Skip to main content

Linear Boolean Classification, Coding and the Critical Problem

Author(s): Abbe, Emmanuel; Alon, N; Bandeira, A; Sandon, C

To refer to this page use:
Full metadata record
DC FieldValueLanguage
dc.contributor.authorAbbe, Emmanuel-
dc.contributor.authorAlon, N-
dc.contributor.authorBandeira, A-
dc.contributor.authorSandon, C-
dc.identifier.citationAbbe, Emmanuel, Alon, Noga, Bandeira, Afonso S, Sandon, Colin. (2016). Linear Boolean Classification, Coding and the Critical Problem. IEEE TRANSACTIONS ON INFORMATION THEORY, 62 (1667 - 1673. doi:10.1109/TIT.2016.2531085en_US
dc.description.abstractThis 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 equivalent to 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 paper 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.en_US
dc.format.extent1667 - 1673en_US
dc.rightsAuthor's manuscripten_US
dc.titleLinear Boolean Classification, Coding and the Critical Problemen_US
dc.typeJournal Articleen_US

Files in This Item:
File Description SizeFormat 
Linear Boolean Classification, Coding and the Critical ProblemBandeira.pdf283.11 kBAdobe PDFView/Download

Items in OAR@Princeton are protected by copyright, with all rights reserved, unless otherwise indicated.