Projections of Tropical Fermat-Weber Points

TL;DR

This paper introduces algorithms to project Fermat-Weber points onto tropical triangles in the tropical projective torus, ensuring the projection preserves Fermat-Weber properties, with improved efficiency demonstrated through experiments.

Contribution

Developed and tested algorithms that reliably project Fermat-Weber points onto tropical triangles, enhancing computational stability and speed in tropical geometry.

Findings

Algorithms outperform random selection in success rate

Algorithm 4 is significantly faster than Algorithm 1

Success rate remains stable across random data sets

Abstract

In the tropical projective torus, it is not guaranteed that the projection of a Fermat-Weber point of a given data set is a Fermat-Weber point of the projection of the data set. In this paper, we focus on the projection on the tropical triangle (the three-point tropical convex hull), and we develop one algorithm (Algorithm 1) and its improved version (Algorithm 4), such that for a given data set in the tropical projective torus, these algorithms output a tropical triangle, on which the projection of a Fermat-Weber point of the data set is a Fermat-Weber point of the projection of the data set. We implement these algorithms in R and test how it works with random data sets. The experimental results show that, these algorithms can succeed with a much higher probability than choosing the tropical triangle randomly, the succeed rate of these two algorithms is stable while data sets are changing randomly, and Algorithm 4 can output the results much faster than Algorithm 1 averagely.