  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.
