@article{bernstein_2017, title={Completion of tree metrics and rank 2 matrices}, volume={533}, ISSN={["1873-1856"]}, DOI={10.1016/j.laa.2017.07.016}, abstractNote={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 × 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.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Bernstein, Daniel Irving}, year={2017}, month={Nov}, pages={1–13} }
@article{bernstein_long_2017, title={L-INFINITY OPTIMIZATION TO LINEAR SPACES AND PHYLOGENETIC TREES}, volume={31}, ISSN={["1095-7146"]}, DOI={10.1137/16m1101027}, abstractNote={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 ultrametrics and tree metrics $l^\infty$-closest 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 three elements and on four 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 $\mathbb{R}^m$ based on the dimension of the set of $l^\infty$-closest points in a linear space.}, number={2}, journal={SIAM JOURNAL ON DISCRETE MATHEMATICS}, author={Bernstein, Daniel Irving and Long, Colby}, year={2017}, pages={875–889} }
@article{bernstein_sullivant_2017, title={Normal Binary Hierarchical Models}, volume={26}, ISSN={["1944-950X"]}, DOI={10.1080/10586458.2016.1142911}, abstractNote={ABSTRACTEach 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.}, number={2}, journal={EXPERIMENTAL MATHEMATICS}, author={Bernstein, Daniel Irving and Sullivant, Seth}, year={2017}, pages={153–164} }
@article{bernstein_sullivant_2017, title={Unimodular binary hierarchical models}, volume={123}, ISSN={["1096-0902"]}, DOI={10.1016/j.jctb.2016.11.003}, abstractNote={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.}, journal={JOURNAL OF COMBINATORIAL THEORY SERIES B}, author={Bernstein, Daniel Irving and Sullivant, Seth}, year={2017}, month={Mar}, pages={97–125} }
@article{bernstein_o'neill_2017, title={Unimodular hierarchical model sand their Graver bases}, volume={8}, DOI={10.18409/jas.v8i2.66}, abstractNote={Given a simplicial complex whose vertices are labeled with positive integers, one can associate a vector configuration whose corresponding toric variety is the Zariski closure of a hierarchical model. We classify all the vertex-weighted simplicial complexes that give rise to unimodular vector configurations. We also provide a combinatorial characterization of their Graver bases.}, number={2}, journal={Journal of Algebraic Statistics}, author={Bernstein, D. I. and O'Neill, C.}, year={2017}, pages={29–43} }
@article{bernstein_ho_long_steel_st john_sullivant_2015, title={BOUNDS ON THE EXPECTED SIZE OF THE MAXIMUM AGREEMENT SUBTREE}, volume={29}, ISSN={["1095-7146"]}, DOI={10.1137/140997750}, abstractNote={We prove lower bounds on the expected size of the maximum agreement subtree of two random binary phylogenetic trees under both the uniform distribution and the Yule--Harding distribution and prove upper bounds under the Yule--Harding distribution. This positively answers a question posed in earlier work. Determining tight upper and lower bounds remains an open problem.}, number={4}, journal={SIAM JOURNAL ON DISCRETE MATHEMATICS}, author={Bernstein, Daniel Irving and Ho, Lam Si Tung and Long, Colby and Steel, Mike and St John, Katherine and Sullivant, Seth}, year={2015}, pages={2065–2074} }
@article{bernstein_grynkiewicz_yerger_2015, title={On three sets with nondecreasing diameter}, volume={338}, ISSN={["1872-681X"]}, DOI={10.1016/j.disc.2015.02.007}, abstractNote={Let [ a , b ] denote the integers between a and b inclusive and, for a finite subset X ⊆ Z , let diam ( X ) = max ( X ) − min ( X ) . We write X < p Y provided max ( X ) < min ( Y ) . For a positive integer m , let f ( m , m , m ; 2 ) be the least integer N such that any 2 -coloring Δ : [ 1 , N ] → { 0 , 1 } has three monochromatic m -sets B 1 , B 2 , B 3 ⊆ [ 1 , N ] (not necessarily of the same color) with B 1 < p B 2 < p B 3 and diam ( B 1 ) ≤ diam ( B 2 ) ≤ diam ( B 3 ) . Improving upon upper and lower bounds of Bialostocki, Erdős and Lefmann, we show that f ( m , m , m ; 2 ) = 8 m − 5 + ⌊ 2 m − 2 3 ⌋ + δ for m ≥ 2 , where δ = 1 if m ∈ { 2 , 5 } and δ = 0 otherwise.}, number={8}, journal={DISCRETE MATHEMATICS}, author={Bernstein, Daniel Irving and Grynkiewicz, David J. and Yerger, Carl}, year={2015}, month={Aug}, pages={1328–1344} }