# A Distance Covariance-based Kernel for Nonlinear Causal Clustering in Heterogeneous Populations

arXiv: https://arxiv.org/abs/2106.03480 · HTML: https://mlsanity.com/paper/2106.03480/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 55 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 5 of 5)

- [A Comprehensive Survey of Graph Embedding: Problems, Techniques and Applications](https://mlsanity.com/paper/1709.07604/) (2017)
- [Causal Structure Discovery from Distributions Arising from Mixtures of DAGs](https://mlsanity.com/paper/2001.11940/) (2020)
- [Partial Distance Correlation with Methods for Dissimilarities](https://mlsanity.com/paper/1310.2926/) (2013)
- [Measurement Dependence Inducing Latent Causal Models](https://mlsanity.com/paper/1910.08778/) (2019)
- [MetaCI: Meta-Learning for Causal Inference in a Heterogeneous Population](https://mlsanity.com/paper/1912.03960/) (2019)

## Built on by (most cited first, 10 of 10)

- [Discovering Mixtures of Structural Causal Models from Time Series Data](https://mlsanity.com/paper/2310.06312/) (2023)
- [Hierarchical and Density-based Causal Clustering](https://mlsanity.com/paper/2411.01250/) (2024)
- [A Transformational Characterization of Unconditionally Equivalent Bayesian Networks](https://mlsanity.com/paper/2203.00521/) (2022)
- [Neuro-Causal Factor Analysis](https://mlsanity.com/paper/2305.19802/) (2023)
- [Coarsening Causal DAG Models](https://mlsanity.com/paper/2601.10531/) (2026)
- [Combinatorial and algebraic perspectives on the marginal independence structure of Bayesian networks](https://mlsanity.com/paper/2210.00822/) (2022)
- [Interventional Causal Discovery in a Mixture of DAGs](https://mlsanity.com/paper/2406.08666/) (2024)
- [Incorporating Expert Knowledge into Bayesian Causal Discovery of Mixtures of Directed Acyclic Graphs](https://mlsanity.com/paper/2510.06735/) (2025)
- [Interpretable Clustering with Adaptive Heterogeneous Causal Structure Learning in Mixed Observational Data](https://mlsanity.com/paper/2509.04415/) (2025)
- [Causal Learning for Heterogeneous Subgroups Based on Nonlinear Causal Kernel Clustering](https://mlsanity.com/paper/2501.11622/) (2025)

## Abstract (arXiv)

We consider the problem of causal structure learning in the setting of heterogeneous populations, i.e., populations in which a single causal structure does not adequately represent all population members, as is common in biological and social sciences. To this end, we introduce a distance covariance-based kernel designed specifically to measure the similarity between the underlying nonlinear causal structures of different samples. Indeed, we prove that the corresponding feature map is a statistically consistent estimator of nonlinear independence structure, rendering the kernel itself a statistical test for the hypothesis that sets of samples come from different generating causal structures. Even stronger, we prove that the kernel space is isometric to the space of causal ancestral graphs, so that distance between samples in the kernel space is guaranteed to correspond to distance between their generating causal structures. This kernel thus enables us to perform clustering to identify the homogeneous subpopulations, for which we can then learn causal structures using existing methods. Though we focus on the theoretical aspects of the kernel, we also evaluate its performance on synthetic data and demonstrate its use on a real gene expression data set.