Sufficient dimension reduction (SDR) is known to be a powerful tool for achieving data reduction and visualization in regression and classification problems. In this work, we study high dimensional SDR problems and propose solutions under a unified minimum discrepancy approach with regularization. When p grows exponentially fast with n, consistency results in both central subspace estimation and variable selection are established simultaneously for important SDR methods. The proposed approach is equipped with a new algorithm to efficiently solve regularized objective functions without the need to invert a large covariance matrix. We further study a unified framework of SDR to high-dimensional survival analysis under weak modeling assumptions. This framework includes many popular survival regression models as special cases, and produces a number of practically useful outputs with theoretical guarantees, including a uniformly consistent Kaplan-Meier type estimator of the conditional distribution function of the survival time and a consistent estimator of the conditional quantile survival time in high dimension. Promising applications of our proposal are demonstrated through simulations and real data analysis on biomedical studies.
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