Minimax Boundary Estimation and Estimation with Boundary
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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 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 ∂ 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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