Sparsification of Phylogenetic Covariance Matrices of $k$-Regular Trees
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| Format: | Preprint |
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2024
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| _version_ | 1866912594682445824 |
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| author | Svihla, Sean P. Lladser, Manuel E. |
| author_facet | Svihla, Sean P. Lladser, Manuel E. |
| contents | Consider a tree $T=(V,E)$ with root $\circ$ and edge length function $\ell:E\to\mathbb{R}_+$. The phylogenetic covariance matrix of $T$ is the matrix $C$ with rows and columns indexed by $L$, the leaf set of $T$, with entries $C(i,j):=\sum_{e\in[i\wedge j,o]}\ell(e)$, for each $i,j\in L$. Recent work [15] has shown that the phylogenetic covariance matrix of a large, random binary tree $T$ is significantly sparsified with overwhelmingly high probability under a change-of-basis with respect to the so-called Haar-like wavelets of $T$. This finding notably 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. Building on the methods of [15], this manuscript further advances their sparsification result to encompass the broader class of $k$-regular trees, for any given $k\ge2$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_17847 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sparsification of Phylogenetic Covariance Matrices of $k$-Regular Trees Svihla, Sean P. Lladser, Manuel E. Populations and Evolution Discrete Mathematics Combinatorics Probability 60C05, 68R05, 65F05, 65F15, 65F55, 92-08, 92C70 G.2.1; G.2.2; G.3; J.2; J.3 Consider a tree $T=(V,E)$ with root $\circ$ and edge length function $\ell:E\to\mathbb{R}_+$. The phylogenetic covariance matrix of $T$ is the matrix $C$ with rows and columns indexed by $L$, the leaf set of $T$, with entries $C(i,j):=\sum_{e\in[i\wedge j,o]}\ell(e)$, for each $i,j\in L$. Recent work [15] has shown that the phylogenetic covariance matrix of a large, random binary tree $T$ is significantly sparsified with overwhelmingly high probability under a change-of-basis with respect to the so-called Haar-like wavelets of $T$. This finding notably 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. Building on the methods of [15], this manuscript further advances their sparsification result to encompass the broader class of $k$-regular trees, for any given $k\ge2$. 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. |
| title | Sparsification of Phylogenetic Covariance Matrices of $k$-Regular Trees |
| topic | Populations and Evolution Discrete Mathematics Combinatorics Probability 60C05, 68R05, 65F05, 65F15, 65F55, 92-08, 92C70 G.2.1; G.2.2; G.3; J.2; J.3 |
| url | https://arxiv.org/abs/2405.17847 |