TY - GEN

T1 - Linear Boolean classification, coding and 'the critical problem'

AU - Abbe, Emmanuel

AU - Alon, Noga

AU - Bandeira, Afonso S.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84906543015&partnerID=8YFLogxK

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U2 - 10.1109/ISIT.2014.6875029

DO - 10.1109/ISIT.2014.6875029

M3 - Conference contribution

AN - SCOPUS:84906543015

SN - 9781479951864

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 1231

EP - 1235

BT - 2014 IEEE International Symposium on Information Theory, ISIT 2014

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2014 IEEE International Symposium on Information Theory, ISIT 2014

Y2 - 29 June 2014 through 4 July 2014

ER -