2018/12/05 by Joshua Lee Mike, Mike, Joshua L., José A. Perea +1
Computer Science · Physics and Astronomy · #05C50 #05C81 #55U10 #Algebraic Topology (math.AT) #Complex Network Analysis Techniques #Data Visualization and Analytics #FOS: Mathematics #Spectral Theory (math.SP) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1812.02139
openalex publication_date 2018/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.