  We introduce a new sparse sliced inverse regression estimator called Cholesky
matrix penalization and its adaptive version for achieving sparsity in
estimating the dimensions of the 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
computationally 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 strength of the Matrix Lasso and the Lasso
sliced inverse regression estimator; it has superior performance in numerical
studies and can be adapted to other sufficient dimension methods in the
literature.
