Adversarial Manifold Estimation

Eddie Aamari (1, 2, 3, 4) , Alexander Knop (5)

1 SU - Sorbonne Université
2 CNRS - Centre National de la Recherche Scientifique
3 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
4 UPCité - Université Paris Cité
5 UC San Diego - University of California [San Diego]

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.

Domaines: Statistiques [math.ST]; Complexité [cs.CC]

Eddie Aamari, Alexander Knop. Adversarial Manifold Estimation. Foundations of Computational Mathematics, 2022. ⟨hal-02995288v2⟩