Tropical Data Science over the Space of Phylogenetic Trees

Abstract

When we apply comparative phylogenetic analyses to genome data, it is a well-known problem and challenge that some of given species (or taxa) often have missing genes. In such a case, we have to impute a missing part of a gene tree from a sample of gene trees. In this short paper we propose a novel method to infer a missing part of a phylogenetic tree using an analogue of a classical linear regression in the setting of tropical geometry. In our approach, we consider a tropical polytope, a convex hull with respect to the tropical metric closest to the data points. We show a condition that we can guarantee that an estimated tree from our method has at most two Robinson-Foulds (RF) distance from the ground truth and computational experiments with simulated data show our method works well.

The tropical geometry of causal inference for extremes

“…In the last five years, work by many authors has successfully used tropical linear algebra and convex geometry to solve important conjectures in classical linear programming [1] and economics [9,79]. Large-scale phylogenetics data [87] along with the revelation that deep neural networks are tropical rational functions [92] have seen the emergence of the exciting field of tropical data science [86]. Informally, the 'tropical approach' to a problem involves two steps.…”

Maximum Inscribed and Minimum Enclosing Tropical Balls of Tropical Polytopes and Applications to Volume Estimation and Uniform Sampling

“…Tropical polytopes [30,23] are gaining popularity as a tropical counterpart of classical polytopes. It is expected that tropical convex geometry has potential to solve many problems [36,39,32,37,38], just as classical convex geometry has many applications. In fact, tropical polytopes over the tropical projective space have been studied thoroughly (for examples, see [30,23,35] and references within) and have been applied in many areas, such as statistics, optimization, and phylogenomics [30,36,39,32,35,11].…”

Imputing Phylogenetic Trees Using Tropical Polytopes over the Space of Phylogenetic Trees

“…Therefore, with the tropical metric with the max-plus algebra, we can conduct statistical analyses using tropical linear algebra, which is an analogue of a classical linear algebra. In fact, there has been much development in statistical learning over the space of phylogenetic trees using tools from tropical geometry [7][8][9][10][11][12].…”