[Paper Review] Tropical Principal Component Analysis and its Application to Phylogenetics

Ruriko Yoshida, Leon Zhang+1 more|arXiv (Cornell University)|Oct 7, 2017

Genomics and Phylogenetic Studies5 references3 citations

TL;DR

This paper introduces two novel tropical principal component analysis (PCA) methods in the context of tropical geometry: one based on Stiefel tropical linear spaces and another on tropical polytopes, both minimizing tropical distance to data points. Applied to phylogenetics, the methods successfully reduce dimensionality of tree-space data, with the tropical polytope approach preserving ultrametric properties and revealing topological structure in Apicomplexa genome data.

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.

Motivation & Objective

• To develop tropical analogues of classical PCA for dimensionality reduction in high-dimensional data within the tropical projective torus.
• To address the challenge of analyzing large-scale phylogenetic datasets by leveraging tropical geometry's structure for tree-space data.
• To provide approximative algorithms for computing tropical PCA that are computationally feasible and preserve key biological properties like ultrametricity.
• To investigate whether tropical PCA structures exhibit nested or hierarchical containment properties analogous to classical PCA.
• To evaluate the performance of both tropical PCA approaches on simulated and empirical phylogenetic datasets, particularly for Apicomplexa genomes.

Proposed method

• Define tropical PCA via Stiefel tropical linear spaces as the closest (in tropical metric) (s−1)-dimensional subspaces to data points in the tropical projective torus.
• Use the tropical metric $ d_{tr}(v,w) = \max_{i<j} |v_i - w_i - v_j + w_j| $ to measure distance between data points and candidate subspaces.
• Formulate the best-fit Stiefel tropical linear space problem as minimizing the sum of squared tropical distances, with an exact solution for $ e $ points in $ \mathbb{R}^e/\mathbb{R}\mathbf{1} $ using tropical volume.
• Propose a heuristic algorithm for general cases based on random sampling of three points to initialize the search for optimal matrices.
• Define tropical PCA via tropical polytopes as the tropical convex hull of $ s $ points minimizing total tropical distance to data.
• Reformulate the best-fit tropical polytope problem as a mixed-integer programming problem and develop an approximative algorithm using iterative refinement and sampling.

Experimental results

Research questions

• RQ1 Can principal component analysis be meaningfully extended to the tropical projective torus using Stiefel tropical linear spaces?
• RQ2 Does the tropical polytope-based PCA preserve key biological properties such as ultrametricity in phylogenetic trees after projection?
• RQ3 How do the two tropical PCA methods compare in performance and interpretability when applied to simulated and real phylogenetic datasets?
• RQ4 Is there a nested or hierarchical containment structure in tropical PCA analogous to classical PCA’s principal component nesting?
• RQ5 Under what conditions is the Stiefel tropical linear space defined by a set of data points contained within the original tropical linear space of tree space?

Key findings

• An exact solution for the best-fit tropical hyperplane (i.e., $ (e-1) $-dimensional space) of $ e $ points in $ \mathbb{R}^e/\mathbb{R}\mathbf{1} $ is derived using tropical volume.
• The tropical polytope PCA approach preserves ultrametricity: projected equidistant trees remain ultrametrics, enabling analysis of tree topology distributions.
• In the Apicomplexa genome dataset, projected tree topologies were distributed across distinct regions of the tropical polytope, indicating meaningful clustering by topology.
• The approximative algorithms for both Stiefel and tropical polytope PCA were successfully applied to simulated and empirical phylogenetic data, demonstrating feasibility.
• Best-fit tropical structures are not unique; in cases where a best-fit Stiefel space does not contain a tropical Fermat-Weber point, an alternative equally good fit exists that does.
• The tropical PCA methods are well-behaved in terms of dimensionality: both Stiefel linear spaces and tropical polytopes of $ s $ points have dimension at most $ s-1 $, avoiding the high-dimensionality issues of BHV metric convex hulls.