Kolmogorov-Arnold causal generative models

Alejandro Almodóvar; Mar Elizo; Patricia A. Apellániz; Santiago Zazo; Juan Parras

cs.LGMar 20, 2026

Kolmogorov-Arnold causal generative models

Authors: Alejandro Almodóvar, Mar Elizo, Patricia A. Apellániz, Santiago Zazo, Juan Parras

Abstract

Causal generative models provide a principled framework for answering observational, interventional, and counterfactual queries from observational data. However, many deep causal models rely on highly expressive architectures with opaque mechanisms, limiting auditability in high-stakes domains. We propose KaCGM, a causal generative model for mixed-type tabular data where each structural equation is parameterized by a Kolmogorov--Arnold Network (KAN). This decomposition enables direct inspection of learned causal mechanisms, including symbolic approximations and visualization of parent--child relationships, while preserving query-agnostic generative semantics. We introduce a validation pipeline based on distributional matching and independence diagnostics of inferred exogenous variables, allowing assessment using observational data alone. Experiments on synthetic and semi-synthetic benchmarks show competitive performance against state-of-the-art methods. A real-world cardiovascular case study further demonstrates the extraction of simplified structural equations and interpretable causal effects. These results suggest that expressive causal generative modeling and functional transparency can be achieved jointly, supporting trustworthy deployment in tabular decision-making settings. Code: https://github.com/aalmodovares/kacgm

Figures & tables

Fig. 1 : Compacted sketch of the proposed causal generative model, KaCGM, including a discrete variable ( $\mathrm{x}_{5}$ ). The true SCM, with unknown structural equations (left), is approximated with additive noise models in which the functions of each endogenous variable are modeled by KANs (right). Several KANs are shown, in which interpretability capabilities decrease with network complexity. As explained in Section IV , ${\mathbf{u}}_{5}$ is modeled implicitly by a logistic-KAN. Closed-form expressions can be extracted from every approximated function.

Fig. 2 : Sensitivity analysis on additivity of exogenous variables. Mean and 95

\%

confidence intervals over 10 realizations of the DGP. The true SCM represents the metrics obtained by data generated by the true SCM. CFlow, as a more flexible model, achieves good approximations and independence even with non-additive noise, while KaCGM deviates from this.

	Continuous variables						Mixed type variables
Model	$\text{RF}_{\text{acc}}^{\text{obs}}$	$\text{MMD}_{\text{obs}}$	$\text{RF}_{\text{acc}}^{\text{int}}$	$\text{MMD}_{\text{int}}$	$\text{MAE}_{\text{CF}}$	p-value	$\text{RF}_{\text{acc}}^{\text{obs}}$	$\text{MMD}_{\text{obs}}$	$\text{RF}_{\text{acc}}^{\text{int}}$	$\text{MMD}_{\text{int}}$	p-value
		$\times 10^{-3}$		$\times 10^{-3}$	$\times 10^{-2}$			$\times 10^{-3}$		$\times 10^{-3}$
KAN	$\mathbf{0.51_{0.0}}$	$16.49_{8.1}$	$\mathbf{0.53_{0.0}}$	$4.79_{4.1}$	$5.10_{2.4}$	$0.061^{*}$	$\mathbf{0.52_{0.02}}$	$1.64_{0.74}$	$0.54_{0.02}$	$9.11_{7.24}$	$0.042$
KAAM	$0.52_{0.0}$	$16.76_{9.6}$	$\mathbf{0.53_{0.0}}$	$4.76_{5.1}$	$4.66_{2.3}$	$0.055^{*}$	$\mathbf{0.52_{0.01}}$	$1.89_{0.93}$	$0.54_{0.02}$	$8.40_{5.63}$	$0.024$
ANM	$0.53_{0.0}$	$\mathbf{8.86_{2.6}}$	$0.54_{0.0}$	$\mathbf{3.02_{2.5}}$	$\mathbf{2.62_{2.5}}$	$\underline{0.500}^{*}$	$0.54_{0.02}$	$\mathbf{1.22_{0.44}}$	$0.56_{0.03}$	$\mathbf{6.22_{5.16}}$	$0.042$
DBCM	$0.56_{0.0}$	$31.50_{23.0}$	$0.55_{0.0}$	$6.79_{5.8}$	$3.98_{1.5}$	$<1\text{e-}3$	$0.58_{0.04}$	$3.43_{1.91}$	$0.57_{0.03}$	$12.53_{9.90}$	$<1\text{e-}3$

TABLE I : Unified results for continuous and discrete experiments. Metrics report averages over the 11 synthetic datasets (

\text{mean}_{\text{std}}

) with

1000

training samples. Lower is better for all metrics. Continuous experiments additionally report counterfactual error (

\text{MAE}_{\text{CF}}

), which is not defined for the discrete setting. The last column of each block reports p-values from post-hoc Wilcoxon tests aggregated over all metrics. The Friedman test p-value is

<1\text{e-}3

for both experiments, indicating significant differences among methods overall. Underline denotes the best method. ∗ indicates no statistically significant difference (

p>0.05

Panel of Figure 4 — Fig. 3 : Boxplot of $\log_{10}$ of observational and interventional MMD of each model across 10 realizations with (a) continuous data and (b) mixed type data, as a function of the number of samples with mixed type data.

Additive

Nonadditive

Model

\text{RF}_{\text{acc}}^{\text{obs}}

\text{MMD}_{\text{obs}}

\text{RF}_{\text{acc}}^{\text{int}}

\text{MMD}_{\text{int}}

\text{MAE}_{\text{CF}}

p-value

\text{RF}_{\text{acc}}^{\text{obs}}

\text{MMD}_{\text{obs}}

\text{RF}_{\text{acc}}^{\text{int}}

\text{MMD}_{\text{int}}

\text{MAE}_{\text{CF}}

p-value

\times 10^{-3}

\times 10^{-3}

\times 10^{-2}

\times 10^{-3}

\times 10^{-3}

\times 10^{-2}

KAN

0.54_{0.02}

1.36_{0.41}

0.54_{0.02}

1.95_{0.67}

3.50_{1.40}

0.180^{*}

0.67_{0.13}

5.27_{9.20}

0.67_{0.14}

6.09_{8.75}

3.42_{1.54}

<1\text{e-}3

KAAM

0.54_{0.01}

1.35_{0.40}

0.55_{0.01}

1.96_{0.74}

3.59_{1.35}

0.180^{*}

0.68_{0.14}

5.97_{9.60}

0.68_{0.14}

6.61_{9.12}

3.56_{1.68}

<1\text{e-}3

ANM

\mathbf{0.53_{0.01}}

\mathbf{1.12_{0.28}}

\mathbf{0.53_{0.01}}

\mathbf{1.30_{0.21}}

\mathbf{1.28_{0.43}}

\underline{0.500}^{*}

0.61_{0.09}

1.71_{0.83}

0.61_{0.09}

2.31_{1.46}

1.76_{1.13}

0.007

DBCM

0.56_{0.01}

2.78_{0.73}

0.56_{0.01}

2.38_{0.54}

3.03_{0.47}

<1\text{e-}3

0.59_{0.04}

2.61_{0.78}

0.60_{0.04}

2.34_{0.60}

2.67_{0.78}

0.157^{*}

TABLE II : Results on the Sachs dataset for additive and nonadditive noise settings. Metrics report averages over runs (

\text{mean}_{\text{std}}

). Lower is better for all metrics. The last column of each block reports p-values from post-hoc Wilcoxon tests aggregated over all metrics. The Friedman test p-value is always

<1\text{e-}3

, indicating significant differences among methods overall. Underline denotes the best method. ∗ indicates no statistically significant difference (

p>0.05

Fig. 4 : Boxplots of

\log_{10}

of observational MMD, interventional MMD and CF MAE on Sachs’ semisynthetic dataset. While in the additive setting, all models achieve similar performance, KaCGM worsen significantly in the nonadditive setting.

Fig. 5 : Observational metrics on the cardiovascular dataset as a function of the simplification stage. MAE denotes predictive error when estimating each node from its parents. DBCM and CFlow cannot compute conditional samples and, therefore, MAE is not reported. HSIC and dHSIC are evaluated only for continuous variables.

Fig. 8 : Interpretability analysis of the ischemia mechanism. (Top) PRPs showing the contribution of each parent variable to the logit for two individuals. (Bottom) PDPs representing the contribution

\Delta

of each variable to the logit. The curves also allow direct computation of average treatment effects (ATE) corresponding to hypothetical interventions on each variable.

	Continuous	Discrete	Sachs Additive	Sachs Nonadditive
KAN	$17.71_{20.5}$	$37.88_{52.5}$	$374.90_{53.51}$	$385.17_{66.42}$
KAAM	$13.79_{5.6}$	$3.36_{2.98}$	$356.39_{20.92}$	$345.11_{49.96}$
ANM	$0.02_{0.0}$	$0.02_{0.02}$	$0.28_{0.25}$	$0.20_{0.21}$
DBCM	$11.19_{8.3}$	$6.74_{6.28}$	$502.45_{120.63}$	$483.95_{134.50}$
CFlow	$3.12_{2.8}$	$2.92_{2.20}$	$1.49_{0.51}$	$9.33_{9.12}$
$\text{KAN}_{\text{mix}}$	$-$	$7.45_{33.93}$	$-$	$-$

TABLE III : Training time in seconds across experiments, reported as averages over datasets or runs (

\text{mean}_{\text{std}}

). Continuous and discrete correspond to the synthetic experiments.

Generative Modeling

Generative modeling learns data distributions to create new samples, from images and text to molecules, simulations, and structured records.

See all 493 papers

TabSCM: A practical Framework for Generating Realistic Tabular Data
Apr 24, 2026Sven Jacob, Bardh Prenkaj, Weijia Shao +1Structural Causal Models Causal Counterfactual Generation
Federal Institute for Occupational Safety and Health (BAuA), Dresden, Germany · School of Computation, Information and Technology, Technical University of Munich, Munich, Germany · Munich Center for Machine Learning (MCML)
Causal Variational Deep Embedding: A Family of Interventional Generators for Confounded Images
Jun 19, 2026Jingyuan Chen, Kangrui Ruan, Junzhe ZhangVariational Autoencoders Structural Causal Models
Department of EECS Syracuse University · AWS AI Lab
CausalWrap: Model-Agnostic Causal Constraint Wrappers for Tabular Synthetic Data
Mar 2, 2026Amir Asiaee, Zhuohui J. Liang, Chao YanSynthetic Data Generation Synthetic Tabular Data Generation
Department of Biostatistics, Vanderbilt University Medical Center, Nashville, TN, USA · Department of Biomedical Informatics, Vanderbilt University Medical Center, Nashville, TN, USA

Apr 24, 2026J/K move · Enter open · S save

TabSCM: A practical Framework for Generating Realistic Tabular Data

Sven Jacob, Bardh Prenkaj, Weijia Shao, Gjergji Kasneci

Federal Institute for Occupational Safety and Health (BAuA), Dresden, Germany · School of Computation, Information and Technology, Technical University of Munich, Munich, Germany · Munich Center for Machine Learning (MCML)

Structural Causal Models Causal Counterfactual Generation Synthetic Tabular Data Generation

Open paper AskPDF arXiv

Most tabular-data generators match marginal statistics yet ignore causal structure, leading downstream models to learn spurious or unfair patterns. We present TabSCM, a mixed-type generator that preserves those causal dependencies. Starting from a Completed Partially Directed Acyclic Graph (CPDAG) found by any causal structure discovery algorithm, TabSCM (i) orients edges to a DAG, (ii) fits root-node marginals with KDE or categorical frequencies, and (iii) learns topologically ordered structural assignments. Such assignments are achieved using conditional diffusion models for continuous variables as child nodes and gradient-boosted trees for categorical ones. Ancestral sampling yields semantically valid records and enables exact counterfactual queries. On seven public datasets, encompassing healthcare, finance, housing, environment, TabSCM matches or surpasses state-of-the-art GAN, diffusion, and LLM baselines in statistical fidelity, downstream utility, and privacy risk, while also cutting rule-violation rates and providing causally meaningful and robust conditional interventions. Because generation is decomposed into explicit equations, it runs up to 583

\times

faster than diffusion-only models and exposes interpretable knobs for fairness auditing and policy simulation, making TabSCM a practical choice for realism, explainability, and causal soundness.

Abstract

Generative Modeling

Explore similar work

TabSCM: A practical Framework for Generating Realistic Tabular Data

Causal Variational Deep Embedding: A Family of Interventional Generators for Confounded Images

CausalWrap: Model-Agnostic Causal Constraint Wrappers for Tabular Synthetic Data