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SCDM: Scalable Causal Discovery in Nonlinear Temporal Systems with Meta-Learning

Jingbo Wang, Kegeng Tang, Zihao Wang, Shaogang Ren

arXiv:2610.29992v1cs.LG

TL;DR

High-dimensional nonlinear time-series causal discovery faces quadratically many directed candidates and costly target-wise fitting or repeated tests. SCDM shares a temporal predictor and causal-strength matrix across target tasks, updating sampled tasks per episode. It is competitive on controlled benchmarks and offers a better runtime–accuracy tradeoff at 1,000 and 3,000 variables.

  • Problem

    Nonlinear time-series causal discovery becomes costly in high-dimensional systems because the directed candidate space grows quadratically and existing methods often require target-wise fitting, repeated tests, or expensive graph search.

    Source: §1 · p. 1

  • Method

    SCDM treats target variables as related tasks, learns a shared temporal predictor and causal-strength matrix, and updates a sampled subset of tasks in each meta-episode.

    Source: §1 · p. 2, §3 · p. 3

  • Results

    SCDM is competitive with strong baselines on controlled benchmarks and has a better runtime–accuracy tradeoff at 1,000 variables, while finishing on 3,000 variables with far less time and memory.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Target-task subsampling lowers training cost without hurting recovery, while the support/query objective yields shared parameters that adapt better to held-out targets than pooled multi-task training.

    Source: §1 · p. 2

    The recovery condition is not unconditional: hidden common causes can make a non-edge score positive, and recovery requires statistical, meta-optimization, and readout errors to stay below the population margin.

    Source: §4.3 · p. 6

Abstract

from arXiv · show

Causal discovery from nonlinear multivariate time series is challenging in high-dimensional systems, where the number of candidate directed relations grows quadratically with the number of variables. Existing methods are often limited by target-wise model fitting, repeated conditional testing, or expensive graph-search procedures. We propose SCDM, a shared-parameter meta-learning framework for scalable temporal causal discovery. SCDM treats each target variable as a task while learning a shared temporal predictor and a shared causal-strength matrix across tasks. Each meta-episode updates only a subset of target-variable tasks, allowing the model to accumulate structural evidence across the full system without exhaustive target-wise optimization. After training, a post-training graph readout converts the learned predictor into continuous directed causal scores for threshold-independent evaluation. We also provide a finite-error recovery principle showing that, under fully observed delayed-system assumptions and a positive risk-gap condition, thresholding SCDM scores recovers the graph when statistical, meta-optimization, and readout errors are below the population margin. Experiments on controlled synthetic, neuroimaging-inspired, high-dimensional, and real-world temporal benchmarks demonstrate competitive causal recovery and improved scalability in large temporal systems.

1 Introduction

High-dimensional nonlinear time-series causal discovery faces a quadratic space of directed candidates and costly target-wise fitting or repeated tests. SCDM shares temporal prediction across target tasks and uses sampled episodic updates, with competitive recovery and a stronger runtime–accuracy tradeoff at large scale.

1 Introduction

  • For d observed variables, the candidate cross-variable edge space has d(d −1) ordered pairs, making nonlinear prediction, target-wise fitting, and repeated conditional tests costly at high dimensions.Many existing approaches still require target-specific optimization or repeated graph evaluation, which is limiting when the edge space contains millions of directed pairs.Source: §1 · p. 1
  • SCDM treats each target variable as a task and learns a shared nonlinear temporal predictor and causal-strength matrix through episodes that update only sampled target tasks.Temporary support adaptation and query losses update shared parameters, accumulating structural evidence without optimizing every target at each update.Source: §1 · p. 2
  • After meta-training, a perturbation-style readout produces continuous directed source–target scores, and a finite-error principle gives a recovery condition for thresholding them.For fully observed delayed systems, recovery holds when statistical, meta-optimization, and readout errors are smaller than the population risk-gap margin.Source: §1 · p. 2
  • SCDM is competitive with strong baselines on controlled benchmarks and has a better runtime–accuracy tradeoff at 1,000 variables, while finishing on 3,000 variables with far less time and memory.The introduction also reports that target-task subsampling lowers training cost without hurting recovery, while the support/query objective adapts better to held-out targets than pooled multi-task training.Source: §1 · p. 2

2 Related Work

Prior temporal causal-discovery work spans predictive, conditional-testing, neural, and scalable-search approaches. SCDM combines shared nonlinear prediction with episodic meta-training across target tasks, using perturbation-style scoring only after training.

2 Related Work

  • Temporal causal discovery includes Granger prediction, VAR and VARLiNGAM models, and conditional-independence methods such as PCMCI and PCMCI+; their linear, component-wise, or repeated-testing structures can be costly in nonlinear high-dimensional systems.The candidate edge space scales as d(d −1), while SCDM replaces repeated target-wise fitting or testing with a shared meta-learned predictor and causal-strength matrix.Source: §2 · p. 2
  • Neural approaches use structured-sparsity predictors, self-explaining networks, temporal convolutions, differentiable graph learning, or perturbation-based dynamic scoring.SCDM shares nonlinear prediction and continuous graph scoring with this literature, but uses perturbation-style scoring as a post-training readout rather than its main scalability mechanism.Source: §2 · p. 3
  • Scalable alternatives include sparsity regularization, variable-wise decomposition, screening, and coarse-to-fine search, whereas SCDM treats target prediction as related tasks with shared parameters.Its support/query episodes sample target tasks and update a shared temporal predictor and causal-strength matrix through query losses, accumulating structural evidence without updating all targets each iteration.Source: §2 · p. 3

3 Problem Formulation

The formulation makes each observed variable a prediction task over lagged history. All tasks share a nonlinear temporal predictor and a global causal-strength matrix, while only temporary task-specific copies are adapted.

3 Problem Formulation

  • For a d-dimensional series, the observed history u_t concatenates the previous p time points, and each target task predicts its variable from that history.The d target variables are treated as related tasks within the same dynamical process.Source: §3 · p. 3
  • All tasks share Θ = {θ, W}: θ parameterizes the nonlinear temporal predictor, and W_j,i represents the strength from source j to target i.Each task adapts only a temporary local copy; no private target-specific head, embedding, or persistent adapted model is introduced.Source: §3 · p. 3
  • The predictor gates each source variable’s lagged coordinates with its shared weight W_j,i, making W a global variable-level directed causal-strength representation.The matrix is later converted into continuous graph scores by the post-training readout.Source: §3 · p. 3

4 Methodology

SCDM learns a shared temporal predictor and causal-strength matrix through episodic target-task adaptation, then reads out directed scores after training. Its recovery result is conditional on a fully observed delayed system and sufficiently small estimation and readout errors.

4.1 Meta-Learning for Causal Discovery in Nonlinear Temporal Systems

  • Each sampled target task adapts shared parameters on support windows, while query losses update the persistent temporal predictor and causal-strength matrix.Task-specific adapted copies are discarded after the episode.Source: Figure 1 · p. 4, §4.1 · pp. 4–5
  • Task subsampling limits each update to B targets, while the support/query split favors shared parameters that perform well after K target-specific steps.The two choices serve distinct roles: subsampling controls training cost, and query losses assess adaptation.Source: §4.1 · p. 5

4.2 Post-training Graph Score Readout

  • After training, SCDM uses source-isolated loss reduction scores or the structural |W_j,i| readout; only the latter is affordable at d ≥ 1000.The scores measure predictive contribution, not intervention effects, and correlated sources can shift an individual edge score to a non-parent.Source: §4.2 · p. 5

4.3 Finite-Error Graph Recovery

  • Under causal sufficiency, correct lag specification, adequate regularity, and a positive population margin, thresholding recovers the delayed graph when statistical, meta-optimization, and readout errors stay below that margin.The theorem separates the data-generating process's margin from SCDM's estimation residuals.Source: §4.3 · p. 6
  • An unobserved common driver can give a non-edge a positive score and invalidate the margin condition; for structural |W_j,i|, the readout gap is unbounded.For that readout, the theorem states a condition rather than a guarantee.Source: §4.3 · p. 6

5 Experiments

Experiments assess causal recovery on controlled benchmarks, then test scalability and structural plausibility at larger scales. Results are competitive at moderate scale, with stronger runtime–accuracy tradeoffs and feasibility as system size grows.

5.1 Experimental Setup

  • The benchmarks separate controlled recovery on known graphs from LargeST assessments of feasibility, forecasting, and spatial plausibility without complete directed causal ground truth.The controlled suite includes NC8, simulated EEG, and ROI-level simulated fMRI; voxel-level fMRI adds a larger known graph but includes latent regional drivers.Source: §5.1 · pp. 6–7

5.2 Causal Recovery on Controlled Benchmarks

  • On NC8, simulated EEG, and simulated fMRI, SCDM is competitive but not uniformly superior, with the leading metric varying by benchmark.It ranks second on both metrics for NC8, leads AUPRC on EEG while GVAR and PCMCI lead AUROC, and leads both metrics on simulated fMRI.Source: §5.2 · p. 7

5.3 High-Dimensional Scalability on Synthetic Voxel-level fMRI

  • At 800 seconds, SCDM reaches 0.645 AUROC and 4.01× enrichment, versus CUTS+ at 0.586 and 1.36× and UnCLe at 0.541 and 1.53×.SCDM leads both metrics from 400 seconds onward, with its margin growing as the budget increases.Source: §5.3 · pp. 8–9, Table 2 · p. 8

5.4 Real-World LargeST Scalability

  • On LargeST, high-scoring directed relations become more spatially local during training, but this plausibility check does not establish that individual edges are causal.The reported median distance falls from about 26 km for random pairs to 9 km after 200 steps and 2.9 km after 1,000 steps.Source: §5.4 · p. 9, Figure 3 · p. 9
  • On 3,000 LargeST sensors and 8,997,000 candidate pairs, SCDM completes training and graph extraction in 56.96 s using 6.9 GB peak GPU memory.UnCLe runs out of memory in its joint phase, while CUTS+, JRNGC, and cMLP do not produce a complete score matrix within 80 minutes; this tests feasibility, not causal accuracy.Source: §5.4 · p. 9, Table 3 · p. 9

5.5 Ablation Study

  • Sampling 32 target tasks per meta-episode cuts training time by 3.1× versus all-target updates and gives the best recovery.The ablation fixes the number of outer updates and varies only the number of sampled target tasks; runtime is training time only.Source: §5.5 · p. 10, Table 4 · p. 10
  • With the same test-time adaptation protocol, SCDM's post-adaptation query MSE is 12.1% below Joint MTL and 31.1% below the episodic variant without inner adaptation.Removing the shared matrix W lowers AUPRC from 0.728 to 0.390; the structural readout performs better on simulated fMRI, while the source-isolated readout performs better on simulated EEG.Source: §5.5 · p. 10
  • In NC8-Collinear, graph-level AUPRC remains between 0.741 and 0.965 as a non-parent correlates with a true parent up to 0.89, yet a non-edge outranks the true edge in 5 of 30 runs.With four of 100 variables withheld, the latent-augmented variant retains observed-block AUPRC of 0.692 on simulated fMRI and 0.470 on EEG.Source: §5.5 · p. 10

6 Conclusion

SCDM shares a temporal predictor and causal-strength matrix across target-variable tasks, updating only a subset of tasks per meta-episode. It is competitive on controlled benchmarks and offers a better runtime–accuracy tradeoff at 1,000 and 3,000 variables, under stated limits on recovery.

6 Conclusion

  • SCDM is competitive on controlled benchmarks and gives a better runtime–accuracy tradeoff at 1,000 and 3,000 variables.The conclusion notes that recovery assumes no hidden common causes, a correct lag order, and no instantaneous effects.Source: §6 · p. 10
  • Target-task subsampling lowers training cost without hurting recovery, while the support/query objective improves shared-parameter adaptation over pooled multi-task training.Source: §6 · p. 10
  • SCDM remains observational: hidden drivers can create spurious edges, and nearly redundant sources can shift scores to correlated non-parents.The latent-variable experiments are described as a controlled diagnostic rather than a general solution; future work includes adaptive lag selection, task sampling, and hidden-confounder modeling.Source: §6 · p. 10

Notation

This block introduces the notation table and outlines the theoretical setting for the recovery analysis. The setup defines parent sets through delayed influence and states identifiability, concentration, meta-optimization, and readout conditions.

Notation

  • Table 5 is identified as the paper’s notation reference.Source: Table 5 · p. 14

A Theoretical Analysis

  • The theoretical analysis is restricted to fully observed, non-latent delayed systems and does not claim recovery with hidden confounding, instantaneous effects, incorrect lag specification, or misspecified predictors.Source: A Theoretical Analysis · p. 14

A.1 Population Target and Assumptions

  • For each target, the parent set contains source variables whose lagged histories directly influence that target.Source: §A.1 · p. 14
  • The assumptions require a correct-enough delayed predictor and a positive population ablation-score margin separating parents from non-parents.They also include temporal concentration and finite meta-optimization and readout residual conditions.Source: §A.1 · pp. 14–15

A.2 Statistical, Meta-optimization, and Readout Errors

This section separates statistical estimation, meta-training, and graph-readout errors in the comparison between practical and population ablation scores. Uniform convergence controls the statistical component, while finite optimization and perturbation readout contribute additional residuals.

A.2 Statistical, Meta-optimization, and Readout Errors

  • The statistical lemma bounds uniform convergence of ablation scores, with the stated simplified rate depending on effective sample size and the number of ordered pairs.The union bound ranges over at most d^2 ordered pairs; the simplified rate applies under bounded complexity and geometric mixing.Source: §A.2 · p. 15
  • Finite meta-training, task-batch size, inner adaptation, and the first-order update contribute to the meta-optimization residual.The cited convergence literature motivates the residual form; the analysis does not claim a new rate for this implementation.Source: §A.2 · p. 16
  • The perturbation readout approximates the ideal source-ablation score, and its fidelity error joins statistical and meta-optimization errors in the total bound.The readout discrepancy vanishes in the infinite-data, fully optimized limit when the predictor is well specified and perturbation faithfully removes source-specific history.Source: §A.2 · p. 16

A.3 Proof of Finite-Error Graph Recovery

The recovery result establishes a thresholding guarantee when total statistical, meta-optimization, and readout error is below the population margin. The interpretation extends this to asymptotic recovery under vanishing errors and clarifies the assumptions and scope of the result.

A.3 Proof of Finite-Error Graph Recovery

  • If total error is below half the population margin, thresholding at half the margin recovers the full graph with probability at least 1 − δ.The guarantee assumes the conditions of the theorem, including delayed-system identifiability and bounded statistical, optimization, and readout errors.Source: §A.3 · p. 16

A.4 Asymptotic Recovery and Interpretation

  • As statistical, meta-optimization, and readout errors vanish, the theorem yields asymptotic full-graph recovery; its condition is sufficient, not a sharp sample-complexity characterization.The bound jointly depends on temporal sample size, function-class complexity, graph dimension through a d^2 union bound, and optimization and readout quality.Source: §A.4 · p. 17
  • The risk-gap target measures the increase in optimal prediction risk when a source’s history is removed, while the perturbation readout approximates this without refitting every ordered pair.The theory attributes identifiability to delayed-system assumptions and the population margin, not to meta-learning alone.Source: §A.4 · p. 17

B Experimental Details and Supplementary Results

  • The appendix consolidates baseline configurations, SCDM training and readout details, dataset protocols, and supplementary experimental results.Source: B Experimental Details and… · p. 17

B.1 Baseline and Implementation Details

This section documents benchmark-specific baseline configurations and SCDM’s training, readout, and evaluation procedures. It also describes the computational cost and dataset split conventions used to support reproducible comparisons.

Evaluation protocol

  • For known-graph datasets, AUROC and AUPRC evaluate continuous scores, while threshold-dependent metrics use a validation-selected threshold applied once to test scores.Graph metrics exclude self-edges; voxel-level fMRI also applies a top-k constraint matching ground-truth sparsity.Source: §B.1 · pp. 18, 20

SCDM implementation

  • SCDM uses a two-hidden-layer ReLU MLP and first-order MAML, sampling B target tasks per update with support and query windows.The predictor widths are dpr →128 →64 →1; each task receives K = 3 inner steps, and the outer optimizer is Adam.Source: §B.1 · pp. 18–19
  • The shared structural matrix W is initialized with dataset-specific ranges, masks self-edges to zero, and is regularized on the sampled target’s incoming-edge column.The penalty is element-wise ℓ1 for r = 1 and group lasso for r > 1.Source: §B.1 · p. 19

Computational cost

  • Each meta-episode costs O(B(KLsup + Lqry)Cf(d, p)), while memory is dominated by W, the first predictor layer, and temporary task adaptations.First-order MAML avoids second-order inner-loop terms; these expressions describe per-update/readout cost, not the number of updates needed to converge.Source: §B.1 · p. 20

Post-training graph readout

  • After training, source-isolated readout compares a target’s baseline loss with loss when only one source history is retained; structural readout instead scores edges using |Wj,i|.Source-isolated scores average across independent support/query draws and are used only for d ≤100; structural readout is used for voxel-level fMRI and LargeST-GLA.Source: §B.1 · p. 20

B.2 Dataset Overview and Evaluation Protocols

The benchmark suite separates controlled graph-recovery tests from high-dimensional scalability and real-world forecasting analyses. Evaluation metrics and dataset roles depend on whether directed ground truth is available.

Benchmark overview

  • NC8, simulated EEG, and ROI-level simulated fMRI provide known graphs for controlled causal-recovery evaluation, while voxel-level fMRI provides a larger known graph for runtime and recovery studies.The simulated neuroimaging benchmarks use Schaefer-100 ROIs; the voxel-level benchmark has 1,000 observed variables.Source: §B.2 · p. 21, Figure 4 · p. 22, Figure 5 · p. 23

Evaluation protocol

  • For datasets with known graphs, AUROC and AUPRC measure threshold-independent recovery, while F1 and SHD use a threshold selected on validation and applied to test data.Self-edges are generally excluded because they represent autoregressive dependencies rather than cross-variable causal discovery.Source: §B.2 · p. 21
  • LargeST is used for real-world scalability, forecasting, and qualitative structural analysis because a complete directed causal ground-truth graph is unavailable.Accordingly, causal AUROC and AUPRC are not reported for LargeST.Source: §B.2 · p. 21, Table 10 · p. 22

B.3 Additional Experimental Results

Supplementary experiments examine runtime–accuracy behavior, feasibility at 3,000 variables, forecasting diagnostics, readout choices, and ablations. Results support scalability and useful structural ranking, while qualifying forecasting and individual-edge interpretations.

B.3 Additional Experimental Results

  • On LargeST-GLA-3000 (3,000 sensors), SCDM completed training and graph extraction in 56.96 s on average with 6.9 GB peak GPU memory, a feasibility result rather than causal-accuracy evidence.Source: §B.3 · pp. 23–24
  • From 400 seconds onward, SCDM has higher AUROC and AUPRC enrichment than UnCLe on the voxel-level fMRI benchmark.At 200 seconds, UnCLe is slightly ahead; plotted means are over three seeds without confidence intervals.Source: §B.3 · p. 23, Figure 6 · p. 24
  • The SCDM-derived graph gives the lowest one-step MAE and RMSE among tested LargeST graph priors, reducing MAE by 0.92 versus Self-only AR.Margins over Random and Distance-matched controls are small, so the result is supplementary evidence of forecast compatibility, not standalone forecasting superiority or causal correctness.Source: §B.3 · p. 25
  • On simulated fMRI, learned |W| has higher AUROC and AUPRC than source-isolated readout, while source-isolated readout has similar AUROC and higher AUPRC on simulated EEG.The better readout depends on dataset, and neither readout implements a causal intervention.Source: §B.3 · p. 25, Table 12 · p. 25
  • In NC8-Collinear, individual edge attribution weakens with source collinearity: reversals first appear at mean absolute correlation 0.572 and occur in 5 of 30 runs overall.Graph-level AUPRC remains between 0.741 and 0.965; individual-edge conclusions warrant care when observed sources are nearly redundant.Source: §B.3 · p. 25, Table 13 · p. 26
  • Moderate task batches balance target coverage and optimization efficiency; using all targets per update increases cost and weakens recovery under the same training budget.The ablation changes only the number of sampled target-variable tasks per meta-episode.Source: §B.3 · pp. 25–26
  • On simulated fMRI, removing subsampling lowers AUROC from 0.867 to 0.747 and AUPRC from 0.728 to 0.552; removing W lowers AUPRC from 0.728 to 0.390.These comparisons use the same training budget and evaluation protocol.Source: §B.3 · p. 26, Table 14 · p. 26

C Latent Recovery and Results

  • The latent-augmented extension is assessed under controlled masking, distinct from the main paper’s fully observed temporal causal-discovery setting.The supplementary experiments test hidden-to-observed influence representation and observed-block recovery under those controlled conditions.Source: C Latent Recovery and Results · p. 27

C.1 Latent-Augmented Setting and Evaluation Protocol

The latent-augmented setting withholds selected variables from training input and adds learnable internal states to represent hidden-to-observed influence. Its evaluations use controlled masking, with separate diagnostics for latent recovery and observed-block recovery.

C.1 Latent-Augmented Setting and Evaluation Protocol

  • The learner receives only observed trajectories; withheld trajectories and the full graph are reserved for evaluation, and latent states are learned internal states rather than supervised targets.The controlled protocol selects withheld variables with no observed parents, focusing on hidden drivers of observed dynamics.Source: §C.1 · p. 27
  • The augmented structural matrix represents observed-to-observed, latent-to-observed, and latent temporal influence, with a zero block reflecting the no-observed-parents construction.Latent states enter each observed target through its WZX column and are gated like observed sources.Source: §C.1 · p. 27
  • Latent-coordinate matching precedes metrics involving latent variables, while continuous scores support AUROC and AUPRC and validation-selected thresholds support F1 and SHD.The full augmented graph and hidden-to-observed block are evaluated against the corresponding ground-truth blocks.Source: §C.1 · pp. 27–28

C.2 Masked NC8 Latent Recovery

  • On masked NC8, the latent-augmented model shows non-trivial hidden-to-observed recovery after coordinate matching, supporting a proof-of-concept rather than unrestricted latent recovery.ZX AUROC remains modest, while ZX AUPRC and thresholded metrics indicate ranking of sparse hidden-to-observed influences above many non-edges.Source: §C.2 · p. 28, Table 16 · p. 28

C.3 Observed-block Recovery under Controlled Masking

  • With four variables withheld, latent augmentation remains competitive on observed-block recovery: it achieves the best XX AUPRC on simulated EEG and best XX AUROC and AUPRC on simulated fMRI.These controlled latent-view experiments assess observed-block recovery, not unrestricted hidden causal recovery.Source: §C.3 · pp. 28–29, Table 17 · p. 29

C.4 Interpretation and Limitations

The paper distinguishes its fully observed causal-discovery claims from controlled latent-setting extensions, then documents the construction and evaluation scope of its simulation benchmarks.

C.4 Interpretation and Limitations

  • Latent coordinates are not identifiable in an absolute sense, so the paper evaluates latent-to-observed structural scores under controlled masking rather than exact latent trajectories.Unrestricted discovery with arbitrary hidden confounding, instantaneous effects, or misspecified latent dimension remains outside scope.Source: §C.4 · p. 29

D Details of Simulation Datasets

  • The appendix constructs controlled temporal datasets with known directed ground-truth graphs, including NC8 and neuroimaging-inspired and voxel-level fMRI benchmarks.It focuses on construction details; usage, splits, and evaluation protocols are summarized separately.Source: D Details of Simulation Datasets · p. 29

D.1 NC8 Benchmark

  • NC8 is an eight-variable nonlinear temporal benchmark with a known lagged graph, evaluated with all variables observed or with two withheld as latent.AUROC and AUPRC are the primary threshold-independent metrics in the fully observed setting; the latent-view extension evaluates observed-to-observed and latent-to-observed recovery.Source: §D.1 · pp. 29–30

D.2 Simulated fMRI Benchmark

  • The simulated fMRI benchmark uses 100 atlas-grounded ROIs and a sparse directed lagged graph informed by distance, hemisphere, and functional-network organization.It is a controlled TVB-inspired, BOLD-like simulation, not a full biophysical TVB simulator or empirical structural-connectivity dataset.Source: §D.2 · pp. 30–31
  • ROI-wise hemodynamic response convolution, variable HRF parameters, and observation noise produce BOLD-like observations from stable nonlinear lagged neural dynamics.The same system supports a full-view setting and a latent-view setting that withholds four of 100 ROIs.Source: §D.2 · pp. 30–31
  • The simulated fMRI method substantially improves over random-prediction, lagged-correlation, and Ridge-VAR diagnostic baselines.Random prediction matches the graph-density baseline, while Ridge-VAR remains close to random under nonlinear dynamics and hemodynamic filtering.Source: §D.2 · p. 31

D.3 Simulated EEG Benchmark

This section presents controlled neuroimaging-inspired benchmarks with known directed graphs for evaluating causal recovery and scalability, while documenting assumptions, limitations, and reproducibility. It also reports compute and runtime information and discusses potential benefits and harms, cautioning against high-stakes use of observational causal scores without validation.

D.3 Simulated EEG Benchmark

  • The simulated EEG benchmark uses 100 atlas-defined cortical source ROIs, a sparse network-modular directed lagged graph, and EEG-like oscillatory and broadband dynamics.It supports full-view evaluation with all 100 sources and a latent-view setting with 96 observed and 4 withheld sources. The simplified 64-channel sensor projection is for visualization and robustness analysis, not a realistic biophysical forward model.Source: §D.3 · pp. 32, 34

D.4 High-dimensional Voxel-level fMRI Benchmark

  • The voxel-level fMRI benchmark uses 1,000 variables and 999,000 candidate directed relations to test scalability while retaining a known synthetic graph for recovery evaluation.The voxel1000_v2 version has T = 800 time points and 7,142 true off-diagonal edges, with maximum lag p = 3; learned edge-strength scores |Wij| rank candidate edges.Source: §D.4 · pp. 35–36

D.5 Recoverability and Evaluation Scope

  • The simulations provide controlled recovery and scalability tests, not fully realistic biomedical simulators or evidence of exact biological realism.NC8 tests directed recovery in a small nonlinear system, simulated fMRI and EEG test medium-dimensional atlas-organized systems, and voxel-level fMRI tests high-dimensional score extraction.Source: §D.5 · p. 36

E Broader Impacts

  • SCDM may support more efficient exploration of structured dynamic relations, but observational scores can reflect predictive association rather than causal mechanisms when assumptions fail.The stated assumptions include delayed causal influence, sufficient observation of relevant variables, correct lag specification, and informative predictive risk gaps; high-stakes use should be validated with domain knowledge, interventions, or external evidence.Source: E Broader Impacts · pp. 36–37

NeurIPS Paper Checklist

  • The paper reports limitations including recovery assumptions, nonuniform performance on small benchmarks, edge-score shifts under collinearity, and the use of LargeST for spatial plausibility rather than causal recovery.The checklist also states that the finite-error recovery result's assumptions, supporting lemmas, proofs, and asymptotic interpretation are provided in the paper and appendices.Source: NeurIPS Paper Checklist · pp. 38–39
  • The authors report that code, configurations, and synthetic-data generation scripts are available, with commands to reproduce tables; LargeST is publicly available from its original authors [Liu et al., 2023].The checklist says experimental details include baselines, metrics, seeds, thresholding, runtime measurement, and high-dimensional evaluation protocols; uncertainty estimates are defined in captions, with Table 3 reporting mean ± SD across three seeds.Source: NeurIPS Paper Checklist · pp. 40–41
  • Compute resources and runtime information are documented across the appendices and experiment sections, including GPU type, CPU memory, job limits, training time, runtime, and peak GPU memory.Appendix B.1 lists an NVIDIA A100 80 GB GPU; Section 5.5 and Appendix B.3 report additional timing and memory details.Source: NeurIPS Paper Checklist · p. 41

10. Broader impacts

  • The experiments use controlled or public benchmark settings and do not involve human subjects, private personal data, or a deployed decision-making system.The paper also states that it does not release high-risk models, scraped personal datasets, or deployed systems with substantial misuse potential.Source: NeurIPS Paper Checklist · p. 41, §10 · pp. 42, 44
  • The paper identifies efficient scientific analysis as a potential benefit, but warns that observational causal scores may mislead or cause harm in high-stakes domains without validation.It presents SCDM as a tool for hypothesis generation and structural analysis, not direct deployment without domain knowledge or external evidence.Source: §10 · p. 42
  • The paper credits baseline methods and dataset assets under their stated licenses, and documents synthetic and neuroimaging-inspired benchmark construction without releasing a new public dataset or pretrained model as a primary asset.LargeST is used under CC BY-NC 4.0; the Schaefer-100 parcellation and Yeo-7 labels are cited where used.Source: §10 · p. 43

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