Sparse Sliced Inverse Regression via Cholesky Matrix Penalization

TL;DR

This paper presents a novel sparse sliced inverse regression method using Cholesky matrix penalization, improving estimation and variable selection of the central subspace with theoretical guarantees and practical efficiency.

Contribution

It introduces a new Cholesky matrix penalization estimator for sparse dimension reduction, with a novel tuning parameter selection criterion and theoretical consistency results.

Findings

Superior numerical performance over existing methods

Theoretical estimation and variable selection consistency

Effective tuning parameter selection via a new criterion

Abstract

We introduce a new sparse sliced inverse regression estimator called Cholesky matrix penalization and its adaptive version for achieving sparsity in estimating the dimensions of the central subspace. The new estimators use the Cholesky decomposition of the covariance matrix of the covariates and include a regularization term in the objective function to achieve sparsity in a computationally efficient manner. We establish the theoretical values of the tuning parameters that achieve estimation and variable selection consistency for the central subspace. Furthermore, we propose a new projection information criterion to select the tuning parameter for our proposed estimators and prove that the new criterion facilitates selection consistency. The Cholesky matrix penalization estimator inherits the strength of the Matrix Lasso and the Lasso sliced inverse regression estimator; it has superior performance in numerical studies and can be adapted to other sufficient dimension methods in the literature.