  Persistent homology is constrained to purely topological persistence while
multiscale graphs account only for geometric information. This work introduces
persistent spectral theory to create a unified low-dimensional multiscale
paradigm for revealing topological persistence and extracting geometric shape
from high-dimensional datasets. For a point-cloud dataset, a filtration
procedure is used to generate a sequence of chain complexes and associated
families of simplicial complexes and chains, from which we construct persistent
combinatorial Laplacian matrices. We show that a full set of topological
persistence can be completely recovered from the harmonic persistent spectra,
i.e., the spectra that have zero eigenvalues, of the persistent combinatorial
Laplacian matrices. However, non-harmonic spectra of the Laplacian matrices
induced by the filtration offer another power tool for data analysis, modeling,
and prediction. In this work, non-harmonic persistent spectra are successfully
devised to analyze the structure and stability of fullerenes and predict the
B-factors of a protein, which cannot be straightforwardly extracted from the
current persistent homology. Extensive numerical experiments indicate the
tremendous potential of the proposed persistent spectral analysis in data
science.
