Institute of Statistical Mathematics, Japan
Title: Kernel methods for topological data analysis
Summary: Topological data analysis (TDA) is an emerging mathematical method for extracting topological information in multi-scale data. In the method, a persistence diagram, 2-D plot for illustrating the topology, is widely used as a descriptor of data. It is able to provide information on which are robust and noisy topological properties in data. In this work, we introduce a kernel method for persistence diagrams to apply statistical methods systematically to TDA. We propose a new positive definite kernel for persistence diagrams, aiming at flexibly distinguishing significant topological properties from noise. We also discuss a computational challenge of TDA caused by its large scale of diagrams, and propose an approximate computation method. As a theoretical background, a stability theorem is proved, which shows that a small change of data points causes only small changes of the distance measure associated with the kernel. Finally, the proposed kernel is applied to several practical problems of data analysis in materials science, showing favorable results in comparison with other existing methods.
This work has been done with Genki Kusano and Yasuaki Hiraoka (Tohoku Univ.).