Shining light on data: Geometric data analysis through quantum dynamics

TL;DR

This paper introduces a novel quantum-inspired framework for geometric data analysis, enabling detailed insights into high-dimensional datasets through wave dynamics and geodesic approximation with proven convergence.

Contribution

It develops the first practical algorithm based on quantum dynamics principles for analyzing data manifolds, with rigorous probabilistic guarantees and applications to real-world datasets.

Findings

Four-fold improvement in dimensionality reduction over existing methods

Successful analysis of population mobility during COVID-19

Detection of anomalous behavior in less than 1.2% of data

Abstract

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.