Distributions of metric n-point invariants and applications to molecular evolution
Abstract
Datasets often arise as samples from a metric measure space whose geometry reflects the data-generating process. We introduce metric n-point invariants, defined as functions of the pairwise distances of n-point configurations, to study the geometry of such spaces. This provides a unified framework for existing concepts, such as the pairwise distance itself, the Gromov product, and Vietoris-Rips persistence diagrams. The distribution of a metric n-point invariant F over random n-point configurations can be used to extract geometric information from the space and is obtained as the pushforward of Gromov's n-th curvature measure μn along F. Our main result is an explicit bound for the expected p-Wasserstein distance between μn and the empirical curvature measure μ̂n of finitely many sampled n-point configurations, and carries over to the associated distributions of Lipschitz metric n-point invariants. From this we derive the sample size needed to achieve a prescribed accuracy and level of confidence in the approximation of the curvature measure. As an application, we use metric n-point invariants to study molecular evolution. We describe how site-wise nucleotide frequencies shape the corresponding distributions in Jukes-Cantor and Halpern-Bruno substitution models, and analyse five million SARS-CoV-2 genomes, revealing increasingly common failures of the four-point condition, constrained nucleotide accessibility consistent with the biased mutation spectrum, and rare cycles in persistent homology reflecting recurrent mutations across lineages. These findings demonstrate the potential of our sampling-based approach to the geometry of data as a scalable, phylogeny-free tool for detecting and quantifying evolutionary structure in large genomic datasets.
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BibTeX
@misc{bleher2026distributions,
title = {{Distributions of metric $n$-point invariants and applications to molecular evolution}},
author = {Michael Bleher and Mario Gómez and Facundo Mémoli and Maximilian Neumann and Andreas Ott},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26070},
eprint = {2610.26070},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26070},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Distributions of metric n-point invariants and applications to molecular evolution
AU - Michael Bleher
AU - Mario Gómez
AU - Facundo Mémoli
AU - Maximilian Neumann
AU - Andreas Ott
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26070
UR - https://arxiv.org/abs/2610.26070
AB - Datasets often arise as samples from a metric measure space whose geometry reflects the data-generating process. We introduce metric n-point invariants, defined as functions of the pairwise distances of n-point configurations, to study the geometry of such spaces. This provides a unified framework for existing concepts, such as the pairwise distance itself, the Gromov product, and Vietoris-Rips persistence diagrams. The distribution of a metric n-point invariant F over random n-point configurations can be used to extract geometric information from the space and is obtained as the pushforward of Gromov's n-th curvature measure μ_n along F. Our main result is an explicit bound for the expected p-Wasserstein distance between μ_n and the empirical curvature measure μ̂_n of finitely many sampled n-point configurations, and carries over to the associated distributions of Lipschitz metric n-point invariants. From this we derive the sample size needed to achieve a prescribed accuracy and level of confidence in the approximation of the curvature measure. As an application, we use metric n-point invariants to study molecular evolution. We describe how site-wise nucleotide frequencies shape the corresponding distributions in Jukes-Cantor and Halpern-Bruno substitution models, and analyse five million SARS-CoV-2 genomes, revealing increasingly common failures of the four-point condition, constrained nucleotide accessibility consistent with the biased mutation spectrum, and rare cycles in persistent homology reflecting recurrent mutations across lineages. These findings demonstrate the potential of our sampling-based approach to the geometry of data as a scalable, phylogeny-free tool for detecting and quantifying evolutionary structure in large genomic datasets.
ER -