Kernel-Based Structural Equation Models for Topology Identification of Directed Networks.

Structural equation models (SEMs) have been widely adopted for inference of causal interactions in complex networks. Recent examples include unveiling topologies of hidden causal networks over which processes, such as spreading diseases, or rumors propagate. The appeal of SEMs in these settings stems from their simplicity and tractability, since they typically assume linear dependencies among observable variables. Acknowledging the limitations inherent to adopting linear models, the present paper put forth nonlinear SEMs, which account for (possible) nonlinear dependencies among network nodes. The advocated approach leverages kernels as a powerful encompassing framework for nonlinear modeling, and an efficient estimator with affordable tradeoffs is put forth. Interestingly, pursuit of the novel kernel-based approach yields a convex regularized estimator that promotes edge sparsity, a property exhibited by most real world networks, and the resulting optimization problem is amenable to proximal-splitting optimization methods. To this end, solvers with complementary merits are developed by leveraging the alternating direction method of multipliers, and proximal gradient iterations. Experiments conducted on simulated data demonstrate that the novel approach outperforms linear SEMs with respect to edge detection errors. Furthermore, tests on a real gene expression dataset unveil interesting new edges that were not revealed by linear SEMs, which could shed more light on regulatory behavior of human genes.