Convex set detection

TL;DR

This paper investigates the minimal detectable size of a segment in a one-dimensional regression setting, analyzing how prior knowledge influences detection capabilities under noisy observations.

Contribution

It provides theoretical insights into the minimal segment size required for accurate detection, considering prior information about segment location.

Findings

Derived bounds on minimal detectable segment size

Analyzed the impact of prior knowledge on detection accuracy

Provided theoretical guidelines for segment detection in noisy data

Abstract

We address the problem of one dimensional segment detection and estimation, in a regression setup. At each point of a fixed or random design, one observes whether that point belongs to the unknown segment or not, up to some additional noise. We try to understand what the minimal size of the segment is so it can be accurately seen by some statistical procedure, and how this minimal size depends on some a priori knowledge about the location of the unknown segment.