Mathematics Faculty Articles

Title

On the Number of Vertices of the Stochastic Tensor Polytope

Document Type

Article

Publication Date

10-2017

Publication Title

Linear and Multilinear Algebra

Keywords

Birkhoff polytope, Birkhoff-von Neumann theorem, Doubly stochastic matrix, Extreme point, Hypermatrix, Multidimensional matrix, Polytope, Stochastic semi-magic cube, Stochastic tensor, Vertex

ISSN

0308-1087

Volume

65

Issue/No.

10

First Page

2064

Last Page

2075

Abstract

This paper studies lower and upper bounds for the number of vertices of the polytope of n x n x n stochastic tensors (i.e. triply stochastic arrays of dimension n). By using known results on polytopes (i.e. the Upper and Lower Bound Theorems), we present some new lower and upper bounds. We show that the new upper bound is tighter than the one recently obtained by Chang et al. [Ann Funct Anal. 2016;7(3):386–393] and also sharper than the one in Linial and Luria’s [Discrete Comput Geom. 2014;51(1);161–170]. We demonstrate that the analog of the lower bound obtained in such a way, however, is no better than the existing ones.

Comments

©2017 Informa UK Limited, trading as Taylor & Francis Group

Additional Comments

National Natural Science Foundation of China grant #s: 11571220, 11531001, 11271256; NNSFC-ISF Research Program #: 11561141001

DOI

10.1080/03081087.2017.1310178

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