  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.
