Sparse Sliced Inverse Regression Via Lasso

TL;DR

This paper introduces Lasso-SIR, a method for sufficient dimension reduction in high-dimensional settings, demonstrating its consistency and optimal convergence under sparsity, with superior empirical performance.

Contribution

The paper proposes Lasso-SIR, a novel sparse sliced inverse regression method that is consistent and achieves optimal convergence rates in high-dimensional regimes.

Findings

Lasso-SIR is consistent under certain sparsity conditions.

Lasso-SIR achieves the optimal convergence rate.

Lasso-SIR outperforms existing methods in numerical studies.

Abstract

For multiple index models, it has recently been shown that the sliced inverse regression (SIR) is consistent for estimating the sufficient dimension reduction (SDR) space if and only if $\rho=\lim\frac{p}{n}=0$, where $p$ is the dimension and $n$ is the sample size. Thus, when $p$ is of the same or a higher order of $n$, additional assumptions such as sparsity must be imposed in order to ensure consistency for SIR. By constructing artificial response variables made up from top eigenvectors of the estimated conditional covariance matrix, we introduce a simple Lasso regression method to obtain an estimate of the SDR space. The resulting algorithm, Lasso-SIR, is shown to be consistent and achieve the optimal convergence rate under certain sparsity conditions when $p$ is of order $o(n^2\lambda^2)$, where $\lambda$ is the generalized signal-to-noise ratio. We also demonstrate the superior performance of Lasso-SIR compared with existing approaches via extensive numerical studies and several real data examples.