Completion of tree metrics and rank-2 matrices

TL;DR

This paper characterizes the independent sets in the algebraic matroid of rank-2 matrices and skew-symmetric matrices, using tropical geometry and phylogenetic trees to inform matrix completion.

Contribution

It provides a novel combinatorial characterization of matroid independence for low-rank matrix completion problems via tropical geometry and phylogenetic tree analysis.

Findings

Characterization of independent sets in algebraic matroids for rank-2 matrices

Reduction of the problem to phylogenetic tree metrics using tropical geometry

Description of prescribed pairwise distances allowing tree metric completion

Abstract

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 $m\times n$ rank-2 matrices and $n\times n$ 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.