Kernel method for persistence diagrams via kernel embedding and weight factor
This provides a tool for researchers in topological data analysis to handle practical data more effectively, though it is incremental as it builds on existing kernel methods.
The paper tackles the challenge of developing a statistical framework for persistence diagrams in topological data analysis by proposing a kernel method that controls the effect of persistence and discounts noisy properties, with results showing advantages over existing methods in practical physics data.
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.