Adaptive estimation of planar convex sets

TL;DR

This paper develops adaptive estimators for unknown convex sets in the plane using noisy support function measurements, achieving optimal convergence rates without smoothness assumptions.

Contribution

It introduces data-driven adaptive estimators for both support function and set estimation that are proven to be optimal and adapt to all convex sets.

Findings

Estimators achieve optimal convergence rates for set estimation.

Estimators adapt to all convex sets without smoothness assumptions.

The approach is applicable to both pointwise and set estimation problems.

Abstract

In this paper, we consider adaptive estimation of an unknown planar compact, convex set from noisy measurements of its support function on a uniform grid. Both the problem of estimating the support function at a point and that of estimating the convex set are studied. Data-driven adaptive estimators are proposed and their optimality properties are established. For pointwise estimation, it is shown that the estimator optimally adapts to every compact, convex set instead of a collection of large parameter spaces as in the conventional minimax theory of nonparametric estimation. For set estimation, the estimators adaptively achieve the optimal rate of convergence. In both these problems, our analysis makes no smoothness assumptions on the unknown sets.