Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The tropical polytope is the set of all weighted tropical Fermat-Weber points

Shelby Cox, Mark Curiel

TL;DR

The paper generalizes tropical Fermat-Weber theory to incorporate positive weights, showing that weighted Fermat-Weber sets FW$(V,\mathbf{w})$ are covector cells of the min-tropical convex hull $\mathrm{tconv}^{\min}(V)$ and that every covector cell can be realized by suitable weights. It reframes the optimization as minimization of a homogeneous tropical signomial $f_{V,\mathbf{w}}$, connects covector decompositions to tropical convexity via the Cayley trick and Newton polytopes, and establishes a constructive method to realize any covector cell through fractional matchings. The results extend to phylogenetics by defining a weighted tropical consensus method that is regular and Pareto/co-Pareto on rooted triples, with FW points corresponding to equidistant trees in tropical tree space. This provides a geometry-driven, weight-aware framework for consensus in tree space and suggests practical avenues for weighted bootstrapping and efficient computation using transportation-like structures.

Abstract

Let $\mathbf{v}_1,\ldots,\mathbf{v}_m$ be points in a metric space with distance $d$, and let $w_1,\ldots,w_m$ be positive real weights. The weighted Fermat-Weber points are those points $\mathbf{x}$ which minimize $\sum w_i d(\mathbf{v}_i, \mathbf{x})$. We extend a result of Comăneci and Joswig, that the set of unweighted Fermat-Weber points agrees with the "central" covector cell of the tropical convex hull of $\mathbf{v}_1,\ldots,\mathbf{v}_m$, to the weighted setting. In particular, we show that for any fixed data points $\mathbf{v}_1, \ldots, \mathbf{v}_m$, and any covector cell of the tropical convex hull of the data, there is a choice of weights that makes that cell the Fermat-Weber set. We similarly extend the method of Comăneci and Joswig for computing consensus trees in phylogenetics.

The tropical polytope is the set of all weighted tropical Fermat-Weber points

TL;DR

The paper generalizes tropical Fermat-Weber theory to incorporate positive weights, showing that weighted Fermat-Weber sets FW are covector cells of the min-tropical convex hull and that every covector cell can be realized by suitable weights. It reframes the optimization as minimization of a homogeneous tropical signomial , connects covector decompositions to tropical convexity via the Cayley trick and Newton polytopes, and establishes a constructive method to realize any covector cell through fractional matchings. The results extend to phylogenetics by defining a weighted tropical consensus method that is regular and Pareto/co-Pareto on rooted triples, with FW points corresponding to equidistant trees in tropical tree space. This provides a geometry-driven, weight-aware framework for consensus in tree space and suggests practical avenues for weighted bootstrapping and efficient computation using transportation-like structures.

Abstract

Let be points in a metric space with distance , and let be positive real weights. The weighted Fermat-Weber points are those points which minimize . We extend a result of Comăneci and Joswig, that the set of unweighted Fermat-Weber points agrees with the "central" covector cell of the tropical convex hull of , to the weighted setting. In particular, we show that for any fixed data points , and any covector cell of the tropical convex hull of the data, there is a choice of weights that makes that cell the Fermat-Weber set. We similarly extend the method of Comăneci and Joswig for computing consensus trees in phylogenetics.
Paper Structure (10 sections, 12 theorems, 24 equations, 4 figures)

This paper contains 10 sections, 12 theorems, 24 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Weighted Asymmetric Tropical Fermat-Weber Problem
  4. Minimizing Tropical Signomials
  5. Tropical Convexity
  6. The weighted tropical Fermat-Weber problem
  7. Applications to Phylogenetics
  8. The Space of Equidistant Trees
  9. Weighted Asymmetric Tropical Consensus
  10. Outlook

Key Result

Theorem 1.1

Given data $\mathbf{v}_1, \ldots, \mathbf{v}_m \in \mathbb{R}^n/\mathbb{R}\mathbbm{1}$ and any real positive weights $w_1, \ldots, w_m$, the weighted Fermat-Weber set with respect to $d_\Delta$ is contained in $\mathrm{tconv}^{\min}\{ \mathbf{v}_1,\ldots,\mathbf{v}_m \}$. More precisely, it is a cov

Figures (4)

  • Figure 1: Left: The regular subdivision $\underline{\mathrm{N}}(f)$ for the tropical polynomial $f = - \frac{2}{3} \odot x_1 \oplus \frac{2}{3} \odot x_1^{1/3} x_2^{2/3} \oplus x_1^{2/3} x_3^{1/3} \oplus \frac{2}{3} \odot x_2 \oplus \frac{2}{3} \odot x_2^{2/3}x_3^{2/3} \oplus x_3$. Overlaid is the normal complex $\overline{\mathrm{NC}}(f_{V, \mathbf{w}})$. Right: $\mathrm{epi}(f_{V, \mathbf{w}}) \cap \mathbbm{1}^\perp_n$ over $\overline{\mathrm{NC}}(f_{V, \mathbf{w}})$.
  • Figure 2: Left to right: the Cayley subdivision for the data $V = \{ \mathbf{v}_1 = (0,0,0), \mathbf{v}_2 = (1,-1,0) \}$, the subdivision (of the support) dual to $f_{V, \frac{1}{2} \mathbbm{1}_2}$, and the subdivision (of the support) dual to $f_{V, (\frac{1}{3}, \frac{2}{3})}$. In the Cayley subdivision, the top star is the point $(\frac{1}{2} \mathbbm{1}_2, \frac{1}{3} \mathbbm{1}_3)$ and the bottom star is the point $(\mathbf{w}, \frac{1}{3} \mathbbm{1}_3)$. In the middle and right pictures, the star is the point $\frac{1}{2} \mathbbm{1}_3$.
  • Figure 3: The tropical line segment $\mathrm{tconv}^{\min}((0,0,0), (1,-1,0))$.
  • Figure 4: Left: vertices in the product of simplices. Right: the corresponding edges of the bipartite subgraph of $K_{2,2}$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Definition 2.1: Comǎneci and Joswig joswig-com-tropical-medians
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Example 2.5
  • Definition 2.6: cf. ETC
  • Lemma 2.7
  • proof
  • ...and 23 more