Geometric Data Analysis Across Scales via Laplacian Eigenvector Cascading

TL;DR

This paper introduces algorithms for constructing multiscale Laplacian eigenfunctions that capture consistent geometric and topological features across data scales, improving stability and efficiency in data analysis.

Contribution

The paper presents novel eigenvector cascading algorithms that accelerate computation and ensure consistency of eigenspaces across multiple data scales.

Findings

Cascading accelerates eigenvector computation.

Eigenvectors are consistent across scales.

Application to TDA mapper reveals stable structures.

Abstract

We develop here an algorithmic framework for constructing consistent multiscale Laplacian eigenfunctions (vectors) on data. Consequently, we address the unsupervised machine learning task of finding scalar functions capturing consistent structure across scales in data, in a way that encodes intrinsic geometric and topological features. This is accomplished by two algorithms for eigenvector cascading. We show via examples that cascading accelerates the computation of graph Laplacian eigenvectors, and more importantly, that one obtains consistent bases of the associated eigenspaces across scales. Finally, we present an application to TDA mapper, showing that our multiscale Laplacian eigenvectors identify stable flair-like structures in mapper graphs of varying granularity.