Adaptive Estimation of Convex Sets and Convex Polytopes from Noisy Data

2012/11/14 by Victor-Emmanuel Brunel, Brunel, Victor-Emmanuel
Engineering · Mathematics · #FOS: Mathematics #Point processes and geometric inequalities #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1211.3224

openalex publication_date 2012/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We estimate convex polytopes and general convex sets in \mathbb R^d,d≥ 2 in the regression framework. We measure the risk of our estimators using a L¹-type loss function and prove upper bounds on these risks. We show that, in the case of polytopes, these estimators achieve the minimax rate. For polytopes, this minimax rate is (ln n)/(n), which differs from the parametric rate for non-regular families by a logarithmic factor, and we show that this extra factor is essential. Using polytopal approximations we extend our results to general convex sets, and we achieve the minimax rate up to a logarithmic factor. In addition we provide an estimator that is adaptive with respect to the number of vertices of the unknown polytope, and we prove that this estimator is optimal in all classes of polytopes with a given number of vertices.

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