Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams

Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data such as shapes, linked twist maps, and material data. Among them, persistent homology is a well-known tool to extract robust topological features, and outputs as persistence diagrams (PDs). However, PDs are point multi-sets which can not be used in machine learning algorithms for vector data. To deal with it, an emerged approach is to use kernel methods, and an appropriate geometry for PDs is an important factor to measure the similarity of PDs. A popular geometry for PDs is the Wasserstein metric.