Sliced Wasserstein Kernel for Persistence Diagrams

Persistence diagrams play a key role in topological data analysis (TDA), in which they are routinely used to describe topological properties of complicated shapes. persistence diagrams enjoy strong stability properties and have proven their utility in various learning contexts. They do not, however, live in a space naturally endowed with a Hilbert structure and are usually compared with non-Hilbertian distances, such as the bottleneck distance. To incorporate persistence diagrams in a convex learning pipeline, several kernels have been proposed with a strong emphasis on the stability of the resulting RKHS distance w.r.t. perturbations of the persistence diagrams. In this article, we use the Sliced Wasserstein approximation of the Wasserstein distance to de ne a new kernel for persistence diagrams, which is not only provably stable but also discriminative (with a bound depending on the number of points in the persistence diagrams) w.r.t. the rst diagram distance between persistence diagrams. We also demonstrate its practicality, by developing an approximation technique to reduce kernel computation time, and show that our proposal compares favorably to existing kernels for persistence diagrams on several benchmarks.