SPARSE SLICED INVERSE REGRESSION VIA CHOLESKY MATRIX PENALIZATION

We introduce a new sparse sliced inverse regression estimator called Cholesky matrix penalization, and its adaptive version, for achieving sparsity when estimating the dimensions of a central subspace. The new estimators use the Cholesky decomposition of the covariance matrix of the covariates and include a regularization term in the objective function to achieve sparsity in a computation-ally efficient manner. We establish the theoretical values of the tuning parameters that achieve estimation and variable selection consistency for the central subspace. Furthermore, we propose a new projection information criterion to select the tuning parameter for our proposed estimators, and prove that the new criterion facilitates selection consistency. The Cholesky matrix penalization estimator inherits the ad-vantages of the matrix lasso and the lasso sliced inverse regression estimator. Fur-thermore, it shows superior performance in numerical studies and can be extended to other sufficient dimension reduction methods in the literature.