  Sliced inverse regression (SIR, Li 1991) is a pioneering work and the most
recognized method in sufficient dimension reduction. While promising progress
has been made in theory and methods of high-dimensional SIR, two remaining
challenges are still nagging high-dimensional multivariate applications. First,
choosing the number of slices in SIR is a difficult problem, and it depends on
the sample size, the distribution of variables, and other practical
considerations. Second, the extension of SIR from univariate response to
multivariate is not trivial. Targeting at the same dimension reduction subspace
as SIR, we propose a new slicing-free method that provides a unified solution
to sufficient dimension reduction with high-dimensional covariates and
univariate or multivariate response. We achieve this by adopting the recently
developed martingale difference divergence matrix (MDDM, Lee & Shao 2018) and
penalized eigen-decomposition algorithms. To establish the consistency of our
method with a high-dimensional predictor and a multivariate response, we
develop a new concentration inequality for sample MDDM around its population
counterpart using theories for U-statistics, which may be of independent
interest. Simulations and real data analysis demonstrate the favorable finite
sample performance of the proposed method.
