Scott Provan | Searcharxiv

arXiv2015

Dynamic Geodesics in Treespace via Parametric Maximum Flow

Shortest paths in treespace, which represent minimal deformations between trees, are unique and can be computed in polynomial time. The ability to quickly compute shortest paths has enabled new approaches for statistical analysis of populations of trees and phylogenetic inference. This paper gives a new algorithm for updating geodesic paths when the end points are dynamic. Such algorithms will be especially useful when optimizing for objectives that are functions of distances from a search point to other points e.g. for finding a tree which has the minimum average distance to a collection of trees. Our method for updating treespace shortest paths is based on parametric sensitivity analysis of the maximum flow subproblems that are optimized when solving for a treespace geodesic.

math.OC↗

arXiv2014

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

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.

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