Hodge Learning: Relating global topology and local features of point clouds.

Topological Data Analysis (TDA) allows us to extract powerful topological and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description of the global structure of the point cloud. However, sometimes, we are interested in translating this global information back to local node-level features, where the individual points have some real-world meaning.

This talk will be about how we can achieve this using the Hodge Laplacian and concepts from TDA, Topological Signal Processing, and Differential Geometry. I will also talk about what we can learn from persistent homology by varying the distance function on the underlying space and analysing the corresponding shifts in the persistence diagrams.