Algebraic boundary of matrices of nonnegative rank at most three

Rob H. Eggermont; Emil Horobeţ; Kaie Kubjas

doi:10.1016/j.laa.2016.06.036

Algebraic boundary of matrices of nonnegative rank at most three

Rob H. Eggermont^*, Emil Horobeţ, Kaie Kubjas

^*Corresponding author for this work

Research output: Contribution to journal › Article › Scientific › peer-review

2 Citations (Scopus)

Abstract

Understanding the boundary of the set of matrices of nonnegative rank at most r is important for applications in nonconvex optimization. The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Gröbner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author.

Original language	English
Pages (from-to)	62-80
Number of pages	19
Journal	Linear Algebra and Its Applications
Volume	508
DOIs	https://doi.org/10.1016/j.laa.2016.06.036
Publication status	Published - 1 Nov 2016
MoE publication type	A1 Journal article-refereed

Keywords

Equivariant Gröbner basis
Mixture model
Nonnegative rank
Stabilization

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10.1016/j.laa.2016.06.036

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abstract = "Understanding the boundary of the set of matrices of nonnegative rank at most r is important for applications in nonconvex optimization. The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Gr{\"o}bner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author.",

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author = "Eggermont, {Rob H.} and Emil Horobe{\c t} and Kaie Kubjas",

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AU - Horobeţ, Emil

AU - Kubjas, Kaie

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N2 - Understanding the boundary of the set of matrices of nonnegative rank at most r is important for applications in nonconvex optimization. The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Gröbner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author.

AB - Understanding the boundary of the set of matrices of nonnegative rank at most r is important for applications in nonconvex optimization. The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Gröbner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author.

KW - Equivariant Gröbner basis

KW - Mixture model

KW - Nonnegative rank

KW - Stabilization

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DO - 10.1016/j.laa.2016.06.036

M3 - Article

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SN - 1873-1856

VL - 508

SP - 62

EP - 80

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Algebraic boundary of matrices of nonnegative rank at most three

Abstract

Keywords

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