Testing to distinguish measures on metric spaces

We study the problem of distinguishing between two distributions on a metric space; i.e., given metric measure spaces $({mathbb X}, d, mu_1)$ and $({mathbb X}, d, mu_2)$, we are interested in the problem of determining from finite data whether or not $mu_1$ is $mu_2$. The key is to use pairwise distances between observations and, employing a reconstruction theorem of Gromov, we can perform such a test using a two sample Kolmogorov--Smirnov test. A real analysis using phylogenetic trees and flu data is presented.