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This is a tutorial and survey paper on various methods for SufficientDimension Reduction (SDR). We cover these methods with both statisticalhigh-dimensional regression perspective and machine learning approach fordimensionality reduction. We start with introducing inverse regression methodsincluding Sliced Inverse Regression (SIR), Sliced Average Variance Estimation(SAVE), contour regression, directional regression, Principal Fitted Components(PFC), Likelihood Acquired Direction (LAD), and graphical regression. Then, weintroduce forward regression methods including Principal Hessian Directions(pHd), Minimum Average Variance Estimation (MAVE), Conditional VarianceEstimation (CVE), and deep SDR methods. Finally, we explain Kernel DimensionReduction (KDR) both for supervised and unsupervised learning. We also showthat supervised KDR and supervised PCA are equivalent.

Sufficient Dimension Reduction for High-Dimensional Regression and Low-Dimensional Embedding: Tutorial and Survey