Completion of tree metrics and rank 2 matrices

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.