article
Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors
published
yes
Liam T
Solus
author 2AADA620-F248-11E8-B48F-1D18A9856A87
Caroline
Uhler
author 49ADD78E-F248-11E8-B48F-1D18A9856A870000-0002-7008-0216
Ruriko
Yoshida
author
CaUh
department
Gaussian Graphical Models: Theory and Applications
project
For a graph G with p vertices the closed convex cone S⪰0(G) consists of all real positive semidefinite p×p matrices whose sparsity pattern is given by G, that is, those matrices with zeros in the off-diagonal entries corresponding to nonedges of G. The extremal rays of this cone and their associated ranks have applications to matrix completion problems, maximum likelihood estimation in Gaussian graphical models in statistics, and Gauss elimination for sparse matrices. While the maximum rank of an extremal ray in S⪰0(G), known as the sparsity order of G, has been characterized for different classes of graphs, we here study all possible extremal ranks of S⪰0(G). We investigate when the geometry of the (±1)-cut polytope of G yields a polyhedral characterization of the set of extremal ranks of S⪰0(G). For a graph G without K5 minors, we show that appropriately chosen normal vectors to the facets of the (±1)-cut polytope of G specify the off-diagonal entries of extremal matrices in S⪰0(G). We also prove that for appropriately chosen scalars the constant term of the linear equation of each facet-supporting hyperplane is the rank of its corresponding extremal matrix in S⪰0(G). Furthermore, we show that if G is series-parallel then this gives a complete characterization of all possible extremal ranks of S⪰0(G). Consequently, the sparsity order problem for series-parallel graphs can be solved in terms of polyhedral geometry.
Elsevier2016
eng
Linear Algebra and Its Applications10.1016/j.laa.2016.07.026
509247 - 275
Solus, Liam T., et al. “Extremal Positive Semidefinite Matrices Whose Sparsity Pattern Is given by Graphs without K5 Minors.” <i>Linear Algebra and Its Applications</i>, vol. 509, Elsevier, 2016, pp. 247–75, doi:<a href="https://doi.org/10.1016/j.laa.2016.07.026">10.1016/j.laa.2016.07.026</a>.
Solus LT, Uhler C, Yoshida R. Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors. <i>Linear Algebra and Its Applications</i>. 2016;509:247-275. doi:<a href="https://doi.org/10.1016/j.laa.2016.07.026">10.1016/j.laa.2016.07.026</a>
Solus LT, Uhler C, Yoshida R. 2016. Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors. Linear Algebra and Its Applications. 509, 247–275.
Solus, Liam T, Caroline Uhler, and Ruriko Yoshida. “Extremal Positive Semidefinite Matrices Whose Sparsity Pattern Is given by Graphs without K5 Minors.” <i>Linear Algebra and Its Applications</i>. Elsevier, 2016. <a href="https://doi.org/10.1016/j.laa.2016.07.026">https://doi.org/10.1016/j.laa.2016.07.026</a>.
L.T. Solus, C. Uhler, R. Yoshida, Linear Algebra and Its Applications 509 (2016) 247–275.
L. T. Solus, C. Uhler, and R. Yoshida, “Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors,” <i>Linear Algebra and Its Applications</i>, vol. 509. Elsevier, pp. 247–275, 2016.
Solus, L. T., Uhler, C., & Yoshida, R. (2016). Extremal positive semidefinite matrices whose sparsity pattern is given by graphs without K5 minors. <i>Linear Algebra and Its Applications</i>. Elsevier. <a href="https://doi.org/10.1016/j.laa.2016.07.026">https://doi.org/10.1016/j.laa.2016.07.026</a>
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