  The single-index model is a statistical model for intrinsic regression where
responses are assumed to depend on a single yet unknown linear combination of
the predictors, allowing to express the regression function as $ \mathbb{E} [ Y
| X ] = f ( \langle v , X \rangle ) $ for some unknown \emph{index} vector $v$
and \emph{link} function $f$. Conditional methods provide a simple and
effective approach to estimate $v$ by averaging moments of $X$ conditioned on
$Y$, but depend on parameters whose optimal choice is unknown and do not
provide generalization bounds on $f$. In this paper we propose a new
conditional method converging at $\sqrt{n}$ rate under an explicit parameter
characterization. Moreover, we prove that polynomial partitioning estimates
achieve the $1$-dimensional min-max rate for regression of H\"older functions
when combined to any $\sqrt{n}$-convergent index estimator. Overall this yields
an estimator for dimension reduction and regression of single-index models that
attains statistical optimality in quasilinear time.
