  Experimental sciences have come to depend heavily on our ability to organize
and interpret high-dimensional datasets. Natural laws, conservation principles,
and inter-dependencies among observed variables yield geometric structure, with
fewer degrees of freedom, on the dataset. We introduce the frameworks of
semiclassical and microlocal analysis to data analysis and develop a novel, yet
natural uncertainty principle for extracting fine-scale features of this
geometric structure in data, crucially dependent on data-driven approximations
to quantum mechanical processes underlying geometric optics. This leads to the
first tractable algorithm for approximation of wave dynamics and geodesics on
data manifolds with rigorous probabilistic convergence rates under the manifold
hypothesis. We demonstrate our algorithm on real-world datasets, including an
analysis of population mobility information during the COVID-19 pandemic to
achieve four-fold improvement in dimensionality reduction over existing
state-of-the-art and reveal anomalous behavior exhibited by less than 1.2% of
the entire dataset. Our work initiates the study of data-driven quantum
dynamics for analyzing datasets, and we outline several future directions for
research.
