Minimax Boundary Estimation and Estimation with Boundary

2021/08/06 by Eddie Aamari, Aamari, Eddie, Catherine Aaron +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Numerical methods in inverse problems #Statistics Theory (math.ST) #Stochastic processes and financial applications

paper · doi:10.48550/arxiv.2108.03135

openalex publication_date 2021/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive non-asymptotic minimax bounds for the Hausdorff estimation of d-dimensional submanifolds M ⊂ ℝ^D with (possibly) non-empty boundary ∂ M. The model reunites and extends the most prevalent C²-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 ∂ M if non-empty. Given n samples, the minimax rates are of order O((log n/n)^2/d) if ∂ M = ∅ and O((log n/n)^2/(d+1)) if ∂ M ≠ ∅, up to logarithmic factors. In the process, we develop a Voronoi-based procedure that allows to identify enough points O((log n/n)^2/(d+1))-close to ∂ M for reconstructing it.

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