Certifiably Optimal Sparse Sufficient Dimension Reduction

Sufficient dimension reduction (SDR) is a popular tool in regression analysis, which replaces the original predictors with a minimal set of their linear combinations. However, the estimated linear combinations generally contain all original predictors, which brings difficulties in interpreting the results, especially when the number of predictors is large. In this paper, we propose a customized branch and bound algorithm, optimal sparse generalized eigenvalue problem (Optimal SGEP), which combines a SGEP formulation of many SDR methods and efficient and accurate bounds allowing the algorithm to converge quickly. Optimal SGEP exactly solves the underlying non-convex optimization problem and thus produces certifiably optimal solutions. We demonstrate the effectiveness of the proposed algorithm through simulation studies.