  Hilbert-Schmidt Independence Criterion (HSIC) has recently been used in the
field of single-index models to estimate the directions. Compared with some
other well-established methods, it requires relatively weaker conditions.
However, its performance has not yet been studied in the high-dimensional
scenario, where the number of covariates is much larger than the sample size.
In this article, we propose a new efficient sparse estimate in HSIC based
single-index model. This new method estimates the subspace spanned by the
linear combinations of the covariates directly and performs variable selection
simultaneously. Due to the non-convexity of the objective function, we use a
majorize-minimize approach together with the linearized alternating direction
method of multipliers algorithm to solve the optimization problem. The
algorithm does not involve the inverse of the covariance matrix and therefore
can handle the large p small n scenario naturally. Through extensive simulation
studies and a real data analysis, we show our proposal is efficient and
effective in the high-dimensional setting. The Matlab codes for this method are
available online.
