Optimal subsampling for estimation of dimension reduction directions

Abstract Sufficient dimension reduction (SDR) techniques have become essential tools in high‐dimensional data analysis. However, many SDR methods rely on iterative procedures, which impose substantial computational costs when applied to large‐scale datasets. To address these challenges, we develop optimal subsampling strategies tailored to two representative SDR approaches: The refined outer product of gradients based on the conditional density functions (rdOPG) and the minimum average (conditional) variance estimation based on the conditional density functions (dMAVE). Our methods are designed to minimize the trace of the asymptotic variance–covariance matrix under an inverse probability weighting framework, thereby achieving an effective trade‐off between computational efficiency and statistical accuracy. We derive explicit forms for the optimal subsampling probabilities and establish the consistency and asymptotic normality of the resulting estimators. Extensive simulation studies and a real‐data application demonstrate that our methods substantially reduce computational burden while achieving improved estimation accuracy compared to uniform subsampling approaches, highlighting their practical value for large‐scale SDR problems.