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We consider the problem of causal structure learning in the setting ofheterogeneous populations, i.e., populations in which a single causal structuredoes not adequately represent all population members, as is common inbiological and social sciences. To this end, we introduce a distancecovariance-based kernel designed specifically to measure the similarity betweenthe underlying nonlinear causal structures of different samples. Indeed, weprove that the corresponding feature map is a statistically consistentestimator of nonlinear independence structure, rendering the kernel itself astatistical test for the hypothesis that sets of samples come from differentgenerating causal structures. Even stronger, we prove that the kernel space isisometric to the space of causal ancestral graphs, so that distance betweensamples in the kernel space is guaranteed to correspond to distance betweentheir generating causal structures. This kernel thus enables us to performclustering to identify the homogeneous subpopulations, for which we can thenlearn causal structures using existing methods. Though we focus on thetheoretical aspects of the kernel, we also evaluate its performance onsynthetic data and demonstrate its use on a real gene expression data set.

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