Point-Level Topological Representation Learning at the Intersection of Topology and Geometry

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, common machine learning applications like classification or applications in single-cell Biology require point-level information and features to be available. In our work, we bridge this gap and propose a novel method to extract node-level topological features from complex point clouds using discrete variants of concepts from algebraic topology and differential geometry.

During the talk, we hope to illuminate how Topological Data Analysis can learn from ideas from Differential Geometry and Geometrical Machine Learning and vice-versa, and hope to paint a promising picture for future research joining the two areas.