arXiv subjects

Daniel Irving Bernstein

Publications and source records attributed to Daniel Irving Bernstein.

24 records · Page 2Linked to original sources

arXiv2017

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 $m\times n$ 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.

math.CO↗

arXiv2017

L-Infinity optimization in tropical geometry and phylogenetics

We investigate uniqueness issues that arise in $l^\infty$-optimization to linear spaces and Bergman fans of matroids. For linear spaces, we give a polyhedral decomposition of $\mathbb{R}^n$ based on the dimension of the set of $l^\infty$-nearest neighbors. This implies that the $l^\infty$-nearest neighbor in a linear space is unique if and only if the underlying matroid is uniform. For Bergman fans of matroids, we show that the set of $l^\infty$-nearest points is a tropical polytope and give an algorithm to compute its tropical vertices. A key ingredient here is a notion of topology that generalizes tree topology. These results have practical implications for distance-based phylogenetic reconstruction using the $l^\infty$-metric. We analyze the possible dimensions of the set of $l^\infty$-nearest equidistant tree metrics to an arbitrary dissimilarity map and the number of tree topologies represented in this set. For both 3 and 4-leaf trees, we decompose the space of dissimilarity maps relative to the tree topologies represented.

math.CO↗

arXiv2017

L-infinity optimization to linear spaces and phylogenetic trees

Given a distance matrix consisting of pairwise distances between species, a distance-based phylogenetic reconstruction method returns a tree metric or equidistant tree metric (ultrametric) that best fits the data. We investigate distance-based phylogenetic reconstruction using the $l^\infty$-metric. In particular, we analyze the set of $l^\infty$-closest ultrametrics and tree metrics to an arbitrary dissimilarity map to determine its dimension and the tree topologies it represents. In the case of ultrametrics, we decompose the space of dissimilarity maps on 3 elements and on 4 elements relative to the tree topologies represented. Our approach is to first address uniqueness issues arising in $l^\infty$-optimization to linear spaces. We show that the $l^\infty$-closest point in a linear space is unique if and only if the underlying matroid of the linear space is uniform. We also give a polyhedral decomposition of $\rr^m$ based on the dimension of the set of $l^\infty$-closest points in a linear space.

math.CO↗

arXiv2016

Unimodular Binary Hierarchical Models

Associated to each simplicial complex is a binary hierarchical model. We classify the simplicial complexes that yield unimodular binary hierarchical models. Our main theorem provides both a construction of all unimodular binary hierarchical models, together with a characterization in terms of excluded minors, where our definition of a minor allows the taking of links and induced complexes. A key tool in the proof is the lemma that the class of unimodular binary hierarchical models is closed under the Alexander duality operation on simplicial complexes.

math.CO↗

arXiv2016

Normal Binary Hierarchical Models

Each simplicial complex and integer vector yields a vector configuration whose combinatorial properties are important for the analysis of contingency tables. We study the normality of these vector configurations including a description of operations on simplicial complexes that preserve normality, constructions of families of minimally nonnormal complexes, and computations classifying all of the normal complexes on up to six vertices. We repeat this analysis for compressed vector configurations, classifying all of the compressed complexes on up to six vertices.

math.CO↗

arXiv2015

Bounds on the Expected Size of the Maximum Agreement Subtree

We prove polynomial upper and lower bounds on the expected size of the maximum agreement subtree of two random binary phylogenetic trees under both the uniform distribution and Yule-Harding distribution. This positively answers a question posed in earlier work. Determining tight upper and lower bounds remains an open problem.

q-bio.PE↗