Manifold learning via quantum dynamics
This work addresses manifold learning and dimensionality reduction for high-dimensional datasets, presenting a novel approach with potential applications in data analysis, though it appears incremental in the context of existing quantum-inspired methods.
The paper tackles the problem of computing geodesics on sampled manifolds for nonlinear dimensionality reduction by introducing an algorithm that simulates quantum dynamics on graph embeddings of data. The method was demonstrated on model manifolds and COVID-19 mobility data, revealing connections between data discretization and quantization.
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.