Subspace Estimation with Automatic Dimension and Variable Selection in Sufficient Dimension Reduction
提出一个统一框架，将多种充分降维方法推广到高维场景，通过二次凸优化同时自动选择子空间维度和重要变量，理论证明一致性，模拟和实例验证有效。
Sufficient dimension reduction (SDR) methods target finding lower-dimensional representations of a multivariate predictor to preserve all the information about the conditional distribution of the response given the predictor. 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 structural dimension selection and coordinate-independent 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. Supplementary materials for this article are available online.