Riemannian Manifold Kernel for Persistence Diagrams
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Algebraic topology methods have recently played an important role for statistical analysis with complicated geometric structured data. Among them, persistent homology is a well-known tool to extract robust topological features, and outputs as persistence diagrams. Unfortunately, persistence diagrams 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. Besides that, geometry for persistence diagrams is also an important factor. A popular geometry for persistence diagrams is the Wasserstein metric. However, Wasserstein distance is not negative definite. Thus, it is limited to build positive definite kernels upon the Wasserstein distance without approximation. In this work, we explore an alternative Riemannian manifold geometry, namely the Fisher information metric. By building upon the geodesic distance on the Riemannian manifold, we propose a positive definite kernel, namely Riemannian manifold kernel. Then, we analyze eigensystem of the integral operator induced by the proposed kernel for kernel machines. Based on that, we conduct generalization error bounds via covering numbers and Rademacher averages for kernel machines using the Riemannian manifold kernel. Additionally, we also show some nice properties for the proposed kernel such as stability, infinite divisibility and comparative time complexity with other kernels for persistence diagrams in term of computation. Throughout experiments with many different tasks on various benchmark datasets, we illustrate that the Riemannian manifold kernel improves performances of other baseline kernels.
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