Adaptive estimation of convex polytopes and convex sets from noisy data

Abstract

We estimate convex polytopes and general convex sets in $\mathbb{R}^{d}$, $d\geq 2$ 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 $\frac{\ln n}{n}$, 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.