Abstract: The sliced inverse regression (SIR) is the most recognized method in sufficient dimension reduction. For high-dimensional multivariate applications, there is promising progress related to the theory and methods of high-dimensional SIR. However, two problems remain in this context. First, choosing the number of slices in an SIR is difficult, and depends on the sample size, distributions of the variables, and other practical considerations. Second, extending SIR from a univariate response to a multivariate response is not trivial. Targeting the same dimension reduction subspace as that of the SIR, we propose a new slicing-free method that provides a unified solution to sufficient dimension reduction for high-dimensional covariates and univariate or multivariate responses. We achieve this by adopting the martingale difference divergence matrix (MDDM) and penalized eigendecomposition algorithms. To establish the consistency of our method for a high-dimensional predictor and a multivariate response, we develop a new concentration inequality for the sample MDDM around its population counterpart using U- statistics theory, which may be of independent interest. Simulations and a real- data analysis demonstrate the favorable finite-sample performance of the proposed method.

Multivariate response, sliced inverse regression, sufficient dimension reduction, U-statistic.