Spectral neighbor joining for reconstruction of latent tree models

Ariel Jaffe, Noah Amsel, Yariv Aizenbud, Boaz Nadler, Joseph T Chang, Yuval Kluger

Research output: Contribution to journalArticlepeer-review

Abstract

A common assumption in multiple scientific applications is that the distribution of observed data can be modeled by a latent tree graphical model. An important example is phylogenetics, where the tree models the evolutionary lineages of a set of observed organisms. Given a set of independent realizations of the random variables at the leaves of the tree, a key challenge is to infer the underlying tree topology. In this work we develop Spectral Neighbor Joining (SNJ), a novel method to recover the structure of latent tree graphical models. Given a matrix that contains a measure of similarity between all pairs of observed variables, SNJ computes a spectral measure of cohesion between groups of observed variables. We prove that SNJ is consistent, and derive a sufficient condition for correct tree recovery from an estimated similarity matrix. Combining this condition with a concentration of measure result on the similarity matrix, we bound the number of samples required to recover the tree with high probability. We illustrate via extensive simulations that in comparison to several other reconstruction methods, SNJ requires fewer samples to accurately recover trees with a large number of leaves or long edges.
Original languageEnglish
Pages (from-to)113-141
Number of pages29
JournalSIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE
Volume3
Issue number1
DOIs
StatePublished - 2021

Keywords

  • 15A18
  • 62H22
  • 62M05
  • 62M15
  • Markov random fields
  • evolutionary trees
  • latent variable models
  • neighbor joining
  • phylogenetics
  • singular values
  • spectral methods
  • tree graphical model

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