# Manifold learning via quantum dynamics

arXiv: https://arxiv.org/abs/2112.11161 · HTML: https://mlsanity.com/paper/2112.11161/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 63 references that are in the map.
- First posted on arXiv 20 Dec 2021.

## Builds on (most cited first, 4 of 4)

- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Spectral Embedding Norm: Looking Deep into the Spectrum of the Graph Laplacian](https://mlsanity.com/paper/1810.10695/) (2018)
- [A Spectral Framework for Anomalous Subgraph Detection](https://mlsanity.com/paper/1401.7702/) (2014)
- [Reconstruction of manifold embeddings into Euclidean spaces via intrinsic distances](https://mlsanity.com/paper/2012.13770/) (2020)

## Abstract (arXiv)

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.