  This paper studies the statistical query (SQ) complexity of estimating
$d$-dimensional submanifolds in $\mathbb{R}^n$. We propose a purely geometric
algorithm called Manifold Propagation, that reduces the problem to three
natural geometric routines: projection, tangent space estimation, and point
detection. We then provide constructions of these geometric routines in the SQ
framework. Given an adversarial $\mathrm{STAT}(\tau)$ oracle and a target
Hausdorff distance precision $\varepsilon = \Omega(\tau^{2 / (d + 1)})$, the
resulting SQ manifold reconstruction algorithm has query complexity $O(n
\operatorname{polylog}(n) \varepsilon^{-d / 2})$, which is proved to be nearly
optimal. In the process, we establish low-rank matrix completion results for
SQ's and lower bounds for randomized SQ estimators in general metric spaces.
