Tropical Principal Component Analysis and Its Application to Phylogenetics

Abstract

Principal component analysis is a widely used method for the dimensionality reduction of a given data set in a high-dimensional Euclidean space. Here we define and analyze two analogues of principal component analysis in the setting of tropical geometry. In one approach, we study the Stiefel tropical linear space of fixed dimension closest to the data points in the tropical projective torus; in the other approach, we consider the tropical polytope with a fixed number of vertices closest to the data points. We then give approximative algorithms for both approaches and apply them to phylogenetics, testing the methods on simulated phylogenetic data and on an empirical dataset of Apicomplexa genomes.

Tropical Ehrhart theory and tropical volume

“…In the study of tropical principal component analysis, the notion is also discussed in [ 37 , § 3.1]. The authors prove that the following notion also extends the tropical determinantal volume to rectangular matrices, but in terms of a sum of tropical distances: Here, is the tropical hyperplane defined by z , and is the generalized Hilbert projective metric (cf.…”

“…Tropical geometry is the study of piecewise-linear objects defined over the -semiring that arises by replacing the classical addition ‘ ’ with ‘ ’ and multiplication ‘ ’ with ‘ .’ While this often focuses on combinatorial properties, see [ 11 , 25 ], we are mainly interested in metric properties. Measuring quantities from tropical geometry turned out to be fruitful for a better understanding of interior point methods for linear programming [ 2 ] and principal component analysis of biological data [ 37 ]. Moreover, it has interesting connections with representation theory [ 29 , 38 ] and computational complexity [ 22 ].…”

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.…”

“…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.…”

Tropical Gaussians: a brief survey

“…Tropical mathematics has found many applications in both pure and applied areas, as documented by a growing number of monographs on its interactions with various other areas of mathematics: algebraic geometry [Baker and Payne 2016;Gross 2011;Huh 2018;Maclagan and Sturmfels 2015], discrete event systems [Baccelli et al 1992;Butkovič 2010], large deviations and calculus of variations [Kolokoltsov and Maslov 1997;Puhalskii 2001], and combinatorial optimization [Joswig ≥ 2020]. At the same time, new applications are emerging in phylogenetics [Monod et al 2018;Yoshida et al 2019;Page et al 2020], statistics [Hook 2017], economics [Baldwin and Klemperer 2019;Crowell and Tran 2016;Elsner and van den Driessche 2004;Gursoy et al 2013;Joswig 2017;Shiozawa 2015;Tran 2013;Tran and Yu 2019], game theory, and complexity theory [Allamigeon et al 2018;Akian et al 2012]. There is a growing need for a systematic study of probability distributions in tropical settings.…”