Completion of tree metrics and rank-2 matrices
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2016
- DOI:
- 10.48550/arXiv.1612.06797
- arXiv:
- arXiv:1612.06797
- Bibcode:
- 2016arXiv161206797B
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Algebraic Geometry;
- 14T05;
- 52B40;
- 52C25
- E-Print:
- Typos fixed and other minor improvements. To appear in Linear Algebra and its Applications