Linear Boolean classification, coding and 'the critical problem'

Emmanuel Abbe, Noga Alon, Afonso S. Bandeira

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

This paper considers the problem of linear Boolean classification, where the goal is to determine in which set, among two given sets of Boolean vectors, an unknown vector belongs to by making linear queries. Finding the least number of queries is formulated as determining the minimal rank of a matrix over GF(2) whose kernel does not intersect a given set S. In the case where S is a Hamming ball, this reduces to finding linear codes of largest dimension. For a general set S, this is an instance of 'the critical problem' posed by Crapo and Rota in 1970, open in general. This work focuses on the case where S is an annulus. As opposed to balls, it is shown that an optimal kernel is composed not only of dense but also of sparse vectors, and the optimal mixture is identified in various cases. These findings corroborate a proposed conjecture that for an annulus of inner and outer radius nq and np respectively, the optimal relative rank is given by the normalized entropy (1 - q)H(p=(1 - q)), an extension of the Gilbert-Varshamov bound.

Original languageEnglish (US)
Title of host publication2014 IEEE International Symposium on Information Theory, ISIT 2014
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages1231-1235
Number of pages5
ISBN (Print)9781479951864
DOIs
StatePublished - Jan 1 2014
Event2014 IEEE International Symposium on Information Theory, ISIT 2014 - Honolulu, HI, United States
Duration: Jun 29 2014Jul 4 2014

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
ISSN (Print)2157-8095

Other

Other2014 IEEE International Symposium on Information Theory, ISIT 2014
CountryUnited States
CityHonolulu, HI
Period6/29/147/4/14

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ASJC Scopus subject areas

  • Theoretical Computer Science
  • Information Systems
  • Modeling and Simulation
  • Applied Mathematics

Cite this

Abbe, E., Alon, N., & Bandeira, A. S. (2014). Linear Boolean classification, coding and 'the critical problem'. In 2014 IEEE International Symposium on Information Theory, ISIT 2014 (pp. 1231-1235). [6875029] (IEEE International Symposium on Information Theory - Proceedings). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ISIT.2014.6875029