Manifold learning via quantum dynamics

TL;DR

This paper presents a novel quantum dynamics-based algorithm for manifold learning and nonlinear dimensionality reduction, leveraging semiclassical analysis and quantum-classical correspondence to analyze sampled data.

Contribution

It introduces a new quantum-inspired approach for computing geodesics on sampled manifolds, enabling improved manifold learning and data analysis techniques.

Findings

Effective geodesic computation on sampled manifolds

Demonstrated clustering on COVID-19 mobility data

Revealed connections between data discretization and quantization

Abstract

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.