Dimensions of Group-based Phylogenetic Mixtures

TL;DR

This paper investigates the geometric properties of group-based phylogenetic mixture models, demonstrating nondefectiveness of certain mixture varieties, which aids in model selection and parameter identifiability.

Contribution

It establishes nondefectiveness of mixture varieties for group-based models when the number of leaves exceeds a specific bound, improving understanding of their geometric structure.

Findings

Mixture varieties are nondefective when n ≥ 2r+5.

Improved bounds for claw trees are provided.

Computational evidence suggests nondefectiveness for small n in 2- and 3-tree mixtures.

Abstract

In this paper we study group-based Markov models of evolution and their mixtures. In the algebreo-geometric setting, group-based phylogenetic tree models correspond to toric varieties, while their mixtures correspond to secant and join varieties. Determining properties of these secant and join varieties can aid both in model selection and establishing parameter identifiability. Here we explore the first natural geometric property of these varieties: their dimension. The expected projective dimension of the join variety of a set of varieties is one more than the sum of their dimensions. A join variety that realizes the expected dimension is nondefective. Nondefectiveness is not only interesting from a geometric point-of-view, but has been used to establish combinatorial identifiability for several classes of phylogenetic mixture models. In this paper, we focus on group-based models where the equivalence classes of identified parameters are orbits of a subgroup of the automorphism group of the group defining the model. In particular, we show that, for these group-based models, the variety corresponding to the mixture of $r$ trees with $n$ leaves is nondefective when $n \geq 2r+5$. We also give improved bounds for claw trees and give computational evidence that 2-tree and 3-tree mixtures are nondefective for small~$n$.