Sliced Wasserstein Kernels for Persistence Diagrams
Joint work with M. Cuturi and S. Oudot
Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they are routinely used to describe topological prop- erties of complicated shapes. PDs enjoy strong stability properties and have proven their utility in various learning contexts. They do not, how- ever, 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 PDs in a convex learn- ing pipeline, several kernels have been proposed with a strong emphasis on the stability of the re- sulting RKHS distance w.r.t. perturbations of the PDs. In this article, we use the Sliced Wasserstein approximation of the Wasserstein distance to define a new kernel for PDs, which is not only provably stable but also discriminative (with a bound depending on the number of points in the PDs) w.r.t. the first diagram distance between PDs. We also demonstrate its practicality, by de- veloping an approximation technique to reduce kernel computation time, and show that our pro- posal compares favorably to existing kernels for PDs on several benchmarks.