L-Infinity Optimization to Linear Spaces and Phylogenetic Trees

Bernstein, Daniel Irving; Long, Colby

doi:10.1137/16m1101027

Cited by 6 publications

(3 citation statements)

References 10 publications

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“…for all pairs i < j, is isomorphic to the tropical projective torus. Under such a mapping, it is then possible to work in R n equipped with the ∞ metric as in [5,8,9].…”

Section: Remark 28mentioning

confidence: 99%

Tropical Geometric Variation of Tree Shapes

Lin

Monod

Yoshida

2022

Discrete Comput Geom

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We study the behavior of phylogenetic tree shapes in the tropical geometric interpretation of tree space. Tree shapes are formally referred to as tree topologies; a tree topology can also be thought of as a tree combinatorial type, which is given by the tree’s branching configuration and leaf labeling. We use the tropical line segment as a framework to define notions of variance as well as invariance of tree topologies: we provide a combinatorial search theorem that describes all tree topologies occurring along a tropical line segment, as well as a setting under which tree topologies do not change along a tropical line segment. Our study is motivated by comparison to the moduli space endowed with a geodesic metric proposed by Billera, Holmes, and Vogtmann (referred to as BHV space); we consider the tropical geometric setting as an alternative framework to BHV space for sets of phylogenetic trees. We give an algorithm to compute tropical line segments which is lower in computational complexity than the fastest method currently available for BHV geodesics and show that its trajectory behaves more subtly: while the BHV geodesic traverses the origin for vastly different tree topologies, the tropical line segment bypasses it.

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“…for all pairs i < j, is isomorphic to the tropical projective torus. Under such a mapping, it is then possible to work in R n equipped with the ∞ metric as in [5,8,9].…”

Section: Remark 28mentioning

confidence: 99%

Tropical Geometric Variation of Tree Shapes

Lin

Monod

Yoshida

2022

Discrete Comput Geom

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“…The image of the set of ultrametrics in the quotient space R ( N 2 ) /R1 is called the space of ultrametrics. This is the image of ultrametrics in the quotient space using the extrinsic metric, via the tropical metric [3].…”

Section: The Tropical Fermat-weber Pointmentioning

confidence: 99%

Tropical Fermat--Weber Points

Lin¹,

Yoshida²

2018

SIAM J. Discrete Math.

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In a metric space, the Fermat-Weber points of a sample are statistics to measure the central tendency of the sample and it is well-known that the Fermat-Weber point of a sample is not necessarily unique in the metric space. We investigate the computation of Fermat-Weber points under the tropical metric on the quotient space R n /R1 with a fixed n ∈ N, motivated by its application to the space of equidistant phylogenetic trees with N leaves (in this case n = N 2 ) realized as the tropical linear space of all ultrametrics. We show that the set of all tropical Fermat-Weber points of a finite sample is always a classical convex polytope, and we present a combinatorial formula for a key value associated to this set. We identify conditions under which this set is a singleton. We apply numerical experiments to analyze the set of the tropical Fermat-Weber points within a space of phylogenetic trees. We discuss the issues in the computation of the tropical Fermat-Weber points.

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“…So from the perspective of the l ∞ -metric, there may be many different evolutionary histories that all explain a given dataset equally well. Colby Long and this author began a study of this, and other related phenomena, in [5]. The main mathematical results therein concern the non-uniqueness of the point in a (non-tropical) linear subspace of R n that is l ∞ -nearest to a given x ∈ R n .…”

Section: Introductionmentioning

confidence: 99%

L-Infinity Optimization to Bergman Fans of Matroids with an Application to Phylogenetics

Bernstein¹

2020

SIAM J. Discrete Math.

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Given a dissimilarity map δ on finite set X, the set of ultrametrics (equidistant tree metrics) which are l ∞ -nearest to δ is a tropical polytope. We give an interior description of this tropical polytope. It was shown by Ardila and Klivans [4] that the set of all ultrametrics on a finite set of size n is the Bergman fan associated to the matroid underlying the complete graph on n vertices. Therefore, we derive our results in the more general context of Bergman fans of matroids. This added generality allows our results to be used on dissimilarity maps where only a subset of the entries are known.

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L-Infinity Optimization to Linear Spaces and Phylogenetic Trees

Cited by 6 publications

References 10 publications

Tropical Geometric Variation of Tree Shapes

Tropical Geometric Variation of Tree Shapes

Tropical Fermat--Weber Points

L-Infinity Optimization to Bergman Fans of Matroids with an Application to Phylogenetics

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