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Abstract:
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Sufficient dimension reduction (SDR) methods target finding lower-dimensional representations of predictors to preserve all the information about the regression or classification analysis. The reduction is commonly achieved by projecting the predictor onto a low-dimensional subspace. The smallest such subspace is known as the Central Subspace (CS) and is the key parameter of interest for most SDR methods. In this article, we propose a unified and flexible framework for estimating the CS in high dimensions. Our approach generalizes a wide range of model-based and model-free SDR methods to high-dimensional settings, where the CS is assumed to involve only a subset of the predictors. We formulate the problem as a quadratic convex optimization so that the global solution is feasible. The proposed estimation procedure simultaneously achieves the dimension selection and variable selection of the CS. Theoretically, our method achieves dimension selection, variable selection, and subspace estimation consistency at a high convergence rate under mild conditions. We demonstrate the effectiveness and efficiency of our method with extensive simulation studies and real data examples.
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