Mohan Sarovar

Akshat Kumar, Mohan Sarovar

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.

Akshat Kumar, Mohan Sarovar

We introduce an algorithm for computing geodesics on sampled manifolds that relies on simulation of quantum dynamics on a graph embedding of the sampled data. Our approach exploits classic results in semiclassical analysis and the quantum-classical correspondence, and forms a basis for techniques to learn the manifold from which a dataset is sampled, and subsequently for nonlinear dimensionality reduction of high-dimensional datasets. We illustrate the new algorithm with data sampled from model manifolds and also by a clustering demonstration based on COVID-19 mobility data. Finally, our method reveals interesting connections between the discretization provided by data sampling and quantization.