High-dimensional Missing Data Dimensionality Reduction Based On Slice Inverse Regression

Sliced inverse regression is a classical method of sufficient dimension reduction,which aims to replace the original covariates with a minimal set of their linear combinations on the premise of not losing the conditional distribution information of response variables given explanatory variables,so as to achieve dimensionality reduction.However,the estimated linear combination often contains all the original covariates,making it difficult to identify the main contributing covariates and explain the results.Especially when the number of covariates is large,it is still complicated to establish a follow-up prediction model after dimension reduction.In this paper,a new convex optimization model for dimension reduction in high dimensions is fitted by combining sliced inverse regression method with elastic net.The model proposed in this paper can simultaneously estimate the central dimension reduction subspace and perform variable selection.The linearized alternating direction of method of multipliers algorithm is improved to solve the model in this paper,and an upper bound of the distance between estimated subspace and the real subspace is established.In studies such as biology,medicine or transportation,data are often collected from a variety of modality.However,a particular challenge with multimodality data is the block-missing.Another major contribution of this paper is to provide an effective dimension reduction method to solve the problem of block-missing multimodality data without imputation.The covariance matrix is divided into a linear combination of several specific matrices and the best estimate of the covariance matrix is obtained based on a quadratic loss function.A convex optimization model based on slice inverse regression and elastic net is applied to the case of block-missing data,which realizes the effective dimension reduction for this type of data.Numerical simulation result shows that the dimension reduction model proposed in this paper can identify important covariates in the high-dimensional environment of two types of data,and is more robust than others.