On dimension reduction in regressions with multivariate responses

This paper is concerned with dimension reduction in regressions with multivariate responses on high-dimensional predictors. A unified method that can be regarded as either an inverse regression approach or a forward regression method is proposed to recover the central dimension reduction subspace. By using Stein's Lemma, the forward regression estimates the first derivative of the conditional characteristic function of the response given the predictors; by using the Fourier method, the inverse regression estimates the subspace spanned by the conditional mean of the predictors given the responses. Both methods lead to an identical kernel matrix, while preserving as much regression information as possible. Illustrative examples of a data set and comprehensive simulations are used to demonstrate the application of our methods.