Statistical Query Complexity of Manifold Estimation
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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(τ) oracle and a target Hausdorff distance precision ε = Ω(τ^2 / (d + 1)), the resulting SQ manifold reconstruction algorithm has query complexity O(n polylog(n) ε^-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.
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