A tutorial on the dynamic Laplacian

TL;DR

This tutorial introduces the dynamic Laplacian, a spectral technique for analyzing data on evolving manifolds over time, highlighting its ability to identify long-lived coherent structures.

Contribution

It consolidates various results on the dynamic Laplacian, explaining its theoretical foundations, computational methods, and applications in a clear, accessible manner.

Findings

Dynamic Laplacian extends spectral methods to time-evolving data.

It effectively identifies long-lived coherent structures.

The SEBA algorithm automates feature separation.

Abstract

Spectral techniques are popular and robust approaches to data analysis. A prominent example is the use of eigenvectors of a Laplacian, constructed from data affinities, to identify natural data groupings or clusters, or to produce a simplified representation of data lying on a manifold. This tutorial concerns the dynamic Laplacian, which is a natural generalisation of the Laplacian to handle data that has a time component and lies on a time-evolving manifold. In this dynamic setting, clusters correspond to long-lived ``coherent'' collections. We begin with a gentle recap of spectral geometry before describing the dynamic generalisations. We also discuss computational methods and the automatic separation of many distinct features through the SEBA algorithm. The purpose of this tutorial is to bring together many results from the dynamic Laplacian literature into a single short document, written in an accessible style.