2016/12/20 by Daniel Irving Bernstein, Bernstein, Daniel Irving · 7 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics #Advanced Topics in Algebra
paper · doi:10.1016/j.laa.2017.07.016
Motivated by applications to low-rank matrix completion, we give a combinatorial characterization of the independent sets in the algebraic matroid associated to the collection of m× n rank-2 matrices and n× n skew-symmetric rank-2 matrices. Our approach is to use tropical geometry to reduce this to a problem about phylogenetic trees which we then solve. In particular, we give a combinatorial description of the collections of pairwise distances between several taxa that may be arbitrarily prescribed while still allowing the resulting dissimilarity map to be completed to a tree metric.