Distributions of metric n-point invariants and applications to molecular evolution

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 μ 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.