Sparsification of Phylogenetic Covariance Matrices of k-Regular Trees

Abstract

Consider a tree T = (V,E) with root ∘ and an edge length function 𝓁:E → ℝ_+. The phylogenetic covariance matrix of T is the matrix C with rows and columns indexed by L, the leaf set of T, with entries C(i,j): = ∑_{e ∈ [i∧ j,o]}𝓁(e), for each i,j ∈ L. Recent work [Gorman & Lladser 2023] has shown that the phylogenetic covariance matrix of a large but random binary tree T is significantly sparsified, with overwhelmingly high probability, under a change-of-basis to the so-called Haar-like wavelets of T. Notably, this finding enables manipulating the spectrum of covariance matrices of large binary trees without the necessity to store them in computer memory but instead performing two post-order traversals of the tree [Gorman & Lladser 2023]. Building on the methods of the aforesaid paper, this manuscript further advances their sparsification result to encompass the broader class of k-regular trees, for any given k ≥ 2. This extension is achieved by refining existing asymptotic formulas for the mean and variance of the internal path length of random k-regular trees, utilizing hypergeometric function properties and identities.