  Based on the theory of reproducing kernel Hilbert space (RKHS) and
semiparametric method, we propose a new approach to nonlinear dimension
reduction. The method extends the semiparametric method into a more generalized
domain where both the interested parameters and nuisance parameters to be
infinite dimensional. By casting the nonlinear dimensional reduction problem in
a generalized semiparametric framework, we calculate the orthogonal complement
space of generalized nuisance tangent space to derive the estimating equation.
Solving the estimating equation by the theory of RKHS and regularization, we
obtain the estimation of dimension reduction directions of the sufficient
dimension reduction (SDR) subspace and also show the asymptotic property of
estimator. Furthermore, the proposed method does not rely on the linearity
condition and constant variance condition. Simulation and real data studies are
conducted to demonstrate the finite sample performance of our method in
comparison with several existing methods.
