  We consider the problem of the uniform (in $L_\infty$) recovery of ridge
functions $f(x)=\varphi(\langle a,x\rangle)$, $x\in B_2^n$, using noisy
evaluations $y_1\approx f(x^1),\ldots,y_N\approx f(x^N)$. It is known that for
classes of functions $\varphi$ of finite smoothness the problem suffers from
the curse of dimensionality: in order to provide good accuracy for the recovery
it is necessary to make exponential number of evaluations. We prove that if
$\varphi$ is analytic in a neighborhood of $[-1,1]$ and the noise is very
small, $\varepsilon\le\exp(-c\log^2n)$, then there is an efficient algorithm
that recovers $f$ with good accuracy using $O(n\log^2n)$ function evaluations.
