Nonlinear independent component analysis with minimal nonlinear distortion

Nonlinear ICA may not result in nonlinear blind source separation, since solutions to nonlinear ICA are highly non-unique. In practice, the nonlinearity in the data generation procedure is usually not strong. Thus it is reasonable to select the solution with the mixing procedure close to linear. In this paper we propose to solve nonlinear ICA with the "minimal nonlinear distortion" principle. This is achieved by incorporating a regularization term to minimize the mean square error between the mixing mapping and the best-fitting linear one. As an application, the proposed method helps to identify linear, non-Gaussian, and acyclic causal models when mild nonlinearity exists in the data generation procedure. Using this method to separate daily returns of a set of stocks, we successfully identify their linear causal relations. The resulting causal relations give some interesting insights into the stock market.