limited data?\n\nWhile UnCLe scales better than several baselines, can the authors provide more detailed runtime and memory usage analysis? This would help in assessing suitability for very high-dimensional settings.", "limitations": "Limitations and potential egative impact are adequately discussed.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper's principal strength lies in its clear identification of a significant gap in current causal discovery literature—namely, the lack of effective, scalable tools for modeling dynamic causal structures. UnCLe is a relevant contribution: its perturbation-based post-hoc analysis offers a novel approach to quantifying time-varying causal influence, and the architectural design with shared TCN-based encoders/decoders enhances scalability and robustness. The results demonstrate UnCLe’s effectiveness, especially in dynamic scenarios where it outperforms the closest dynamic baseline, GVAR, both qualitatively (e.g., motion phase interpretation) and quantitatively. Overall, the paper is well-written. Although Python code and datasets are provided in the supplementary material, I did not verify the reproducibility of the experiments.\n\nRegarding weaknesses, while the model architecture is novel in its integration of existing components (e.g., TCNs, perturbation-based attribution), it does not introduce fundamentally new neural paradigms. Furthermore, the paper could engage more deeply with theoretical guarantees (e.g., identifiability, spurious effects), and the interpretability of semantic channels. Lastly, the dynamic causal graphs are evaluated largely through visual or intuitive means (e.g., MoCap phases). More rigorous quantitative evaluation on dynamic benchmarks would further strengthen the case.", "was_revised": true, "final_justification": "After reading the rebuttal and the reply of the author(s), my score remain unchanged.", "rebuttal": "### **Response to Reviewer T7tD**\n\nWe thank you for your positive assessment and for providing valuable feedback to strengthen our paper.\n\n**Weaknesses:**\n\n*   **W1: On the lack of fundamentally new neural paradigms.**\n    We agree that UnCLe does not introduce a fundamentally new neural paradigm. Its novelty lies in the specific architectural integration of established components (like TCNs) and a perturbation-based analysis, tailored specifically to tackle the challenging and under-explored problem of scalable *dynamic* causal discovery.\n\n*   **W2: On the need for more theoretical guarantees and interpretability.**\n    (This point is also raised by Reviewers SUxp and 9Mey). We agree that more theoretical guarantees, such as for identifiability, are an important next step. Please see our detailed response to **Reviewer SUxp, Weakness W1**. Regarding channel interpretability, we provide a detailed analysis in **Appendix I (\"Interpreting Channel-wise Causal Contributions\")**, which shows how different channels capture different facets of the underlying causal structure.\n\n*   **W3: On the need for more rigorous quantitative evaluation on dynamic benchmarks.**\n    We appreciate this suggestion. In the paper, we provide quantitative evaluation on the dynamic SEM (dSEM) dataset via AUROC/AUPRC and on the MoCap dataset via a missing rate metric for skeletal connections. We also complement this with qualitative analysis of the MoCap phases. We will enhance our qualitative analysis by incorporating more specific biomechanics literature, as suggested by Reviewer K7ML, and will continue to explore additional quantitative evaluation methods for dynamic graphs.\n\n**Questions:**\n\n*   **Q1: On clarifying the link between Dependency Matrices and true causal effects.**\n    We provide a detailed analysis of this in Appendix I (\"Interpreting Channel-wise Causal Contributions\"), using the Lorenz#1 dataset as an example. In brief, each channel's Dependency Matrix learns a specific \"semantic\" mode of interaction between variables. The true causal mechanism is modeled by the collective action of all these interaction modes. Aggregating them allows us to recover a comprehensive view of the static causal graph.\n\n*   **Q2: On the sensitivity of UnCLe to the choice of perturbation strategy.**\n    This is a great question. We experimented with three strategies: permutation (our default), zero-masking, and random noise injection. Permutation performed best. We reason that zero-masking disrupts the data distribution, affecting model stability, while noise injection does not fully remove the original signal. Permutation uniquely preserves the marginal distribution while effectively nullifying predictive temporal information. We believe this is a valuable point and will add a summary of this sensitivity analysis to the final version.\n\n*   **Q3: On avoiding spurious correlations in high-dimensional, limited-data settings.**\n    This is a key challenge in causal discovery. UnCLe mitigates this risk through its parameter-sharing TCN autoencoder, which allows the model to learn robust, common representations from the entire dataset, effectively increasing the data available for learning each component. This enhances model stability. We tested this scenario with the Lorenz#3 dataset (N=100, T=500), where UnCLe outperformed all baselines, suggesting our approach is relatively robust in such settings.\n\n*   **Q4: On providing more detailed runtime and memory analysis for scalability.**\n    We agree this is important for assessing practical applicability. UnCLe's scalability stems from its parameter-sharing design and parallelizable computations. In the final version, we will expand the \"Time Efficiency\" subsection to include a table or plot detailing UnCLe's runtime and memory usage across different data scales (N and T), and supplement the existing Figure 4 with efficiency comparisons on other datasets.\n\nThank you for your constructive and detailed feedback."}, {"review_id": "mlMkbLcV0D", "reviewer": "Reviewer_9Mey", "summary": "This paper proposes the UnCLe algorithm, a Granger causal discovery method that can extract dynamic causal structures. The UnCLe algorithm first uses TCN to encode the original variable into a latent representation, and then searches for a dynamic causal dependency matrix in the latent space. It should be noted that a reconstruction loss supervises the latent representation.", "questions": "1. How can we ensure the identifiability of the causal dependency structure discovered by the proposed method in latent space? I will be happy to improve the score if this issue is solved.\n\n2. The dynamic causal dependency matrix is ​​defined as $C$ learnable $N \\times N$ matrices. This may cause problems when dealing with high-dimensional time series, as it significantly increases the number of parameters that need to be learned.\n\n3. The learning strategy of Ψ seems to need further explanation. Is it a randomly initialized matrix? Is there any additional design?", "limitations": "yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n1. This method proposes a dynamic inter-slice causal structure algorithm based on Granger causality, which better matches the real dynamic system.\n\n2. This method can obtain the causal dependencies of dynamic and static structures at the same time.\n\n**Weaknesses**\n\n1. Eq. (3) and Eq. (4) are Granger causal models that model lag 1, and it seems that the possibility of extending the model to higher orders is not explored.\n\n2. The author provides a reasonable modeling method and mathematical expressions, but there is no theoretical analysis of the identifiability of cause and effect, especially since this article will learn causal relationships in the latent space.", "was_revised": false, "final_justification": "", "rebuttal": "### **Response to Reviewer 9Mey**\n\nWe thank you for your time and for the valuable questions, which help us clarify important aspects of our method.\n\n**Weaknesses:**\n\n*   **W1: On not exploring higher-order lags.**\n    This is an excellent point. We conducted an ablation study on the Lorenz#1 dataset to investigate the effect of increasing the autoregressive lag in the Dependency Matrices. The results are as follows:\n\n| lag | AUROC (P) | AUPRC (P) | AUROC (A) | AUPRC (A) |\n|:-:|:---------------:|:---------------:|:---------------:|:---------------:|\n| 1 | **.999**($\\pm$.002) | **.996**($\\pm$.008) | **.994**($\\pm$.007) | **.962**($\\pm$.054) |\n| 2 | .961($\\pm$.042) | .878($\\pm$.089) | .956($\\pm$.021) | .822($\\pm$.071) |\n| 4 | .877($\\pm$.076) | .670($\\pm$.121) | .940($\\pm$.041) | .783($\\pm$.099) |\n\n\nIncreasing the lag decreased performance. We hypothesize this is because the TCN's large receptive field already encodes long-range historical information into the latent representation $z_t$. Adding explicit higher-order lags in the linear prediction step introduces parameter redundancy, making the model more prone to overfitting on spurious, non-primary relationships, while also increasing computational cost. Thank you for raising this; we will add this experiment to the final version.\n\n**Questions:**\n\n*   **Q1: On the identifiability of the causal structure in latent space.**\n    (This is a core question also raised by Reviewers SUxp and T7tD). We agree that this is a critical theoretical point. Please see our detailed response to **Reviewer SUxp, Weakness W1**. In summary, while our work does not provide a formal proof of identifiability, which remains an open challenge for many deep learning methods, we provide extensive empirical evidence for UnCLe's effectiveness and consider this a key direction for future theoretical work.\n\n*   **Q2: On the parameter size of the Dependency Matrices in high-dimensional settings.**\n    The parameter count of the Dependency Matrices does grow quadratically with the number of variables $N$ (as $C \\times N \\times N$). However, the number of channels $C$ is a small, constant hyperparameter. More importantly, UnCLe's architecture is highly parallelizable. In cases where memory is a constraint on very large-scale data, the batch size $B$ can be reduced to allow the model to run, trading a small amount of speed for feasibility. This is detailed further in Appendix L (\"Implementation of Temporal Perturbation\").\n\n*   **Q3: On the learning strategy for the Dependency Matrices ($\\Psi$).**\n    Our apologies for not detailing this sufficiently. The Dependency Matrices $\\Psi$ are learnable parameters that are optimized jointly with the rest of the network. They are initialized from a normal distribution $N(0, \\sqrt{3/N})$, where $N$ is the number of time series. Their values are updated based on their role in the prediction task (via the $L_{Pred}$ loss) and are encouraged towards sparsity by the $L_{L1}$ regularization term. We will add these details to the methodology section in the final version.\n\nThank you again for your insightful questions and feedback."}]

Author rebuttal:

Meta-review: This paper introduces UnCLe, a novel and scalable deep learning framework
for dynamic causal discovery in non-linear temporal systems. The method
uses an "Uncoupler-Recoupler" architecture to learn disentangled latent
representations and infers time-varying causal graphs by analyzing
prediction errors from a post-hoc perturbation analysis.

The reviewers were in strong agreement on several key strengths of the
work. They praised the paper for addressing a significant and
under-explored problem, noting that the ability to model dynamic, rather
than static, causal relationships is a critical step forward for
understanding complex real-world systems. The reviewers found the proposed
method to be novel, well-motivated, and clearly presented. A major point of
consensus was the extensive and compelling empirical evaluation across a
diverse set of synthetic and real-world benchmarks (including fMRI and
human motion capture), where UnCLe consistently outperforms
state-of-the-art baselines.

The primary concern raised during the review process was the lack of formal
identifiability guarantees—a well-known and challenging open problem for
many deep learning-based causal discovery methods. The authors forthrightly
acknowledged this limitation. Other concerns regarding sensitivity to
hyperparameters and model design choices were effectively addressed in the
author rebuttal, often with new ablation studies.

Overall, despite the theoretical limitations, the paper's significant
contribution to the important problem of dynamic causality, backed by
strong and comprehensive empirical results, makes a compelling case for
acceptance. UnCLe represents a significant practical advance and is likely
to be of high interest to the community.
Decision: Accept (poster)