A Geometric Approach to Subset Selection and Sparse Sufficient Dimension Reduction

Sufficient dimension reduction methods allow to estimate lower dimensional subspaces while retaining most of the information about the regression of a response variable on a set of predictors. However, it may happen that only a subset of predictors is actually required. We propose a geometric approach to subset selection by imposing sparsity constraints on certain coefficients which determine the estimated directions. This method can be applied to most existing dimension reduction methods, such as sliced inverse regression and sliced average variance estimation, and may help to improve the estimation accuracy and facilitate interpretation.