Adversarial Manifold Estimation

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 problem to three natural routines: projection, tangent space estimation, and point detection. then provide constructions these routines SQ framework. Given an adversarial $\mathrm{STAT}(\tau)$ oracle target Hausdorff distance precision $\varepsilon = \Omega(\tau^{2 / (d + 1)})$, resulting manifold reconstruction has $O(n \operatorname{polylog}(n) \varepsilon^{-d 2})$, which is proved be nearly optimal. In process, we establish low-rank matrix completion results for SQ's lower bounds randomized estimators general metric spaces.