Reviews: Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport

This paper proposes a new method for the clustering of persistence diagrams using recent techniques in optimal transport. The problem is quite important; clustering provides a sensible way to group data according to their topological characterizations. It is also very challenging due to the Wasserstein distance between the persistence diagrams. This paper proposes to (1) approximate the Wasserstein distance between diagrams using the regularized optimal transport, and (2) treat the computation of the Frechet means as another optimal transport problem, and find the optimal solution using gradient descent. Several major technical challenges are addressed, include: 1) the Wasserstein distance may involve matching points with the a diagonal line. The proposed method is compared with the state-of-the-art (Hera) and is shown to be more efficient.