Testing convexity of figures under the uniform distribution

Piotr Berman, Meiram Murzabulatov, Sofya Raskhodnikova

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We consider the following basic geometric problem: Given 𝜖 ∈ (0, 1/2), a 2-dimensional black-and-white figure is ∊- far from convex if it differs in at least an ∊ fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is ∊- far from convex are needed to detect a violation of convexity with constant probability? This question arises in the context of designing property testers for convexity. We show that Θ(𝜖 -4/3 ) uniform samples (and the same running time) are necessary and sufficient for detecting a violation of convexity in an ∊-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our algorithm beats the Ω(𝜖 −3∕2 ) lower bound by Schmeltz [32] on the number of samples required for learning convex figures under the uniform distribution. It demonstrates that, with uniform samples, we can check if a set is approximately convex much faster than we can find an approximate representation of a convex set.

Original languageEnglish (US)
Pages (from-to)413-443
Number of pages31
JournalRandom Structures and Algorithms
Volume54
Issue number3
DOIs
StatePublished - May 2019

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

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