## The 24th Statistical Machine Learning Seminar (2015.7.10)

Time: July 10 (Fri), 2015. 13:30-

Place: Seminar Room 2

Speaker:
Dr. Ruriko Yoshida (University of Kentucky; Visiting Assoc. Prof.
at ISM)

Title:
Extremal Positive Semidefinite Matrices for Weakly Bipartite Graphs

Abstract:

For a graph $G$ with $p$ vertices the cone of concentration matrices
consists of all real positive semidefinite $p\times p$ 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. For a
weakly bipartite graph $G$, we show that the normal vectors to the facets of
the $(\pm1)$-cut polytope of $G$ specify the off-diagonal entries of
extremal matrices in $\K_G$. We also prove that the constant term of the
linear equation of each facet-supporting hyperplane is the rank of its
corresponding extremal matrix in $\K_G$. Furthermore, we show that if $G$ is
series-parallel then this gives a complete characterization of all possible
extremal ranks of $\K_G$, consequently solving the sparsity order problem
for series-parallel graphs.
This is joint work with Liam Solus and Caroline Uhler.