Adversarial Manifold Estimation

This paper studies the statistical query (SQ) complexity of estimating d-dimensional submanifolds in 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 STAT(tau) oracle and a target Hausdorff distance precision epsilon = Omega(tau(2/(d+1))), the resulting SQ manifold reconstruction algorithm has query complexity (O) over tilde (n epsilon(-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.