Killian Meehan (07/10/24): Topological Node2vec: improving graph embeddings with persistent homology

Title: Topological Node2vec: improving graph embeddings with persistent homology
Abstract: We discuss some theory, applications, and heuristics regarding implementation of topology into the machine learning task of graph embeddings. Persistent homology defines topology for point clouds (or weighted graphs) by examining the pairwise distances (edge weights) between points (nodes). Yet, when using graph embedding methods to transform the pairwise relationships of weighted edges into those of euclidean distances, we find that topology is frequently distorted or destroyed in even simple synthetic examples. We demonstrate this problem as well as its correction via the introduction of a novel topological loss term, which showcases some new results and proposals in the realm of optimal transport as it relates to TDA.