Minimax Boundary Estimation and Estimation with Boundary

TL;DR

This paper establishes non-asymptotic minimax bounds for Hausdorff estimation of d-dimensional manifolds with or without boundary, extending existing models and introducing a Voronoi-based boundary reconstruction method.

Contribution

It derives new minimax rates for manifold and boundary estimation and proposes a Voronoi-based procedure for boundary reconstruction.

Findings

Minimax rates of order (log n / n)^{2/d} for manifolds without boundary.

Minimax rates of order (log n / n)^{2/(d+1)} for manifolds with boundary.

A Voronoi-based method effectively identifies points near the boundary for reconstruction.

Abstract

We derive non-asymptotic minimax bounds for the Hausdorff estimation of $d$-dimensional submanifolds $M \subset \mathbb{R}^D$ with (possibly) non-empty boundary $\partial M$. The model reunites and extends the most prevalent $\mathcal{C}^2$-type set estimation models: manifolds without boundary, and full-dimensional domains. We consider both the estimation of the manifold $M$ itself and that of its boundary $\partial M$ if non-empty. Given $n$ samples, the minimax rates are of order $O\bigl((\log n/n)^{2/d}\bigr)$ if $\partial M = \emptyset$ and $O\bigl((\log n/n)^{2/(d+1)}\bigr)$ if $\partial M \neq \emptyset$, up to logarithmic factors. In the process, we develop a Voronoi-based procedure that allows to identify enough points $O\bigl((\log n/n)^{2/(d+1)}\bigr)$-close to $\partial M$ for reconstructing it.