Projected Gradient Descent Method for Tropical Principal Component Analysis over Tree Space

Tropical Principal Component Analysis (PCA) is an analogue of the classical PCA in the setting of tropical geometry, and applied it to visualize a set of gene trees over a space of phylogenetic trees, which is a union of lower-dimensional polyhedral cones in an Euclidean space with dimension m ( m − 1 ) / 2 , where m is the number of leaves. In this paper, we introduce a projected gradient descent method to estimate the tropical principal polytope over the space of phylogenetic trees, and we apply it to an Apicomplexa dataset. With computational experiments against Markov Chain Monte Carlo (MCMC) samplers, we show that our projected gradient descent method yields a lower sum of tropical distances between observations and their projections onto the estimated best-fit tropical polytope, compared with the MCMC-based approach.