Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kolmogorov-Arnold causal generative models

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

Kolmogorov-Arnold causal generative models

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
Paper Structure (25 sections, 2 theorems, 20 equations, 8 figures, 3 tables, 5 algorithms)

This paper contains 25 sections, 2 theorems, 20 equations, 8 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Background
  4. Structural Causal Models
  5. Causal generative models
  6. Kolmogorov-Arnold Networks
  7. Kolmogorov-Arnold causal generative model
  8. KaCGM as ANM
  9. Mixed type $\text{{KaCGM}\xspace}_\text{mix}$
  10. Towards interpretability
  11. Experiments
  12. Validation metrics
  13. Sensitivity analysis
  14. Performance on synthetic datasets
  15. Experiment on continuous data
  16. ...and 10 more sections

Key Result

Proposition 4.1

Consider the true SCM, $\mathcal{M}$, in which one variable, $\mathrm{x}\xspace_j\xspace$ is discrete, with observational probability mass function $p\xspace(\mathrm{x}\xspace_j\xspace\;|\;{\mathrm{pa}(j\xspace)\xspace})$. Let $p\xspace_{\theta\xspace}(\mathrm{x}\xspace_j\xspace\;|\;{\mathrm{pa}(j\x with $\mathrm{u}\xspace_j\xspace \sim \mathcal{U}\xspace(0,1)$. Then i)$\mathrm{x}\xspace_j\xspace

Figures (8)

  • Figure 1: Compacted sketch of the proposed causal generative model, KaCGM, including a discrete variable ($\mathrm{x}\xspace_5$). 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 \ref{['sec:method']}, ${\mathbf{u}}\xspace_5$ is modeled implicitly by a logistic-KAN. Closed-form expressions can be extracted from every approximated function.
  • Figure 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.
  • Figure 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.
  • Figure 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.
  • Figure 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.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3: do-operator pearl2009causality
  • Proposition 4.1: mixed CGM equivalence with SCMs
  • Proposition 4.2: Counterfactual identifiability in mixed GCM
  • proof : Proof of \ref{['prop:interventional_mixed']}
  • proof : Proof of \ref{['prop:counterfactual_mixed']}