Observable Covariance and Principal Observable Analysis for Data on Metric Spaces

Datasets consisting of objects such as shapes, networks, images, or signals overlaid on such geometric objects permeate data science. Such datasets are often equipped with metrics that quantify the similarity or divergence between any pair of elements turning them into metric spaces $(X,d)$, or a metric measure space $(X,d,\mu)$ if data density is also accounted for through a probability measure $\mu$. This paper develops a Lipschitz geometry approach to analysis of metric measure spaces based on metric observables; that is, 1-Lipschitz scalar fields $f \colon X \to \mathbb{R}$ that provide reductions of $(X,d,\mu)$ to $\mathbb{R}$ through the projected measure $f_\sharp (\mu)$. Collectively, metric observables capture a wealth of information about the shape of $(X,d,\mu)$ at all spatial scales. In particular, we can define stable statistics such as the observable mean and observable covariance operators $M_\mu$ and $\Sigma_\mu$, respectively. Through a maximization of variance principle, analogous to principal component analysis, $\Sigma_\mu$ leads to an approach to vectorization, dimension reduction, and visualization of metric measure data that we term principal observable analysis. The method also yields basis functions for representation of signals on $X$ in the observable domain.