  For the single index model $y=f(\beta^{\tau}x,\epsilon)$ with Gaussian
design, %satisfying that rank $var(\mathbb{E}[x\mid y])=1$ where $f$ is unknown
and $\beta$ is a sparse $p$-dimensional unit vector with at most $s$ nonzero
entries, we are interested in testing the null hypothesis that $\beta$, when
viewed as a whole vector, is zero against the alternative that some entries of
$\beta$ is nonzero. Assuming that $var(\mathbb{E}[x \mid y])$ is non-vanishing,
we define the generalized signal-to-noise ratio (gSNR) $\lambda$ of the model
as the unique non-zero eigenvalue of $var(\mathbb{E}[x \mid y])$. We show that
if $s^{2}\log^2(p)\wedge p$ is of a smaller order of $n$, denoted as
$s^{2}\log^2(p)\wedge p\prec n$, where $n$ is the sample size, one can detect
the existence of signals if and only if gSNR$\succ\frac{p^{1/2}}{n}\wedge
\frac{s\log(p)}{n}$. Furthermore, if the noise is additive (i.e.,
$y=f(\beta^{\tau}x)+\epsilon$), one can detect the existence of the signal if
and only if gSNR$\succ\frac{p^{1/2}}{n}\wedge \frac{s\log(p)}{n} \wedge
\frac{1}{\sqrt{n}}$. It is rather surprising that the detection boundary for
the single index model with additive noise matches that for linear regression
models.
  These results pave the road for thorough theoretical analysis of
single/multiple index models in high dimensions.
