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

Sparse Sliced Inverse Regression Via Lasso