GEOTOP-A | Hodge Laplacians on Sequences | Hannah Santa Cruz

Hodge Laplacians have been previously proposed as a natural tool for understanding higher-order interactions in networks and directed graphs. In this talk, we will cover a Hodge-theoretic approach to spectral theory and dimensionality reduction for probability distributions on sequences and simplicial complexes. We will introduce a feature space based on the Laplacian eigenvectors associated to a set of sequences, and will see these eigenvectors capture the underlying geometry of our data. Furthermore, we will show this Hodge theory has desirable properties with respect to natural null-models, where the underlying vertices are independent. Specifically, we will see the appropriate Hodge Laplacian has an integer spectrum with high multiplicities, and describe its eigenspaces. Finally, we will cover a simple proof showing the underlying cell complex of sequences has trivial reduced homology.