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 $μ_n$ along $F$. Our main result is an explicit bound for the expected $p$-Wasserstein distance between $μ_n$ and the empirical curvature measure $\hatμ_{n}$ 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.