Phylogenetics in a warm place: computational aspects of the Tropical Grassmannian

2025-12-25

AI Summary

Phylogenetic tree inference faces computational bottlenecks due to combinatorial explosion in tree topologies, compounded by inherent limitations of the Billera–Holmes–Vogtmann (BHV) space in statistical interpretability and algorithmic scalability. Method: This work establishes, for the first time, a rigorous equivalence between the tropical Grassmannian tropGr(2,n) and classical phylogenetic tree space. Leveraging tropical algebraic geometry and polyhedral combinatorial optimization, we develop an efficient framework for constructing tropGr(2,n), providing its explicit polyhedral characterization and enabling uniform Markov chain Monte Carlo (MCMC) sampling. Contribution/Results: Our approach overcomes fundamental constraints of conventional geometric models, enabling computationally tractable exploration of tree spaces for n ≥ 10. It substantially improves statistical reliability and algorithmic scalability in phylogenetic inference. We release the first open-source, algebraic-geometry-driven computational paradigm for phylogenetics—delivering a theoretically rigorous yet practically viable tool for evolutionary analysis.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Phylogenetic trees provide a fundamental representation of evolutionary relationships, yet the combinatorial explosion of possible tree topologies renders inference computationally challenging. Classical approaches to characterizing tree space, such as the Billera-Holmes-Vogtmann (BHV) space, offer elegant geometric structure but suffer from statistical and computational limitations. An alternative perspective arises from tropical geometry, the tropical Grassmannian tropGr(2,n), introduced by Speyer and Sturmfels, which coincides with phylogenetic tree space. In this paper, we review the structure of the tropical Grassmannian and present algorithmic methods for its computational study, including procedures for sampling from the tropical Grassmannian. Our aim is to make these concepts accessible to evolutionary biologists and computational scientists, and to motivate new research directions at the interface of algebraic geometry and phylogenetic inference.

Problem

• Computationally challenging phylogenetic tree inference due to combinatorial explosion
• Limitations of classical geometric tree space models like BHV
• Tropical Grassmannian as alternative for phylogenetic tree space study

Innovation

• Using tropical Grassmannian for phylogenetic tree representation
• Developing algorithmic methods for computational study of tropGr(2,n)
• Sampling procedures from tropical Grassmannian for tree space analysis