License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.37
URN: urn:nbn:de:0030-drops-75867
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Chen, Xi ; Freilich, Adam ; Servedio, Rocco A. ; Sun, Timothy

Sample-Based High-Dimensional Convexity Testing

LIPIcs-APPROX-RANDOM-2017-37.pdf (0.7 MB)


In the problem of high-dimensional convexity testing, there is an unknown set S in the n-dimensional Euclidean space which is promised to be either convex or c-far from every convex body with respect to the standard multivariate normal distribution. The job of a testing algorithm is then to distinguish between these two cases while making as few inspections of the set S as possible.

In this work we consider sample-based testing algorithms, in which the testing algorithm only has access to labeled samples (x,S(x)) where each x is independently drawn from the normal distribution. We give nearly matching sample complexity upper and lower bounds for both one-sided and two-sided convexity testing algorithms in this framework. For constant c, our results show that the sample complexity of one-sided convexity testing is exponential in n, while for two-sided convexity testing it is exponential in the square root of n.

BibTeX - Entry

  author =	{Xi Chen and Adam Freilich and Rocco A. Servedio and Timothy Sun},
  title =	{{Sample-Based High-Dimensional Convexity Testing}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{37:1--37:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Klaus Jansen and Jos{\'e} D. P. Rolim and David Williamson and Santosh S. Vempala},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-75867},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.37},
  annote =	{Keywords: Property testing, convexity, sample-based testing}

Keywords: Property testing, convexity, sample-based testing
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)
Issue Date: 2017
Date of publication: 11.08.2017

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