Tropical Machine Learning Models and Applications to Phylogenetic Trees

Aliatimis, George and Grant, James A. and Boyaci, Burak and Yoshida, Ruriko (2026) Tropical Machine Learning Models and Applications to Phylogenetic Trees. PhD thesis, Lancaster University.

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Abstract

Classification of gene trees is an important task both in the analysis of multi-locus phylogenetic data, and assessment of the convergence of Markov Chain Monte Carlo (MCMC) analyses used in Bayesian phylogenetic tree reconstruction. The logistic regression model is one of the most popular classification models in statistical learning, thanks to its computational speed and interpretability. However, it is not appropriate to directly apply the standard logistic regression model to a set of phylogenetic trees, as the space of phylogenetic trees is non-Euclidean and thus contradicts the standard assumptions on covariates. It is well-known in tropical geometry and phylogenetics that the space of phylogenetic trees is a tropical linear space in terms of the max-plus algebra. Therefore, in this thesis, we propose an analogue approach of the logistic regression model in the setting of tropical geometry. Our proposed method outperforms classical logistic regression in terms of Area under the ROC Curve in numerical examples, including with data generated by the multi-species coalescent model. Theoretical properties such as statistical consistency are proved and generalization error rates are derived. Finally, our classification algorithm is proposed as an MCMC convergence criterion for Mr Bayes. Unlike the convergence metric used by Mr Bayes which is only dependent on tree topologies, our method is sensitive to branch lengths and therefore provides a more robust metric for convergence. In a test case, it is illustrated that the tropical logistic regression can differentiate between two independently run MCMC chains, even when the standard metric cannot. Building upon this tropical geometric foundation, deep neural networks show great success when input vectors are in an Euclidean space. However, those classical neural networks show a poor performance when inputs are phylogenetic trees, which can be written as vectors in the tropical projective torus. Here we propose tropical embedding to transform a vector in the tropical projective torus to a vector in the Euclidean space via the tropical metric. We introduce a tropical neural network where the first layer is a tropical embedding layer and the following layers are the same as the classical ones. We prove that a tropical neural network is a universal approximator and we derive a backpropagation rule for deep tropical neural networks. Then we provide TensorFlow 2 codes for implementing a tropical neural network in the same fashion as the classical one, where the weights initialization problem is considered according to the extreme value statistics. We apply our method to empirical data including sequences of hemagglutinin for influenza virus from New York. Finally we show that a tropical neural network can be interpreted as a generalization of a tropical logistic regression. While the first two parts focus on statistical learning within tree space geometry, the final part of this thesis addresses the underlying statistical noise in phylogenomic data. In phylogenomics, species-tree methods must contend with two major sources of noise: stochastic gene-tree variation under the multispecies coalescent model (MSC) and finite-sequence substitutional noise. Fast agglomerative methods such as GLASS, STEAC, and METAL combine multi-locus information via distance-based clustering. We derive the exact covariance matrix of these pairwise distance estimates under a joint MSC-plus-substitution model and leverage it for reliable confidence estimation, and we algebraically decompose it into components attributable to coalescent variation versus sequence-level stochasticity. Our theory identifies parameter regimes where one source of variance greatly exceeds the other. For both very low and very high mutation rates, substitutional noise dominates, while coalescent variance is the primary contributor at intermediate mutation rates. Moreover, the interval over which coalescent variance dominates becomes narrower as the species-tree height increases. These results imply that in some settings one may legitimately ignore the weaker noise source when designing methods or collecting data. In particular, when gene-tree variance is dominant, adding more loci is most beneficial, while when substitution noise dominates, longer sequences or imputation are needed. Finally, leveraging the derived covariance matrix, we implement a Gaussian-sampling procedure to generate split support values for METAL trees and demonstrate empirically that this approach yields more reliable confidence estimates than traditional bootstrapping.