Adversarial Manifold Estimation

TL;DR

This paper introduces a geometric algorithm for estimating submanifolds in high-dimensional space within the statistical query model, achieving nearly optimal query complexity and establishing new bounds for SQ estimators.

Contribution

The paper presents Manifold Propagation, a geometric SQ algorithm for submanifold estimation, with nearly optimal query complexity and new theoretical bounds in metric spaces.

Findings

Query complexity is nearly optimal at O(n polylog(n) ε^{-d/2})

Established low-rank matrix completion results for SQs

Proved lower bounds for randomized SQ estimators in metric spaces

Abstract

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.