Given a point cloud in Euclidean space and a fixed length scale, we can create simplicial complexes (called Rips complexes) to represent that point cloud using the pairwise distances between the points. By tracking how the homology classes evolve as we increase that length scale, we summarise the topology and the geometry of the “shape” of the point cloud in what is called the persistent homology of its Rips filtration. A major obstacle to more widespread take up of persistent homology as a data analysis tool is the long computation time and, more importantly, the large memory requirements needed to store the filtrations of Rips complexes and compute its persistent homology. We bypass these issues by finding a “Reduced Rips Filtration” which has the same degree-1 persistent homology but with dramatically fewer simplices.

The talk is based off joint work is with Musashi Koyama, Facundo Memoli and Vanessa Robins.