Extrinsic Analysis on BHV4

2026-05-29

AI Summary

This study addresses the challenge of extrinsic statistical analysis in the four-leaf Billera–Holmes–Vogtmann tree space (BHV₄) by proposing a novel approach that combines the Spiky Projective Excavated Dodecahedron (SPED) representation with the Veronese–Whitney embedding. The method yields, for the first time, a symmetry-driven natural extrinsic metric on BHV₄ and provides an analytical solution for the extrinsic mean. Applied to phylogenetic trees from four yeast clades, the framework enables efficient and accurate computation of extrinsic means, offering both a new theoretical foundation and a practical pathway for statistical analysis in tree spaces.

Abstract

One investigates the extrinsic statistical analysis on the space of Billera- Holmes-Vogtmann tree space with four leaves (T4 or BHV4) based on its recently proposed novel representation (see [1])- the Spiky Projective ExcavatedDodecahedron (SPED). Due to the symmetry of the SPED, the Veronese- Whitney (VW) embeddingwe consider here produces a natural extrinsicmetric for a statistical analysis on BHV4. one derives the exact solution for the VW extrinsic mean and applies this novel method on a yeast genome dataset to study the phylogenetic trees of four distinct yeast clades.

Problem

• BHV4
• extrinsic statistical analysis
• tree space
• Veronese-Whitney embedding
• phylogenetic trees

Innovation

• BHV4
• SPED
• Veronese-Whitney embedding
• extrinsic mean
• phylogenetic trees