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Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams
Tam Le · Makoto Yamada

Thu Dec 06 02:00 PM -- 04:00 PM (PST) @ Room 210 #81

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, \textit{persistent homology} is a well-known tool to extract robust topological features, and outputs as \textit{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 \textit{Wasserstein metric}. However, Wasserstein distance is not \textit{negative definite}. Thus, it is limited to build positive definite kernels upon the Wasserstein distance \textit{without approximation}. In this work, we rely upon the alternative \textit{Fisher information geometry} to propose a positive definite kernel for PDs \textit{without approximation}, namely the Persistence Fisher (PF) kernel. Then, we analyze eigensystem of the integral operator induced by the proposed kernel for kernel machines. Based on that, we derive generalization error bounds via covering numbers and Rademacher averages for kernel machines with the PF kernel. Additionally, we show some nice properties such as stability and infinite divisibility for the proposed kernel. Furthermore, we also propose a linear time complexity over the number of points in PDs for an approximation of our proposed kernel with a bounded error. Throughout experiments with many different tasks on various benchmark datasets, we illustrate that the PF kernel compares favorably with other baseline kernels for PDs.

Author Information


I completed my Ph.D. in 09/2015, and officially obtained a Ph.D. degree from Kyoto University in 01/2016, under the supervision of Professor Marco Cuturi and Professor Akihiro Yamamoto. Then, I worked as a post-doctoral researcher at the Nagoya Institute of Technology and National Institute of Materials Science, Japan between 01/2016 and 08/2017. After that, I have been working as a postdoctoral researcher in RIKEN AIP, Japan since 09/2017.

Makoto Yamada (Kyoto University / RIKEN AIP)

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