Nonlinear Causal Discovery for High Dimensional Data: A Kernelized Trace Method

Causal discovery for high-dimensional observations is a useful tool in many fields such as climate analysis and financial market analysis. A linear Trace method has been proposed to identify the causal direction between two linearly coupled high-dimensional observations X and Y. However, in reality, the relations between X and Y are usually nonlinear and consequently the linear Trace method may fail. In this paper, we propose a method to infer the nonlinear causal relations for two high-dimensional observations X and Y. The idea is to map the observations to high dimensional Reproducing Kernel Hilbert Space (RKHS) such that the nonlinear relations become simple linear ones. We show that the linear Trace condition holds for the causal direction but it is violated for the anti-causal direction in RKHS. Based on this theoretical result, we develop a simple algorithm to infer the causal direction for nonlinearly coupled causal pairs. Synthetic data and real world data experiments are conducted to show the effectiveness of our proposed method.