A RKHS-based semiparametric approach to nonlinear dimension reduction

A nonlinear dimension reduction method, the generalized semiparametric kernel sliced inverse regression (GSKSIR for short), was proposed, developed based on the theory of reproducing kernel Hilbert Space (RKHS) and the semiparametric method. The method extends the classical semiparametric method into a more generalized semiparametric domain, and is capable of handling infinite dimensional interested a parameter spaces. With this method, both spaces of nuisance parameters and parameters of interests can be infinitely dimensional, the corresponding generalized nuisance tangent space orthogonal complement was derived, estimation equation for the purpose of dimension reduction was constructed, and optimization of the target function could be achieved based on RKHS theory and regularization method, which leads to a nonlinear estimated sufficient reduced dimension subspace with efficient properties. Furthermore, this new method does not impose the linearity design conditions (LDC) required by methods such as the sliced inverse regression (SIR) and the kernel SIR, and so on, and thus, is more general and can be more widely applied. Finally, a Monte Carlo simulation was conducted, and the results demonstrate the excellent finite sample properties of this new method.