  Let $(\bX, Y)$ be a random pair taking values in $\mathbb R^p \times \mathbb
R$. In the so-called single-index model, one has $Y=f^{\star}(\theta^{\star
T}\bX)+\bW$, where $f^{\star}$ is an unknown univariate measurable function,
$\theta^{\star}$ is an unknown vector in $\mathbb R^d$, and $W$ denotes a
random noise satisfying $\mathbb E[\bW|\bX]=0$. The single-index model is known
to offer a flexible way to model a variety of high-dimensional real-world
phenomena. However, despite its relative simplicity, this dimension reduction
scheme is faced with severe complications as soon as the underlying dimension
becomes larger than the number of observations ("$p$ larger than $n$"
paradigm). To circumvent this difficulty, we consider the single-index model
estimation problem from a sparsity perspective using a PAC-Bayesian approach.
On the theoretical side, we offer a sharp oracle inequality, which is more
powerful than the best known oracle inequalities for other common procedures of
single-index recovery. The proposed method is implemented by means of the
reversible jump Markov chain Monte Carlo technique and its performance is
compared with that of standard procedures.
