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

Jingbo Wang; Kegeng Tang; Zihao Wang; Shaogang Ren

cs.LGOct 21, 2026

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

Authors: Jingbo Wang, Kegeng Tang, Zihao Wang, Shaogang Ren

Organizations: University of Tennessee at Chattanooga

Abstract

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.

Figures & tables

Figure 1: Overview of SCDM. Each target variable is treated as a task; sampled tasks undergo temporary support adaptation, and their query losses update the shared temporal predictor and causal-strength matrix.

Method	NC8		Simulated EEG		Simulated fMRI
Method	AUROC $\uparrow$	AUPRC $\uparrow$	AUROC $\uparrow$	AUPRC $\uparrow$	AUROC $\uparrow$	AUPRC $\uparrow$
UnCLe	0.947 $\pm$ 0.014	0.756 $\pm$ 0.080	0.571 $\pm$ 0.008	0.155 $\pm$ 0.008	0.661 $\pm$ 0.012	0.268 $\pm$ 0.006
PCMCI	0.909 $\pm$ 0.003	0.594 $\pm$ 0.022	0.743 $\pm$ 0.012	0.390 $\pm$ 0.018	0.569 $\pm$ 0.012	0.160 $\pm$ 0.010
TCDF	0.740 $\pm$ 0.019	0.415 $\pm$ 0.020	0.685 $\pm$ 0.005	0.224 $\pm$ 0.007	0.585 $\pm$ 0.019	0.178 $\pm$ 0.020
GVAR	0.750 $\pm$ 0.013	0.413 $\pm$ 0.041	0.750 $\pm$ 0.004	0.344 $\pm$ 0.019	0.619 $\pm$ 0.017	0.219 $\pm$ 0.019
CUTS+	0.795 $\pm$ 0.025	0.354 $\pm$ 0.035	0.589 $\pm$ 0.013	0.160 $\pm$ 0.007	0.532 $\pm$ 0.009	0.141 $\pm$ 0.009

Table 1: Causal discovery performance on low- and medium-dimensional temporal benchmarks with known ground-truth graphs. We report AUROC and AUPRC as threshold-independent metrics. Results are shown as mean

\pm

95% confidence interval. Best results are shown in bold and second-best results are underlined .

Figure 2: Qualitative causal recovery on the atlas-grounded simulated fMRI benchmark. (a) Schaefer-100 ROI layout with Yeo-network labels. (b) Network-sorted ground-truth directed graph. (c) Continuous causal scores recovered by SCDM, shown in the same source–target ordering as (b). (d) Representative high-confidence recovered interactions mapped back to ROI space.

Method	200 s	400 s	600 s	800 s
UnCLe (lag 1)	0.573 / 2.05 $\times$	0.556 / 1.85 $\times$	0.556 / 1.85 $\times$	0.541 / 1.53 $\times$
CUTS+ ( $L=3$ )	0.571 / 1.31 $\times$	0.563 / 1.27 $\times$	0.577 / 1.33 $\times$	0.586 / 1.36 $\times$
SCDM	0.566 / 1.96 $\times$	0.566 / 1.96 $\times$	0.605 / 2.47 $\times$	0.645 / 4.01 $\times$

Table 2: Fixed-budget results on voxel-level fMRI (

d=1000

; mean over three seeds). Each entry is AUROC / AUPRC enrichment over random (random AUPRC

=0.0071

). At each budget we evaluate the latest checkpoint finished within that budget, so equal values at consecutive budgets mean that no new checkpoint finished in between. SCDM uses the structural readout

|W_{j,i}|

. Best value per budget in bold .

Figure 3: Spatial-plausibility diagnostic on LargeST-GLA. (a) Locations of the sensors used in this experiment; this panel shows geography only. (b) Representative high-scoring directed relations from SCDM. Each arrow points from the inferred source sensor to the inferred target sensor; only relations shorter than 10 km are drawn to reduce clutter, and arrow curvature is for readability only. (c) Distribution of the geographic distance between the two sensors of each top-scoring relation. The median distance falls from about 26 km for random sensor pairs to 9 km after 200 training steps and 2.9 km after 1,000 steps, so high-scoring relations become more local as training proceeds.

Method	Outcome	Time to outcome (s)	Peak GPU (GB)
SCDM	Completed	$56.96\pm 4.23$	6.9
UnCLe	Out of memory	$349.25\pm 13.30$	15.6
CUTS+	Time limit	$>4800$	72.7
JRNGC	Time limit	$>4800$	14.3
cMLP	Time limit	$>4800$	38.1

Table 3: Feasibility on LargeST-GLA-3000 (3,000 sensors,

8{,}997{,}000

candidate pairs). One A100 80 GB GPU per run, 80-minute limit, three seeds. A run counts as completed only if it writes directed scores for all candidate pairs. Time to outcome is reported as mean

\pm

standard deviation (SD) across three seeds.

Task batch $B$	AUROC $\uparrow$	AUPRC $\uparrow$	F1 $\uparrow$	SHD $\downarrow$	Train (s) $\downarrow$	Speedup $\uparrow$
10	$0.819\pm 0.007$	$0.656\pm 0.017$	$0.541\pm 0.030$	$841.2\pm 30.9$	$\mathbf{226.9\pm 2.5}$	$9.0{\times}$
32	$\mathbf{0.867\pm 0.006}$	$\mathbf{0.728\pm 0.019}$	$\mathbf{0.675\pm 0.025}$	$\mathbf{770.6\pm 22.2}$	$668.4\pm 7.2$	$3.1{\times}$
64	$0.822\pm 0.007$	$0.680\pm 0.012$	$0.588\pm 0.015$	$785.2\pm 14.9$	$1301.0\pm 24.8$	$1.6{\times}$
100	$0.747\pm 0.008$	$0.552\pm 0.015$	$0.392\pm 0.033$	$1020.8\pm 29.9$	$2049.9\pm 13.3$	$1.0{\times}$

Table 4: Task-batch-size ablation on the 100-variable simulated fMRI benchmark. All variants use the full method with the structural readout and differ only in the number of target tasks per meta-episode. Results are mean

\pm

95% CI over five seeds; runtime reports training time only.

Appendix figures & tables32 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Definition
$T$	Number of time points
$d$	Number of observed variables, which is also the number of target tasks
$p$	Observed-history order: the input at time $t$ is $x_{t-1},\ldots,x_{t-p}$
$\ell$	Lag index, $\ell=1,\ldots,p$
$\mathrm{loss}(\cdot,\cdot)$	Pointwise prediction loss (squared error in all experiments)
$\theta$	Shared temporal predictor parameters

Appendix

Table 5: Notation used throughout the paper.

Method	Temporal setting	NC8	Sim. EEG	Sim. fMRI	How chosen
SCDM	observed-history order $p$	16	5	3	True maximum lag
UnCLe	latent AR lag / kernel	1 / 8	3 / 5	3 / 5	NC8: UnCLe benchmark configuration; others: cover the true lag
cMLP	observed lag order	16	3	3	NC8: UnCLe benchmark configuration; others: lag 3
GVAR	observed lag order	16	3	3	As for cMLP
TCDF	kernel / dilation / max delay	16 / – / –	4 / 4 / 5	4 / 4 / 5	Receptive field covers the true lag
PCMCI	maximum lag $\tau_{\max}$	16	5	3	True maximum lag

Appendix

Table 6: Temporal settings of each method. These quantities are not directly comparable: SCDM’s

p

and the observed lag orders set how many past observations enter the model, UnCLe’s lag acts on latent representations after a convolutional encoder, and convolution kernels set a receptive field. The true maximum lag is 16 for NC8, 5 for simulated EEG, and 3 for simulated fMRI.

Dataset	$d$	$p$	$B$	$L_{\mathrm{sup}}$	$L_{\mathrm{qry}}$	$\lambda_{W}$	$N$	Split
NC8	8	16	8	64	64	0.05	1,000	70/15/15
Simulated fMRI	100	3	32	64	64	0.05	5,000	70/15/15
Simulated EEG	100	5	32	64	64	0.05	3,000	70/15/15
Voxel-level fMRI	1,000	3	32	32	32	0.03	500	70/15/15
LargeST-GLA-1500	1,500	3	64	64	64	$10^{-4}$	200	60/20/20
LargeST-GLA-3000	3,000	3	64	64	64	$10^{-4}$	1,000	60/20/20

Appendix

Table 7: Dataset-specific hyperparameters for SCDM.

d

: number of observed variables;

p

: observed-history order;

B

: target tasks per outer update;

L_{\mathrm{sup}}

L_{\mathrm{qry}}

: support/query window length;

\lambda_{W}

: sparsity coefficient (same in support and query losses);

N

: number of outer updates; Split: train/validation/test fraction.

Dataset	$\bm{d}$	$\bm{q}$	GT graph	Primary role
NC8	8	0	Yes	Controlled causal recovery
Masked NC8	6	2	Yes	Controlled latent diagnostic
Simulated fMRI	100	0	Yes	Neuroimaging causal recovery
Latent-view simulated fMRI	96	4	Yes	Latent-view observed-block recovery
Simulated EEG	100	0	Yes	Source-space causal recovery
Latent-view simulated EEG	96	4	Yes	Latent-view observed-block recovery

Appendix

Table 8: Summary of datasets used in the experiments. The symbol

d

denotes the number of observed variables provided to the learner, and

q

denotes withheld or latent variables when applicable. For LargeST, chronological train/validation/test splits are used and the complete directed causal ground truth is unavailable.

Dataset group	Split protocol
NC8 and masked NC8	Training, validation, and test sequences or replicas are separated before model selection and final reporting.
Simulated fMRI / EEG and latent-view variants	200/50/50 train/validation/test samples are used for controlled recovery experiments.
Voxel-level fMRI	Train, validation, and test intervals are separated; ground-truth graph evaluation is performed on held-out test scores.
LargeST-GLA	Chronological train/validation/test intervals are used to avoid future-to-past leakage.

Appendix

Table 9: Dataset split protocol. Synthetic datasets use train, validation, and test partitions for model fitting, threshold selection, and final reporting. LargeST uses chronological splits to avoid leakage from future intervals.

Dataset group	Causal recovery metrics	Auxiliary use
NC8, simulated fMRI, simulated EEG	AUROC, AUPRC, F1, SHD against known directed graphs	Runtime and controlled ablations
Masked NC8 and latent-view neuroimaging benchmarks	Block-level recovery diagnostics under controlled masking	Latent-extension analysis
Voxel-level fMRI	AUROC/AUPRC against known high-dimensional synthetic graph	Runtime–performance tradeoff and scalability analysis
LargeST-GLA	No causal AUROC/AUPRC because complete directed ground truth is unavailable	Real-world scalability, forecasting, and qualitative graph visualization

Appendix

Table 10: Evaluation role of each dataset group. Controlled synthetic datasets support causal recovery metrics because directed ground-truth graphs are available. LargeST is used for real-world scalability, forecasting, and qualitative structural analysis because complete directed causal ground truth is unavailable.

Figure 4: Overview of the simulated fMRI benchmark. The benchmark uses Schaefer-100 cortical ROIs grouped by Yeo-7 functional networks. The visualization summarizes the atlas-based ROI layout, the network-sorted directed ground-truth graph, representative BOLD-like time series, and the structured lagged effective-connectivity pattern used for evaluation. This figure illustrates the construction of the controlled TVB-inspired, atlas-grounded fMRI simulation rather than empirical subject-level fMRI connectivity.

Figure 5: Overview of the simulated EEG benchmark. The benchmark is constructed as a Schaefer-100 source-space EEG-like simulation with Yeo-network-modular directed source interactions. The visualization summarizes the source ROI organization, the network-sorted source-level ground-truth graph, representative oscillatory source signals, and the synthetic 64-channel source-to-sensor projection used for qualitative visualization. The projection is a simplified source-mixing model and is not intended to represent a subject-specific BEM/FEM EEG forward solution.

Figure 6: Runtime–accuracy tradeoff on the high-dimensional voxel-level fMRI benchmark under fixed wall-clock budgets. Each point reports the mean over three random seeds, without confidence intervals. Left: AUROC computed from continuous causal scores. Right: AUPRC enrichment over random, defined as AUPRC normalized by the random-ranking baseline. UnCLe is slightly ahead at 200 seconds; from 400 seconds on, SCDM has higher AUROC and AUPRC enrichment.

Graph prior	MAE $\downarrow$	RMSE $\downarrow$	$\Delta$ MAE vs. Self-only $\uparrow$
Self-only AR	20.60	32.35	—
Random	19.77 $\pm$ 0.06	31.11 $\pm$ 0.16	+0.83
Distance-kNN	20.37	31.98	+0.23
Road-network	20.68	32.91	$-$ 0.08
UnCLe	19.90	31.38	+0.70
Permuted SCDM	19.92 $\pm$ 0.33	31.44 $\pm$ 0.77	+0.68

Appendix

Table 11: Supplementary one-step graph-conditioned forecasting on LargeST-GLA-1500 active sensors. All variants use the same graph-masked RidgeVAR forecaster with

k=5

, lag

p=3

, and fixed ridge regularization

\alpha=1.0

. Active sensors are selected using train-only standard deviation

\sigma_{\mathrm{train}}\geq 10

(

d=1489

Dataset	Readout	AUROC $\uparrow$	AUPRC $\uparrow$
Simulated fMRI	Learned $\|W\|$ score	$0.867\pm 0.006$	$0.728\pm 0.019$
Simulated fMRI	Source-isolated perturbation readout	$0.832\pm 0.003$	$0.693\pm 0.010$
Simulated EEG	Learned $\|W\|$ score	$0.713\pm 0.053$	$0.426\pm 0.086$
Simulated EEG	Source-isolated perturbation readout	$0.717\pm 0.037$	$0.476\pm 0.067$

Appendix

Table 12: Readout ablation on the simulated fMRI and simulated EEG benchmarks. Both readouts are applied to SCDM outputs. The learned

|W|

score evaluates whether the trained causal-strength matrix itself contains useful structural information, while the source-isolated perturbation readout evaluates an alternative post-training graph-extraction procedure. Results are reported as mean

\pm

95% confidence interval.

Driver $a$	Mean $\|\mathrm{corr}(X_{1},X_{2})\|$	Condition number	AUPRC $\uparrow$	Reversals $\downarrow$
0.0	0.037	8.45	0.741	0/5
0.2	0.252	9.91	0.960	0/5
0.4	0.572	12.76	0.955	1/5
0.6	0.750	17.09	0.947	1/5
0.8	0.841	21.81	0.965	1/5
1.0	0.892	26.69	0.915	2/5

Appendix

Table 13: NC8-Collinear stress test (five seeds per level). Reversals count runs in which the non-edge

X_{2}\to X_{3}

scores above the true edge

X_{1}\to X_{3}

Variant	Task subsample	Shared $W$	AUROC $\uparrow$	AUPRC $\uparrow$	F1 $\uparrow$	SHD $\downarrow$
Full SCDM	✓	✓	$\mathbf{0.867\pm 0.006}$	$\mathbf{0.728\pm 0.019}$	$\mathbf{0.675}$	$\mathbf{770.6}$
w/o task subsampling	–	✓	$0.747\pm 0.008$	$0.552\pm 0.015$	$0.392$	$1020.8$
w/o shared $W$ (MLP-only)	✓	–	$0.732\pm 0.012$	$0.390\pm 0.022$	$0.401$	$1486$

Appendix

Table 14: Component ablation on the Schaefer100 TVB-inspired simulated fMRI benchmark. All variants use the same training budget and evaluation protocol. AUROC and AUPRC are reported as mean

\pm

95% confidence interval; F1 and SHD are reported as means.

Training variant	Inner steps in training	Post-adaptation query MSE $\downarrow$
Joint MTL (sampled targets)	–	4.305
Episodic, no inner adaptation	$K=0$	5.487
Full SCDM	$K=3$	3.782

Appendix

Table 15: Held-out-target adaptation. All variants share architecture, data, initialization, training budget, and test-time adaptation; only the training objective differs.

Aggregate and latent-to-observed recovery using structural scores
Evaluation target	AUROC $\uparrow$	AUPRC $\uparrow$	F1 $\uparrow$	SHD $\downarrow$
Full augmented graph	$0.926\pm 0.008$	$0.634\pm 0.026$	$0.598\pm 0.057$	$4.87\pm 0.49$
Latent $\to$ observed (ZX)	$0.606\pm 0.040$	$0.680\pm 0.028$	$0.648\pm 0.059$	$3.80\pm 0.44$
Observed-block score diagnostic
Evaluation target	Score readout	AUROC $\uparrow$
Observed $\to$ observed (XX)	Structural score $\|W\|$	$0.943\pm 0.007$

Appendix

Table 16: Controlled latent recovery on masked NC8. Upper block: full augmented graph and latent-to-observed (ZX) recovery with structural scores. Lower block: observed-to-observed (XX) recovery with the structural score and with the source-isolated readout. Mean

\pm

95% CI over five graph replicas and three seeds.

Method	Simulated EEG		Simulated fMRI
Method	XX AUROC $\uparrow$	XX AUPRC $\uparrow$	XX AUROC $\uparrow$	XX AUPRC $\uparrow$
UnCLe	0.566 $\pm$ 0.005	0.146 $\pm$ 0.002	0.626 $\pm$ 0.033	0.228 $\pm$ 0.033
PCMCI	0.745 $\pm$ 0.011	0.388 $\pm$ 0.019	0.566 $\pm$ 0.015	0.156 $\pm$ 0.012
TCDF	0.681 $\pm$ 0.006	0.213 $\pm$ 0.012	0.590 $\pm$ 0.009	0.179 $\pm$ 0.011
GVAR	0.747 $\pm$ 0.003	0.339 $\pm$ 0.016	0.619 $\pm$ 0.014	0.214 $\pm$ 0.014
CUTS+	0.589 $\pm$ 0.001	0.154 $\pm$ 0.000	0.527 $\pm$ 0.012	0.137 $\pm$ 0.006

Appendix

Table 17: Observed-block causal recovery under controlled latent-view masking on simulated EEG and fMRI. Results are reported as mean

\pm

95% confidence interval. Best results are shown in bold and second-best results are underlined .

Statistic	NC8	Masked NC8
Total variables in full system	8	8
Observed variables provided to learner	8	6
Withheld / latent variables	0	2
Time points	2000	2000
Ground-truth graph available	Yes	Yes
Maximum lag	16	16

Appendix

Table 18: Statistics of the NC8 benchmark and its controlled latent-view variant.

Component	Setting
Parcellation	Schaefer-100 cortical atlas
Number of ROIs	100
Coordinate space	MNI152
Functional labels	Yeo-7 canonical networks
Hemisphere labels	Left/right hemisphere labels retained
ROI metadata	ROI name, hemisphere, Yeo network, and MNI centroid

Appendix

Table 19: Atlas and anatomical metadata used in the simulated fMRI benchmark.

Component	Implementation
Node definition	Cortical ROIs from the Schaefer-100 atlas
Topology type	Sparse directed lagged effective-connectivity graph
Structural prior	Distance-, hemisphere-, and network-informed edge sampling
Spatial constraint	Nearby ROIs have higher prior probability while long-range edges are retained
Network constraint	Within-network and selected cross-network interactions are allowed
Hemisphere constraint	Both intra-hemispheric and inter-hemispheric edges are retained

Appendix

Table 20: Ground-truth graph construction in the simulated fMRI benchmark.

Parameter	Value
Dynamics type	Stable nonlinear lagged ROI dynamics
Self autoregression	0.50
Coupling scale	0.25 before stability rescaling
Spectral-radius target	0.87
Process noise standard deviation	0.045
Slow drive standard deviation	0.015

Appendix

Table 21: Neural source dynamics used in the simulated fMRI benchmark.

Parameter	Value
Observation type	BOLD-like ROI time series
TR	2.0 s
HRF type	Double-gamma
HRF peak delay	Uniform(4.5, 6.5) s
HRF undershoot delay	Uniform(12.0, 18.0) s
HRF undershoot scale	Uniform(0.15, 0.30)

Appendix

Table 22: Hemodynamic observation model for the simulated fMRI benchmark.

Statistic	Full-view	Latent-view
Total variables in full system	100	100
Observed variables provided to learner	100	96
Withheld / latent variables	0	4
Time points per sample	1000	1000
TR	2.0 s	2.0 s
Ground-truth graph available	Yes	Yes

Appendix

Table 23: Full-view and latent-view versions of the simulated fMRI benchmark.

Method	AUROC / AUPRC	Interpretation
Random baseline	0.500 / 0.123	Expected from graph density
Lagged correlation max-lag	0.673 / 0.285	Simple temporal baseline has signal but remains limited
Ridge-VAR	0.526 / 0.133	Linear VAR remains near random
Ours, 5000 steps	0.823 / 0.669	Strong improvement over simple baselines

Appendix

Table 24: Recoverability diagnostics for the simulated fMRI benchmark.

Component	Setting
Parcellation	Schaefer-100 cortical atlas
Number of source ROIs	100
Functional labels	Yeo-7 canonical networks
Hemisphere labels	Left/right hemisphere labels retained
ROI metadata	ROI name, hemisphere, Yeo network, and atlas coordinate information
Use in simulation	Network-aware source graph construction

Appendix

Table 25: Atlas and source-space metadata used in the simulated EEG benchmark.

Component	Implementation
Node definition	Cortical source ROIs from the Schaefer-100 atlas
Topology type	Sparse directed lagged source effective-connectivity graph
Structural prior	Yeo-network-modular edge sampling
Within-network structure	Denser directed interactions within the same functional network
Cross-network structure	Sparser directed interactions across functional networks
Lag structure	Directed effects occur over lags 1–5

Appendix

Table 26: Ground-truth source graph construction in the simulated EEG benchmark.

Parameter	Value
Dynamics type	Stable lagged source dynamics with oscillatory components
Lag order	1–5
Time points per sample	2000
Frequency structure	Theta, alpha, and beta-band source activity
Broadband component	$1/f$ -like background activity
Nonstationarity	Slow amplitude modulation and ROI-level variability

Appendix

Table 27: Source dynamics and spectral diagnostics for the simulated EEG benchmark.

Component	Setting
Number of synthetic sensors	64
Sensor layout	Synthetic 10–10-like layout
Leadfield / mixing matrix	leadfield_synthetic.npy , shape $64\times 100$
Sensor positions	sensor_positions.npy , shape $64\times 3$
Full sensor signal	EEG.npy , generated from all 100 source ROIs
Observed-only sensor signal	EEG_obs96.npy , generated from 96 observed source ROIs

Appendix

Table 28: Synthetic source-to-sensor projection used in the simulated EEG benchmark.

Statistic	Full-view	Latent-view
Total source ROIs in full system	100	100
Observed variables provided to learner	100	96
Withheld / latent variables	0	4
Time points per sample	2000	2000
Ground-truth graph available	Yes	Yes
Graph type	Directed lagged source EC	Directed lagged source EC

Appendix

Table 29: Full-view and latent-view versions of the simulated EEG benchmark.

Index	ROI name
5	7Networks_LH_Vis_6
58	7Networks_RH_SomMot_1
86	7Networks_RH_Cont_Cing_1
41	7Networks_LH_Default_PFC_1

Appendix

Table 30: Withheld ROIs in the latent-view simulated EEG setting.

Block	Description
A_xx.npy	Directed edges among observed source ROIs, $X\to X$
A_xz.npy	Directed edges from observed source ROIs to hidden source ROIs, $X\to Z$
A_zx.npy	Directed edges from hidden source ROIs to observed source ROIs, $Z\to X$
A_zz.npy	Directed edges among hidden source ROIs, $Z\to Z$

Appendix

Table 31: Ground-truth graph blocks in the latent-view simulated EEG setting.

Method	AUROC / AUPRC	Interpretation
Random baseline	0.500 / 0.1149	Expected from graph density
Lagged correlation	0.6292 / 0.1826	Simple temporal association has partial signal
Ridge-VAR	0.6983 / 0.2668	Linear VAR recovers part of the lagged structure
Ours, 3000 steps	0.6933 / 0.4598	Stronger sparse-edge ranking in AUPRC

Appendix

Table 32: Recoverability diagnostics for the simulated EEG benchmark. The “Ours” row is a one-seed smoke diagnostic on the full 100-source non-latent setting and is not a replacement for the final multi-seed experimental table.

Statistic	Value
Dataset version	voxel1000_v2
Dataset type	Controlled high-dimensional BOLD-like temporal simulation
Observed variables	$1000$
Latent drivers	$4$
Time points	$800$
Spatial layout	$10\times 10\times 10$ voxel grid

Appendix

Table 33: At-a-glance summary of the high-dimensional voxel-level fMRI benchmark used in the scalability experiments.

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Oct 2, 2026

Mapping and Advancing the Scalability-Accuracy Frontier of Nonlinear Causal Discovery

Scalable nonlinear causal discovery requires methods that combine flexible mechanism estimators with efficient search over large graph spaces. Several algorithmic families have been proposed to address this challenge, yet their accuracy-runtime trade-offs remain poorly understood. We empirically compare the four major approaches: differentiable structure learning, amortized structure learning, score-matching, and combinatorial search. Our results reveal complementary bottlenecks: differentiable and amortized methods scale well but exhibit an accuracy gap, score-matching methods can be accurate in low dimensions but degrade quickly for increasing feature sizes, and combinatorial methods remain accurate but are slowed by repeated and redundant local scoring. Motivated by this bottleneck, we develop SPADE, a spline-based score-evaluation scheme that compiles sufficient statistics once and reuses them throughout combinatorial search. Under bounded indegree, its Gaussian variant reduces algorithmic complexity from O(nd^3) to O(nd^2+d^3). Empirically, SPADE shifts the observed scalability-accuracy frontier by orders of magnitude: it solves 100-variable problems with 160K samples in seconds and 1600-variable problems with 2.5K samples in minutes, while retaining high structural accuracy across synthetic and real-world benchmarks. These results reveal a substantial shift in the practical scale of combinatorial search and highlight the importance of evaluating scalable causal-discovery methods along the full accuracy-runtime frontier.

Hendrik Suhr, Sascha Xu, Jilles Vreeken

CISPA Helmholtz Center for Information Security

Score-Based Causal Discovery Causal Discovery

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Abstract

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Mapping and Advancing the Scalability-Accuracy Frontier of Nonlinear Causal Discovery