Adaptive Estimation of Convex Sets and Convex Polytopes from Noisy Data

TL;DR

This paper develops adaptive estimators for convex sets and polytopes in high-dimensional regression, achieving near-minimax rates and optimality with respect to the number of vertices, under noisy data conditions.

Contribution

It introduces new estimators that adapt to the complexity of convex polytopes, achieving minimax rates and optimality in various classes, extending to general convex sets.

Findings

Estimators achieve the minimax rate of (ln n)/n for polytopes.

The extra logarithmic factor is proven to be necessary.

The adaptive estimator is optimal for polytopes with a fixed number of vertices.

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 that, in the case of polytopes, these estimators achieve the minimax rate. For 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 polytopes with a given number of vertices.