Tropical mixtures of star tree metrics (original) (raw)
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Tree Topologies along a Tropical Line Segment
Vietnam Journal of Mathematics, 2022
Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.
Tropical Geometric Variation of Tree Shapes
Discrete & Computational Geometry
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 ...
The space of tropically collinear points is shellable
Collectanea mathematica, 2009
The space T d,n of n tropically collinear points in a fixed tropical projective space TP d−1 is equivalent to the tropicalization of the determinantal variety of matrices of rank at most 2, which consists of real d × n matrices of tropical or Kapranov rank at most 2, modulo projective equivalence of columns. We show that it is equal to the image of the moduli space M 0,n (TP d−1 , 1) of n-marked tropical lines in TP d−1 under the evaluation map. Thus we derive a natural simplicial fan structure for T d,n using a simplicial fan structure of M 0,n (TP d−1 , 1) which coincides with that of the space of phylogenetic trees on d + n taxa. The space of phylogenetic trees has been shown to be shellable by Trappmann and Ziegler. Using a similar method, we show that T d,n is shellable with our simplicial fan structure and compute the homology of the link of the origin. The shellability of T d,n has been conjectured by Develin in .
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SIAM Journal on Discrete Mathematics
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2010
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A tropical ball is a ball defined by the tropical metric over the tropical projective torus. In this paper we show several properties of tropical balls over the tropical projective torus and also over the space of phylogenetic trees with a given set of leaf labels. Then we discuss its application to the K nearest neighbors (KNN) algorithm, a supervised learning method used to classify a high-dimensional vector into given categories by looking at a ball centered at the vector, which contains K vectors in the space.
Tropical fans and the moduli spaces of tropical curves
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We give a rigorous definition of tropical fans (the "local building blocks for tropical varieties") and their morphisms. For a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point -a statement that can be viewed as one of the important first steps of tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some R r ) together with the evaluation and forgetful morphisms. Using our results this gives new, easy, and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any R r ) through given points are independent of the points.
Trees of metric compacta and trees of manifolds
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We present a construction, called the limit of a tree system of spaces (or, less formally, a tree of spaces). The construction is designed to produce compact metric spaces that resemble fractals, out of more regular spaces, such as closed manifolds, compact polyhedra, compact Menger manifolds, etc. Such spaces are potential candidates to be homeomorphic to ideal boundaries of infinite groups. A very special case of this construction, trees of manifolds (known also as Jakobsche spaces), has been studied in the literature. We present here a different approach, much more general, and, as we believe, much more convenient for establishing various basic properties of the resulting spaces, in a more general setting. Already in the case of trees of manifolds, using this approach we clarify, correct and extend so far known results and properties.
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We give an affirmative answer to a conjecture proposed by Tevelev in characteristic 0 case: any variety contains a sch\"on very affine open subvariety. Also we show that any fan supported on the tropicalization of a sch\"on very affine variety produces a sch\"on compactification. Using toric schemes over a discrete valuation ring, we extend tropical compatifications to the non-constant coefficient case.