  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.

Spectral Theory;Algebraic Topology