  Single-Index Models are high-dimensional regression problems with planted
structure, whereby labels depend on an unknown one-dimensional projection of
the input via a generic, non-linear, and potentially non-deterministic
transformation. As such, they encompass a broad class of statistical inference
tasks, and provide a rich template to study statistical and computational
trade-offs in the high-dimensional regime.
  While the information-theoretic sample complexity to recover the hidden
direction is linear in the dimension $d$, we show that computationally
efficient algorithms, both within the Statistical Query (SQ) and the Low-Degree
Polynomial (LDP) framework, necessarily require $\Omega(d^{k^\star/2})$
samples, where $k^\star$ is a "generative" exponent associated with the model
that we explicitly characterize. Moreover, we show that this sample complexity
is also sufficient, by establishing matching upper bounds using a partial-trace
algorithm. Therefore, our results provide evidence of a sharp
computational-to-statistical gap (under both the SQ and LDP class) whenever
$k^\star>2$. To complete the study, we provide examples of smooth and Lipschitz
deterministic target functions with arbitrarily large generative exponents
$k^\star$.
