DAGAF: A directed acyclic generative adversarial framework for joint structure learning and tabular data synthesis

Hristo Petkov; Calum MacLellan; Feng Dong

cs.LGApr 5, 2026

DAGAF: A directed acyclic generative adversarial framework for joint structure learning and tabular data synthesis

Authors: Hristo Petkov, Calum MacLellan, Feng Dong

Organizations: Department of Computer and Information Sciences, University of Strathclyde, 16 Richmond Street, Glasgow, G1 1XQ, Lanarkshire, United Kingdom.

Abstract

Understanding the causal relationships between data variables can provide crucial insights into the construction of tabular datasets. Most existing causality learning methods typically focus on applying a single identifiable causal model, such as the Additive Noise Model (ANM) or the Linear non-Gaussian Acyclic Model (LiNGAM), to discover the dependencies exhibited in observational data. We improve on this approach by introducing a novel dual-step framework capable of performing both causal structure learning and tabular data synthesis under multiple causal model assumptions. Our approach uses Directed Acyclic Graphs (DAG) to represent causal relationships among data variables. By applying various functional causal models including ANM, LiNGAM and the Post-Nonlinear model (PNL), we implicitly learn the contents of DAG to simulate the generative process of observational data, effectively replicating the real data distribution. This is supported by a theoretical analysis to explain the multiple loss terms comprising the objective function of the framework. Experimental results demonstrate that DAGAF outperforms many existing methods in structure learning, achieving significantly lower Structural Hamming Distance (SHD) scores across both real-world and benchmark datasets (Sachs: 47%, Child: 11%, Hailfinder: 5%, Pathfinder: 7% improvement compared to state-of-the-art), while being able to produce diverse, high-quality samples.

Figures & tables

Figure 1: Pipeline of the DAGAF algorithm

Figure 2: A Visual Representation of DAGAF. (a) The optimization structure under ANM and LiNGAM, where input data is processed to reconstruct

\tilde{\mathbf{X}}

using multiple loss terms, excluding

L_{\text{KLD}}

in the LiNGAM case. (b) The extended framework integrating ANM, LiNGAM, and PNL, where an additional inversion function

g^{-1}

is introduced to compute

L_{\text{PNL}}

, unifying the optimization process. (c) The synthetic data generation process, illustrating how the framework enables structured data synthesis while preserving underlying causal relationships.

Model	SHD (5000 linear samples)
Model	d=10	d=20	d=50	d=100
DAG-Notears	8.6 $\pm$ 7.2	13.8 $\pm$ 9.6	41.8 $\pm$ 29.4	102.8 $\pm$ 53.2
DAG-Notears-MLP	4.6 $\pm$ 4.3	7.6 $\pm$ 6.3	29.6 $\pm$ 18.5	74 $\pm$ 30.6
DAG-GNN	6 $\pm$ 6.9	11.4 $\pm$ 8.2	33.6 $\pm$ 21.2	85.4 $\pm$ 46.4
GAE	5.5 $\pm$ 4.9	10.3 $\pm$ 7.2	31.3 $\pm$ 13.8	80.2 $\pm$ 24.6
GraN-DAG	3.4 $\pm$ 5.2	6.4 $\pm$ 7.5	25.2 $\pm$ 14.6	68.4 $\pm$ 25.8

Table 1: DAG structures recovered from linear data

Model	SHD (5000 non-linear-1 samples)
Model	d=10	d=20	d=50	d=100
DAG-Notears	11.4 $\pm$ 4.5	28.2 $\pm$ 10.2	55 $\pm$ 23.1	105.6 $\pm$ 48.3
DAG-Notears-MLP	5.2 $\pm$ 1.8	15.4 $\pm$ 4.6	43.8 $\pm$ 15.4	86.2 $\pm$ 29.8
DAG-GNN	9.2 $\pm$ 3.3	23.4 $\pm$ 8.4	50.2 $\pm$ 19.5	98.6 $\pm$ 37.6
GAE	8.6 $\pm$ 2.2	20 $\pm$ 5.7	47.5 $\pm$ 10.2	92.3 $\pm$ 18.9
GraN-DAG	4 $\pm$ 2.4	11.2 $\pm$ 6.5	36.4 $\pm$ 11.9	72.8 $\pm$ 21.7

Table 2: DAG structures recovered from non-linear-1 data

Model	SHD (5000 non-linear-2 samples)
Model	d=10	d=20	d=50	d=100
DAG-Notears	10.4 $\pm$ 3.9	22.4 $\pm$ 8.1	47.6 $\pm$ 21.2	112.8 $\pm$ 57.8
DAG-Notears-MLP	5.4 $\pm$ 1.5	13.8 $\pm$ 4.3	30.4 $\pm$ 15.7	85.6 $\pm$ 35.6
DAG-GNN	8.4 $\pm$ 3.2	19.2 $\pm$ 7.7	36.2 $\pm$ 18.6	91.8 $\pm$ 49.3
GAE	7.3 $\pm$ 1.8	17.4 $\pm$ 5.1	33.7 $\pm$ 13.7	88.4 $\pm$ 26.6
GraN-DAG	4.2 $\pm$ 2.1	11.6 $\pm$ 5.6	25.2 $\pm$ 14.5	71.6 $\pm$ 29.7

Table 3: DAG structures recovered from non-linear-2 data

Model	SHD (5000 post-non-linear-1 samples)
Model	d=10	d=20	d=50	d=100
DAG-GNN	13.7 $\pm$ 9.2	21.7 $\pm$ 10.4	63.7 $\pm$ 31.2	118.6 $\pm$ 50.1
GAE	12.3 $\pm$ 8.1	19.1 $\pm$ 8.8	56.2 $\pm$ 24.6	101.3 $\pm$ 37.4
CAREFL	11.8 $\pm$ 6.4	18.5 $\pm$ 7.9	52.1 $\pm$ 22.8	97.2 $\pm$ 34.9
DAG-NF	11.2 $\pm$ 5.3	16.2 $\pm$ 6.1	47.3 $\pm$ 19.5	92.5 $\pm$ 31.3
DAG-WGAN	10.5 $\pm$ 4.7	15.6 $\pm$ 5.8	44.5 $\pm$ 17.7	88.7 $\pm$ 29.6

Table 4: DAG structures recovered from post-non-linear-1 data

Model	SHD (5000 post-non-linear-2 samples)
Model	d=10	d=20	d=50	d=100
DAG-GNN	10.8 $\pm$ 8.7	16.1 $\pm$ 11.9	37.1 $\pm$ 30.3	128.3 $\pm$ 48.2
GAE	9.1 $\pm$ 6.3	14.3 $\pm$ 9.5	31.5 $\pm$ 24.8	105.7 $\pm$ 34.4
CAREFL	8.3 $\pm$ 5.8	13.5 $\pm$ 8.3	29.8 $\pm$ 22.4	92.1 $\pm$ 32.3
DAG-NF	7.7 $\pm$ 5.5	12.8 $\pm$ 7.4	28.4 $\pm$ 21.7	84.8 $\pm$ 28.5
DAG-WGAN	7.2 $\pm$ 5.2	11.4 $\pm$ 6.2	25.2 $\pm$ 18.6	76.5 $\pm$ 27.6

Table 5: DAG structures recovered from post-non-linear-2 data

Datasets	Nodes	DAG-WGAN	DAG-WGAN+	DAG-GNN	DAGAF
Datasets	Nodes	SHD
Child	20	20	19	30	17
Alarm	37	36	35	55	43
Hailfinder	56	73	66	71	63
Pathfinder	109	196	194	218	181

Table 6: DAG structures recovered from benchmark data

Model	Sachs Dataset
Model	SHD
DAG-WGAN	17
DAG-WGAN+	15
DAG-NF	15
DAG-GNN	25
GAE	20

Table 7: DAG structures recovered from real data

Panel of Figure 3 — Figure 3: Comparison of the correlation matrices for real (left) and synthetic (right) features reveals that the statistical correlations across the feature space for both real and synthetic data are nearly identical, in both the ANM (first row) and the PNL (second row) case.

Panel of Figure 4 — Figure 4: Principal Component Analysis (PCA) between the original and synthetic samples for both the ANM (left) and the PNL (right) case. We observe both the input and the synthetic samples have similar clusters and outliers. The results indicate that the implicitly generated distribution resembles the original distribution in both mean and standard deviation, making them indistinguishable from each other.

Panel of Figure 5 — Figure 5: Feature importance comparison between real (left) and synthetic (right) data, in both the ANM (first row) and the PNL (second row) case. The synthetic features with their relevance are indistinguishable from the original ones, allowing for their application in regression tasks.

Figure 6: Visualizing the Wasserstein distance between the original and synthetic data over the course of the augmented Lagrangian algorithm. The significant discrepancy between the real and the generated samples (165-170 and from 300 epochs onward) occurs because of fluctuations in the SHD, courtesy of the parameter-tuning for the continuous optimization approach. Conversely, the lowest SHD is detected when the Wasserstein Distance is at its lower conversions (50-150 and 175 - 275 epochs).

Panel of Figure 7 — Figure 7: Visualizing the distributions of the real and synthetic features, we plotted x5 against x8 (left), x3 against x6 (right), in the case of ANM, and x3 against x4 for the PNL case. The joint and marginal distributions are accurately modeled with no significant differences between the real and synthetic features.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Loss function	SHD
Loss function	Sachs	ECOLI70	MAGIC-IRRI	ARTH150
w/o recon loss	21	115	163	377
recon loss (MSE)	14	91	117	288
recon loss (NLL)	16	106	132	320
MSE + MMD	10	57	80	189
NLL + MMD	14	91	117	288

Appendix

Table 8: DAGAF ablation study

Hyper-parameters	Sachs Dataset
Hyper-parameters	SHD
lr = 3e-3, dropout = 0.5, z-size = 1, batch-size = 100	9
lr = 3e-3, dropout = 0.0, z-size = 1, batch-size = 100	10
lr = 3e-3, dropout = 0.5, z-size = 2, batch-size = 100	10
lr = 3e-3, dropout = 0.5, z-size = 5, batch-size = 100	11
lr = 3e-3, dropout = 0.5, z-size = 1, batch-size = 500	9

Appendix

Table 9: DAGAF sensitivity analysis

Panel of Figure 8 — Figure 8: Further examples of the synthetic joint and marginal distributions for our method on the dataset presented in Section 5.4 . We observe multiple cases with different distribution shapes. Additionally, we depict one case of severe mode collapse (bottom) in the produced data from DAGAF.

Figure 9: Remaining examples of feature importances to supplement the results in Section 5.4 . We observe some failure cases, where the synthetic features differ significantly from their real counterparts.

Feature	$p$ -value
x1	7.7952e-07
x2	0.5004
x3	0.1683
x4	0.0020
x5	0.8563
x6	0.9127

Appendix

Table 10: Mann-Whitney t-test results for all real and synthetic features to supplement Figure 7 . We observe some failure cases, where the real and synthetic features differ significantly (

p<0.05

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Synthetic tabular data generation addresses data scarcity and privacy constraints in a variety of domains. Tabular Prior-Data Fitted Network (TabPFN), a recent foundation model for tabular data, has been shown capable of generating high-quality synthetic tabular data. However, TabPFN is autoregressive: features are generated sequentially by conditioning on the previous ones, depending on the order in which they appear in the input data. We demonstrate that when the feature order conflicts with causal structure, the model produces spurious correlations that impair its ability to generate synthetic data and preserve causal effects. We address this limitation by integrating causal structure into TabPFN's generation process through two complementary approaches: Directed Acyclic Graph (DAG)-aware conditioning, which samples each variable given its causal parents, and a partially directed acyclic graph (PDAG)-based strategy for scenarios with partial causal knowledge. We evaluate these approaches on controlled benchmarks and six CSuite datasets, assessing structural fidelity, distributional quality, and Average Treatment Effect (ATE) preservation. Across most settings, DAG-aware conditioning improves the quality and stability of synthetic data relative to vanilla TabPFN. Under partial causal knowledge, the oracle partially directed acyclic graph (oracle-PDAG), which orients only the edges into the colliders, shows moderate gains, while the benefit of a Completed Partially Directed Acyclic Graph (CPDAG) discovered from data depends on how well the causal structure is recovered. These results indicate that reliable causal structure, even partial, can be injected into TabPFN at inference time, without parameter updates, to improve synthetic data quality.

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