Tropical Fermat--Weber Points

Abstract

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.

L-Infinity Optimization to Linear Spaces and Phylogenetic Trees

“…Example 3.3. Figure 5 depicts three ultrametrics in the set of closest ultrametrics to δ = (2,4,6,8,10,12). The subdominant ultrametric is δ U = (2, 4, 6, 4, 6, 6), d(δ, U 4 ) = 3, and the canonical closest ultrametric (pictured far left) is δ c = (5,7,9,7,9,9).…”

“…We will perturb u to construct a dissimilarity map δ where the set of l ∞ -closest points to δ has dimension two. The triple (C(AB)) is compatible with T (u) and so we let δ = (5,7,9,7,9,9) + (e AC − e BC ) = (5,8,9,6,9,9).…”

“…Example 3.13. Let δ 1 = (4, 8,12,9,21,22) and δ 2 = (4, 8,12,9,13,14). Both dissimilarity maps satisfy δ i 12 < δ i 13 < δ i 23 < δ i 14 < δ i 24 < δ i 34 .…”

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

“…In tropical geometry, one redefines arithmetic over the real numbers so that the sum of two numbers is their maximum and the product is their sum (in the usual sense). There are strong connections between phylogenetics and tropical geometry [3,4,14,15,19,20] so the l ∞ -metric is a natural choice to measure best fit for phylogenetic reconstruction. An algorithm of Chepoi and Fichet computes an ultrametric l ∞ -nearest to a given dissimilarity map in polynomial time [6] but this is generally not the only l ∞ -nearest ultrametric.…”

“…The take-home message is that, according to this criteria, tropical convexity is the superior convexity theory. In a sequel [15], Lin and Yoshida study the non-uniqueness of the tropical Fermat-Weber point of a set of ultrametrics. In another sequel [20], Yoshida, Zhang, and Zhang develop a theory of tropical principal component analysis.…”