Shining light on data: Geometric data analysis through quantum dynamics
This work addresses the challenge of interpreting complex datasets in experimental sciences by providing a new quantum-inspired method for geometric data analysis, which is foundational and not incremental.
The paper tackles the problem of extracting fine-scale geometric structure from high-dimensional datasets by introducing a novel uncertainty principle based on quantum dynamics, leading to a tractable algorithm for approximating wave dynamics and geodesics with rigorous convergence rates. It demonstrates a four-fold improvement in dimensionality reduction on real-world data, such as COVID-19 mobility information, revealing anomalous behavior in less than 1.2% of the dataset.
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.