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

Alex Markham, Richeek Das, Moritz Grosse-Wentrup

Keywords: graphical causal models, kernels, clustering, structure learning, distance correlation, heterogeneous populations, whole-graph embeddings

TL;DR: We define a new kernel using distance correlation, prove its corresponding kernel space and distance measure are isometric to Hamming distance between causal ancestral graphs, and demonstrate its use in clustering.

Abstract: 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.