Adaptive estimation of convex polytopes and convex sets from noisy data

Abstract

We estimate convex polytopes and general convex sets in R d , d ≥ 2 in the regression framework. We measure the risk of our estimators using a L 1 -type loss function and prove upper bounds on these risks. We show, in the case of convex polytopes, that these estimators achieve the minimax rate. For convex 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 convex polytopes with a given number of vertices.

Methods for Estimation of Convex Sets

“…Throughout the section, we assume that G is a convex body included in [0, 1] d . In [Bru13], the function f is the indicator function of G: f (x) = 1 for x ∈ G, f (x) = 0 otherwise. The design points X 1 , .…”

“…, X n are i.i.d., uniformly distributed in [0, 1] d and the ξ i 's are sub-Gaussian, i.e., E e tξ 1 ≤ e σ 2 t 2 2 , for all t ∈ R, where σ > 0 need not be known. [Bru13] considers a least squares estimatorĜ n ∈ argmin C∈N A(C), where N is a n −2 (d+1) -net of K (1 − 2Y i )1 X i ∈C . The following upper bound is shown in [Bru13], in which P G stands for the joint distribution of the sample with f (⋅) = 1 ⋅∈G .…”

“…[Bru13] considers a least squares estimatorĜ n ∈ argmin C∈N A(C), where N is a n −2 (d+1) -net of K (1 − 2Y i )1 X i ∈C . The following upper bound is shown in [Bru13], in which P G stands for the joint distribution of the sample with f (⋅) = 1 ⋅∈G .…”

“…Theorem 10 ( [Bru13]). There exist three positive constants C 1 , C 2 , C 3 that depend on d and σ 2 and a positive integer n 0 that depends on d only, such that the following holds:…”

“…Another open question is whether G n (orG n ) is adaptive to polytopal supports. In [Bru13], it is shown that the minimax rate on the class P r of polytopes with at most vertices is of the order (ln n) n: Like in density support estimation, is the error ofĜ n (orG n ) of that order when the true support G is a polytope, up to logarithmic factors?…”