Tropical Principal Component Analysis and Its Application to Phylogenetics
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.
KeywordsDimensionality reduction Phylogenomics Tropical geometry
R. Y. was supported by Research Initiation Proposals from the Naval Postgraduate School and NSF Division of Mathematical Sciences 1622369. L. Z. was supported by an NSF Graduate Research Fellowship. X. Z. was supported by travel funding from the Department of Statistics at the University of Kentucky. The authors thank Bernd Sturmfels (UC Berkeley and MPI Leipzig) for many helpful conversations. The authors also thank Daniel Howe (University of Kentucky) for his input on Apicomplexa tree topologies.
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