Certifiably Optimal Sparse Sufficient Dimension Reduction

TL;DR

This paper introduces a novel branch and bound algorithm called Optimal SGEP that efficiently finds certifiably optimal sparse solutions for sufficient dimension reduction, improving interpretability in high-dimensional regression.

Contribution

It develops a customized algorithm that exactly solves the non-convex sparse generalized eigenvalue problem for SDR, ensuring optimal and interpretable solutions.

Findings

Algorithm converges quickly with accurate bounds.

Certifiably optimal solutions are achieved for SDR problems.

Simulation studies demonstrate effectiveness and efficiency.

Abstract

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.