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.

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