Relative Optimality Conditions and Algorithms for Treespace Fr\'{e}chet Means

TL;DR

This paper develops new mathematical conditions and algorithms for computing the Fréchet mean in treespace, which is crucial for phylogenetic and anatomical data analysis, by analyzing the function's differential properties along geodesics.

Contribution

It introduces a decomposition theorem for the derivative of the Fréchet function in treespace and formulates optimality conditions for algorithms to verify relative optimality.

Findings

Decomposition theorem for the derivative along geodesics in treespace

Optimality conditions for the Fréchet mean computation

Algorithmic framework for verifying relative optimality

Abstract

Recent interest in treespaces as well-founded mathematical domains for phylogenetic inference and statistical analysis for populations of anatomical trees has motivated research into efficient and rigorous methods for optimization problems on treespaces. A central problem in this area is computing an average of phylogenetic trees, which is equivalently characterized as the minimizer of the Fr\'echet function. The Fr\'echet mean can be used for statistical inference and exploratory data analysis: for example it can be leveraged as a test statistic to compare groups via permutation tests, or to find trends in data over time via kernel smoothing. By analyzing the differential properties of the Fr\'echet function along geodesics in treespace we obtained a theorem describing a decomposition of the derivative along a geodesic. This decomposition theorem is used to formulate optimality conditions which are used as a logical basis for an algorithm to verify relative optimality at points where the Fr\'echet function gradient does not exist.