  Topological data analysis is an emerging mathematical concept for
characterizing shapes in multi-scale data. In this field, persistence diagrams
are widely used as a descriptor of the input data, and can distinguish robust
and noisy topological properties. Nowadays, it is highly desired to develop a
statistical framework on persistence diagrams to deal with practical data. This
paper proposes a kernel method on persistence diagrams. A theoretical
contribution of our method is that the proposed kernel allows one to control
the effect of persistence, and, if necessary, noisy topological properties can
be discounted in data analysis. Furthermore, the method provides a fast
approximation technique. The method is applied into several problems including
practical data in physics, and the results show the advantage compared to the
existing kernel method on persistence diagrams.
