Abstract
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.
Duc Nguyen
Associate Professor of Mathematics
Duc Nguyen develops mathematical and AI frameworks for molecular bioscience, drug discovery, and scientific computing. His group blends differential geometry, graph theory, and machine learning to build high-fidelity models for biomolecular systems, with notable wins in the D3R Grand Challenges and collaborations with Pfizer and Bristol Myers Squibb. Supported by multiple NSF awards, he has advised students and postdocs across theory and applications of AI-driven drug design.